https://peda.net/id/0a1b1724049 2019-11-11T17:15:04+02:00 https://peda.net/:static/532/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/9369bcd21d2 2019-12-13T00:27:27+02:00 ​<a href="https://peda.net/p/kirin_porsti/ma/ma13s/teht%C3%A4v%C3%A4t/laskulappu/maa13laskulappu-ods#top" class="service">MAA13Laskulappu.ods</a>​ 2019-12-13T00:27:26+02:00 https://peda.net/id/31ece7c01be 2019-12-11T09:55:03+02:00 <div>428</div> <div>a)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(2%7B%2C%7D3%5Cright)%3D2%5E3-3%5E2%2B5%3D8-9%2B5%3D4" alt="f\left(2{,}3\right)=2^3-3^2+5=8-9+5=4"/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-2%7B%2C%7D1%5Cright)%3D%5Cleft(-2%5Cright)%5E3-1%5E2%2B5%3D-8-1%2B5%3D-4%7B%2C%7D%5C%20On" alt="f\left(-2{,}1\right)=\left(-2\right)^3-1^2+5=-8-1+5=-4{,}\ On"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(3%7B%2C%7D-1%5Cright)%3D3%5E3-%5Cleft(-1%5Cright)%5E2%2B5%3D9-1%2B5%3D13%7B%2C%7D%5C%20Ei%5C%20ole" alt="f\left(3{,}-1\right)=3^3-\left(-1\right)^2+5=9-1+5=13{,}\ Ei\ ole"/><br/> <div>c)</div> <div> z-akselilla x- ja y-koordinaatti ovat 0.  </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(0%7B%2C%7D0%5Cright)%3D0%5E3-0%5E2%2B5%3D5" alt="f\left(0{,}0\right)=0^3-0^2+5=5"/></div> <div><span> Kuvaaja leikkaa z-akselin pisteessä (0, 0, 5).  </span><br/> <div>d) <br/> <a href="https://peda.net/p/kirin_porsti/ma/ma13s/teht%C3%A4v%C3%A4t/4-2/428d-png#top" title="428d.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma13s/teht%C3%A4v%C3%A4t/4-2/428d-png:file/photo/b6e8993d4f31df3f407f58517851fd854a3fbaf3/428d.PNG" alt="" title="428d.PNG" class="inline" loading="lazy"/></a><br/> <div>430</div> <div>a) </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(6%7B%2C%7D-10%5Cright)%3D-18" alt="f\left(6{,}-10\right)=-18"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(14%7B%2C%7D-5%5Cright)%3D-15%7B%2C%7D%5C%20Suurempi" alt="f\left(14{,}-5\right)=-15{,}\ Suurempi"/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(10%7B%2C%7D-10%5Cright)%3D-20" alt="f\left(10{,}-10\right)=-20"/></div> <div>c) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=Esim.%5C%20f%5Cleft(6%7B%2C%7D-15%5Cright)%7B%2C%7D%5C%20f%5Cleft(8%7B%2C%7D-15%5Cright)%7B%2C%7D%5C%20f%5Cleft(10%7B%2C%7D-15%5Cright)" alt="Esim.\ f\left(6{,}-15\right){,}\ f\left(8{,}-15\right){,}\ f\left(10{,}-15\right)"/><br/> <div> </div> </div> <div>434</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D_xf%5Cleft(x%7B%2C%7Dy%5Cright)%3D3-0%2B0%3D3" alt="D_xf\left(x{,}y\right)=3-0+0=3"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=D_yf%5Cleft(x%7B%2C%7Dy%5Cright)%3D0-7%2B0%3D-7" alt="D_yf\left(x{,}y\right)=0-7+0=-7"/><br/> <div> </div> <div> <div>439</div> <div>442</div> <div>448</div> </div> </div> </div> 2019-12-11T09:52:35+02:00 https://peda.net/id/b0d0414e1ab 2019-12-10T15:39:58+02:00 403 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=y%3D2x%2B4" alt="y=2x+4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7By-4%7D%7B2%7D%3D%5Cfrac%7By%7D%7B2%7D-2%3D%5Cfrac%7B1%7D%7B2%7Dy-2" alt="x=\frac{y-4}{2}=\frac{y}{2}-2=\frac{1}{2}y-2"/></div> <div>Sijoitetaan y:n paikalle x, funktiot ovat tällöin täsmälleen samat</div> <div>joten ne ovat toistensa käänteisfunktiot.</div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7Bx%2B1%7D" alt="y=\sqrt[]{x+1}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=y%5E2%3Dx%2B1" alt="y^2=x+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3Dy%5E2-1" alt="x=y^2-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-1%5Cne%20x%5E2%2B1" alt="x^2-1\ne x^2+1"/> <div>funktit eivät ole toistensa käänteisfunktiot</div> <br/> 405 <div>a) 4</div> <div>b) 1</div> <span>c) x=-1</span> <div>d) x=2</div> <div>e) -1</div> <br/> <div>412</div> <div>a)</div> <div>Funktio f kuvaaja on suora, jonka kulmakerroin on 0,5. Funktio on kasvava, joten se saa pienimmän arvonsa kohdassa x=0.</div> <div>f(0)=2<br/> <div>Funktio ei saavuta suurinta arvoaan, mutta saavuttaa kasvavuuden ja jatkuvuuden peusteella kaikki arvot lukuun <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow6%7Df%5Cleft(x%5Cright)%3D5" alt="\lim_{x\rightarrow6}f\left(x\right)=5"/>saakka.</div> <div>Funktion f arvojoukko on [2,5[<br/> <a href="https://peda.net/p/kirin_porsti/ma/ma13s/teht%C3%A4v%C3%A4t/4-1/412-a-png#top" title="412 a.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma13s/teht%C3%A4v%C3%A4t/4-1/412-a-png:file/photo/b406fc05b8040e2581c0ebce74a7156a2d9f46c6/412%20a.PNG" alt="" title="412 a.PNG" class="inline" loading="lazy"/></a></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=y%3D0%7B%2C%7D5x%2B2" alt="y=0{,}5x+2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3Df%5E%7B-1%7D%5Cleft(x%5Cright)%3D%5Cfrac%7By-2%7D%7B0%7B%2C%7D5%7D%3D%5Cfrac%7B1%7D%7B0%7B%2C%7D5%7Dy-%5Cfrac%7B2%7D%7B0%7B%2C%7D5%7D%3D2y-4%7B%2C%7D%5C%200%5Cle%20x%3C6" alt="x=f^{-1}\left(x\right)=\frac{y-2}{0{,}5}=\frac{1}{0{,}5}y-\frac{2}{0{,}5}=2y-4{,}\ 0\le x&lt;6"/></div> Käänteisfunktion määrittelyjoukko on sama kuin funktion f arvojoukko</div> <div>c) Käänteisfunktio arvojoukko on funktion f määrittelyjoukko, eli [0,6[<br/> <div><a href="https://peda.net/p/kirin_porsti/ma/ma13s/teht%C3%A4v%C3%A4t/4-1/412-b-png#top" title="412 b.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma13s/teht%C3%A4v%C3%A4t/4-1/412-b-png:file/photo/e5066886ae9d7426d79ff1415ed938d91091ae18/412%20b.PNG" alt="" title="412 b.PNG" class="inline" loading="lazy"/></a></div> </div> <br/> 413<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Cfrac%7Bx%2B2%7D%7Bx-3%7D" alt="y=\frac{x+2}{x-3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%5Cleft(x-3%5Cright)%3Dx%2B2" alt="y\left(x-3\right)=x+2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2B2%3Dyx-3y" alt="x+2=yx-3y"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-yx%3D-3y-2" alt="x-yx=-3y-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cleft(1-y%5Cright)%3D-3y-2" alt="x\left(1-y\right)=-3y-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-3y-2%7D%7B1-y%7D" alt="x=\frac{-3y-2}{1-y}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B3y-2%7D%7By-1%7D" alt="x=\frac{3y-2}{y-1}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow3%2B%7D%5Cfrac%7Bx%2B2%7D%7Bx-3%7D%3D%5Cinfty" alt="\lim_{x\rightarrow3+}\frac{x+2}{x-3}=\infty"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow3-%7D%5Cfrac%7Bx%2B2%7D%7Bx-3%7D%3D%5Cfrac%7Bx%5Cleft(1%2B%5Cfrac%7B2%7D%7Bx%7D%5Cright)%7D%7Bx%5Cleft(1-%5Cfrac%7B3%7D%7Bx%7D%5Cright)%7D%3D%5Cfrac%7B1%2B0%7D%7B1-0%7D%3D1" alt="\lim_{x\rightarrow3-}\frac{x+2}{x-3}=\frac{x\left(1+\frac{2}{x}\right)}{x\left(1-\frac{3}{x}\right)}=\frac{1+0}{1-0}=1"/><br/> <div>Jatkuvan ja vähenevän funktion f arvojoukko on ]1, ∞[, joten tämä on käänteisfunktion määrittelyjoukko. </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5E%7B-1%7D%5Cleft(x%5Cright)%3D%5Cfrac%7B3x%2B2%7D%7Bx-1%7D%7B%2C%7D%5C%20x%3E1" alt="f^{-1}\left(x\right)=\frac{3x+2}{x-1}{,}\ x&gt;1"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff; color: #333333; font-family: 'Times New Roman'; font-size: 17px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"--> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5E%7B-1%7D%5Cleft(f%5Cleft(x%5Cright)%5Cright)%3D%5Cfrac%7B3x%2B2%7D%7Bx-1%7D%3D%5Cfrac%7B3%5Ccdot%5Cfrac%7Bx%2B2%7D%7Bx-3%7D%2B2%7D%7B%5Cfrac%7Bx%2B2%7D%7Bx-3%7D-1%7D%3D%5Cfrac%7B3%5Ccdot%5Cfrac%7Bx%2B2%7D%7Bx-3%7D-2%5Ccdot%5Cfrac%7Bx-3%7D%7Bx-3%7D%7D%7B%5Cfrac%7Bx%2B2%7D%7Bx-3%7D-%5Cfrac%7Bx-3%7D%7Bx-3%7D%7D%3D%5Cfrac%7B%5Cfrac%7B5x%7D%7Bx-3%7D%7D%7B%5Cfrac%7B5%7D%7Bx-3%7D%7D%3D%5Cfrac%7B5x%7D%7B5%7D%3Dx" alt="f^{-1}\left(f\left(x\right)\right)=\frac{3x+2}{x-1}=\frac{3\cdot\frac{x+2}{x-3}+2}{\frac{x+2}{x-3}-1}=\frac{3\cdot\frac{x+2}{x-3}-2\cdot\frac{x-3}{x-3}}{\frac{x+2}{x-3}-\frac{x-3}{x-3}}=\frac{\frac{5x}{x-3}}{\frac{5}{x-3}}=\frac{5x}{5}=x"/></div> <br/> 416<br/> <div>a) Koska<img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D5x%5E4%2B3x%5E2%2B1%3E0" alt="f'\left(x\right)=5x^4+3x^2+1&gt;0"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"-->kaikkialla, funktio on kasvava ja sillä on käänteisfunktio.  </div> <div>b) <img src="https://math-demo.abitti.fi/math.svg?latex=f%5E%7B-1%7D%5Cleft(-2%5Cright)%3D-1" alt="f^{-1}\left(-2\right)=-1"/>, koska <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-1%5Cright)%3D-2" alt="f\left(-1\right)=-2"/><br/> <a href="https://peda.net/p/kirin_porsti/ma/ma13s/teht%C3%A4v%C3%A4t/4-1/416-b-png#top" title="416 b.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma13s/teht%C3%A4v%C3%A4t/4-1/416-b-png:file/photo/d0a83c43e8f6d6730f6ff8474dbba9f1841bac3f/416%20b.PNG" alt="" title="416 b.PNG" class="inline" loading="lazy"/></a></div> <div>c) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(f%5E%7B-1%7D%5Cright)%27%5Cleft(-2%5Cright)%3D%5Cfrac%7B1%7D%7Bf%27%5Cleft(-1%5Cright)%7D%3D%5Cfrac%7B1%7D%7B5%5Ccdot%5Cleft(-1%5Cright)%5E4%2B3%5Ccdot%5Cleft(-1%5Cright)%5E2%2B1%7D%3D%5Cfrac%7B1%7D%7B9%7D" alt="\left(f^{-1}\right)'\left(-2\right)=\frac{1}{f'\left(-1\right)}=\frac{1}{5\cdot\left(-1\right)^4+3\cdot\left(-1\right)^2+1}=\frac{1}{9}"/></div> <div>d) Tangentin kulmakerroin on derivaatan arvo, eli <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B9%7D" alt="\frac{1}{9}"/></div> 2019-12-09T21:40:01+02:00 https://peda.net/id/11a9e0b4173 2019-12-05T11:03:13+02:00 <div>331</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0A%5Cfrac%7B1%7D%7Bx%5E4%7D%7B%2C%7D%26kun%5C%20x%3C-1%5C%20tai%5C%20x%3E1%5C%5C%0Aa%7B%2C%7D%26%5C%20kun%5C%20-1%5Cle%20x%5Cle1%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\begin{cases}&#10;\frac{1}{x^4}{,}&amp;kun\ x&lt;-1\ tai\ x&gt;1\\&#10;a{,}&amp;\ kun\ -1\le x\le1&#10;\end{cases}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <div>Jotta f(x) olisi tiheysfunktio , on oltava f(x)≥0 </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E4%5Cge0" alt="x^4\ge0"/>kaikilla x. Siten <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7Bx%5E4%7D%3E0" alt="\frac{1}{x^4}&gt;0"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"-->, kun x &lt; -1 tai x &gt; 1</div> <div>Lisäksi valitaan a siten, että a ≥ 0, kun -1 ≤ x ≤ 1. </div> <div>Jotta f(x) olisi tiheysfnktio, on myös oltava </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7Df%5Cleft(x%5Cright)dx%3D1" alt="\int_{-\infty}^{\infty}f\left(x\right)dx=1"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7Df%5Cleft(x%5Cright)dx%3D%5Cint_%7B-%5Cinfty%7D%5E%7B-1%7D%5Cfrac%7B1%7D%7Bx%5E4%7Ddx%2B%5Cint_%7B-1%7D%5E1adx%2B%5Cint_1%5E%7B%5Cinfty%7D%5Cfrac%7B1%7D%7Bx%5E4%7Ddx" alt="\int_{-\infty}^{\infty}f\left(x\right)dx=\int_{-\infty}^{-1}\frac{1}{x^4}dx+\int_{-1}^1adx+\int_1^{\infty}\frac{1}{x^4}dx"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Clim_%7Bt%5Crightarrow%5Cinfty%7D%5Cint_t%5E%7B-1%7Dx%5E%7B-4%7Ddx%2B%5Cint_%7B-1%7D%5E1adx%2B%5Clim_%7Bs%5Crightarrow%5Cinfty%7D%5Cint_1%5Esx%5E%7B-4%7Ddx" alt="=\lim_{t\rightarrow\infty}\int_t^{-1}x^{-4}dx+\int_{-1}^1adx+\lim_{s\rightarrow\infty}\int_1^sx^{-4}dx"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Clim_%7Bt%5Crightarrow-%5Cinfty%7D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!t%7D%5E%7B-1%7D-%5Cfrac%7B1%7D%7B3x%5E3%7D%2B%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5E1ax%2B%5Clim_%7Bs%5Crightarrow%5Cinfty%7D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!1%7D%5E3-%5Cfrac%7B1%7D%7B3x%5E3%7D" alt="=\lim_{t\rightarrow-\infty}\bigg/_{\!\!\!\!\!t}^{-1}-\frac{1}{3x^3}+\bigg/_{\!\!\!\!\!{-1}}^1ax+\lim_{s\rightarrow\infty}\bigg/_{\!\!\!\!\!1}^3-\frac{1}{3x^3}"/></div> <div> </div> <br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Clim_%7Bt%5Crightarrow-%5Cinfty%7D%5Cleft(%5Cfrac%7B1%7D%7B3%7D%2B%5Cfrac%7B1%7D%7B3t%5E2%7D%5Cright)%2B%5Cleft(a-a%5Cleft(a%5Ccdot-1%5Cright)%5Cright)%2B%5Clim_%7Bs%5Crightarrow%5Cinfty%7D%5Cleft(-%5Cfrac%7B1%7D%7B3s%5E3%7D%2B%5Cfrac%7B1%7D%7B3%7D%5Cright)" alt="=\lim_{t\rightarrow-\infty}\left(\frac{1}{3}+\frac{1}{3t^2}\right)+\left(a-a\left(a\cdot-1\right)\right)+\lim_{s\rightarrow\infty}\left(-\frac{1}{3s^3}+\frac{1}{3}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B1%7D%7B3%7D%2Ba%2Ba%2B%5Cfrac%7B1%7D%7B3%7D%3D%5Cfrac%7B2%7D%7B3%7D%2B2a" alt="=\frac{1}{3}+a+a+\frac{1}{3}=\frac{2}{3}+2a"/> <div><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%7D%7B3%7D%2B2a%3D1" alt="\frac{2}{3}+2a=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2a%3D1-%5Cfrac%7B2%7D%7B3%7D" alt="2a=1-\frac{2}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D%5Cfrac%7B1%7D%7B6%7D" alt="a=\frac{1}{6}"/> </div> <div>Tällöin </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%5Cbegin%7Bcases%7D%0A%5Cfrac%7B1%7D%7Bx%5E4%7B%2C%7D%7D%26kun%5C%20x%3C-1%5C%20tai%5C%20x%3E1%5C%5C%0A%5Cfrac%7B1%7D%7B6%7D%7B%2C%7D%26kun%5C%20-1%5Cle%20x%5Cle1%0A%5Cend%7Bcases%7D" alt="f\left(x\right)\begin{cases}&#10;\frac{1}{x^4{,}}&amp;kun\ x&lt;-1\ tai\ x&gt;1\\&#10;\frac{1}{6}{,}&amp;kun\ -1\le x\le1&#10;\end{cases}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(0%5Cle%20X%5Cle2%5Cright)%3D%5Cint_0%5E2f%5Cleft(x%5Cright)dx" alt="P\left(0\le X\le2\right)=\int_0^2f\left(x\right)dx"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cint_0%5E1%5Cfrac%7B1%7D%7B6%7Ddx%2B%5Cint_1%5E2%5Cfrac%7B1%7D%7Bx%5E4%7Ddx" alt="=\int_0^1\frac{1}{6}dx+\int_1^2\frac{1}{x^4}dx"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B11%7D%7B24%7D%5Capprox0%7B%2C%7D46" alt="=\frac{11}{24}\approx0{,}46"/></div> <div><br/> <div>332</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(t%5Cright)%5Cbegin%7Bcases%7D%0A6%5Cleft(0%7B%2C%7D25-%5Cleft(t-1%7B%2C%7D5%5Cright)%5E2%5Cright)%7B%2C%7D%26kun%5C%201%5Cle%20t%5Cle2%5C%5C%0A0%26muualla%0A%5Cend%7Bcases%7D" alt="f\left(t\right)\begin{cases}&#10;6\left(0{,}25-\left(t-1{,}5\right)^2\right){,}&amp;kun\ 1\le t\le2\\&#10;0&amp;muualla&#10;\end{cases}"/></div> Merkitään todennäköisyyttä, että vähintään 1300 tuntia,</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%5Cge1%7B%2C%7D3%5Cright)" alt="P\left(X\ge1{,}3\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%5Cge1%7B%2C%7D3%5Cright)%3D%5Cint_%7B1%7B%2C%7D3%7D%5E%7B%5Cinfty%7Df%5Cleft(t%5Cright)dt" alt="P\left(X\ge1{,}3\right)=\int_{1{,}3}^{\infty}f\left(t\right)dt"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cint_%7B1%7B%2C%7D3%7D%5E2f%5Cleft(t%5Cright)dt%2B%5Cint_2%5E%7B%5Cinfty%7Df%5Cleft(t%5Cright)dt" alt="=\int_{1{,}3}^2f\left(t\right)dt+\int_2^{\infty}f\left(t\right)dt"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cint_%7B1%7B%2C%7D3%7D%5E26%5Cleft(0%7B%2C%7D25-%5Cleft(t-1%7B%2C%7D5%5Cright)%5E2dt%2B%5C%20%5Cint_2%5E%7B%5Cinfty%7D0dt%5Cright)" alt="=\int_{1{,}3}^26\left(0{,}25-\left(t-1{,}5\right)^2dt+\ \int_2^{\infty}0dt\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D0%7B%2C%7D784" alt="=0{,}784"/></div> <div>Todennäköisyys on noin 0,78</div> <div><br/> <div>333</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3DP%5Cleft(X%5Cle%20x%5Cright)%3D%5Cint_%7B-%5Cinfty%7D%5Exf%5Cleft(t%5Cright)dt" alt="F\left(x\right)=P\left(X\le x\right)=\int_{-\infty}^xf\left(t\right)dt"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%5Cbegin%7Bcases%7D%0A0%7B%2C%7D%26kun%5C%20x%5Cle1%5C%5C%0A%5Cfrac%7B1%7D%7B2x%5Csqrt%5B%5D%7Bx%7D%7D%7B%2C%7D%26kun%5C%20x%3E1%0A%5Cend%7Bcases%7D" alt="f\left(x\right)\begin{cases}&#10;0{,}&amp;kun\ x\le1\\&#10;\frac{1}{2x\sqrt[]{x}}{,}&amp;kun\ x&gt;1&#10;\end{cases}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--> </div> <div>Kun x≤1, niin </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-%5Cinfty%7D%5Exf%5Cleft(t%5Cright)dt%3D%5Cint_%7B-%5Cinfty%7D%5Ex0dt%3D0" alt="\int_{-\infty}^xf\left(t\right)dt=\int_{-\infty}^x0dt=0"/></div> Kun x&gt;1, niin</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-%5Cinfty%7D%5Exf%5Cleft(t%5Cright)dt%3D%5C%20%5Cint_%7B-%5Cinfty%7D%5E1f%5Cleft(t%5Cright)dt%2B%5Cint_1%5Exf%5Cleft(t%5Cright)dt%3D0%2B%5Cint_1%5Ex%5Cfrac%7B1%7D%7B2t%5Csqrt%5B%5D%7Bt%7D%7Ddt%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!1%7D%5Ex-%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7Bt%7D%7D%3D%5Cfrac%7B%5Csqrt%5B%5D%7Bx%7D-1%7D%7B%5Csqrt%5B%5D%7Bx%7D%7D%3D1-%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7Bx%7D%7D" alt="\int_{-\infty}^xf\left(t\right)dt=\ \int_{-\infty}^1f\left(t\right)dt+\int_1^xf\left(t\right)dt=0+\int_1^x\frac{1}{2t\sqrt[]{t}}dt=\bigg/_{\!\!\!\!\!1}^x-\frac{1}{\sqrt[]{t}}=\frac{\sqrt[]{x}-1}{\sqrt[]{x}}=1-\frac{1}{\sqrt[]{x}}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%5Cbegin%7Bcases%7D%0A0%7B%2C%7D%26kun%5C%20x%5Cle1%5C%5C%0A1-%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7Bx%7D%7D%7B%2C%7D%26kun%5C%20x%3E1%0A%5Cend%7Bcases%7D" alt="F\left(x\right)\begin{cases}&#10;0{,}&amp;kun\ x\le1\\&#10;1-\frac{1}{\sqrt[]{x}}{,}&amp;kun\ x&gt;1&#10;\end{cases}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%5Cge4%5Cright)%3D1-F%5Cleft(4%5Cright)%3D1-%5Cleft(1-%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7B4%7D%7D%5Cright)%3D%5Cfrac%7B1%7D%7B2%7D" alt="P\left(X\ge4\right)=1-F\left(4\right)=1-\left(1-\frac{1}{\sqrt[]{4}}\right)=\frac{1}{2}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(-1%3CX%5Cle2%7B%2C%7D25%5Cright)%3DF%5Cleft(2%7B%2C%7D25%5Cright)-F%5Cleft(-1%5Cright)%3D1-%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7B2%7B%2C%7D25%7D%7D-0%3D%5Cfrac%7B1%7D%7B3%7D" alt="P\left(-1&lt;X\le2{,}25\right)=F\left(2{,}25\right)-F\left(-1\right)=1-\frac{1}{\sqrt[]{2{,}25}}-0=\frac{1}{3}"/><br/> <div> </div> <div>335</div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3DP%5Cleft(X%5Cle%20x%5Cright)%3D%5Cint_%7B-%5Cinfty%7D%5Exf%5Cleft(t%5Cright)dt" alt="F\left(x\right)=P\left(X\le x\right)=\int_{-\infty}^xf\left(t\right)dt"/> <br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%5Cbegin%7Bcases%7D%0A%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7B8%7Dx%7B%2C%7D%26kun%5C%200%5Cle%20x%5Cle4%5C%5C%0A0%26mulloin%0A%5Cend%7Bcases%7D" alt="f\left(x\right)\begin{cases}&#10;\frac{1}{2}-\frac{1}{8}x{,}&amp;kun\ 0\le x\le4\\&#10;0&amp;mulloin&#10;\end{cases}"/> </div> <div>a)</div> <div>Kun x≤0, niin</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-%5Cinfty%7D%5Exf%5Cleft(t%5Cright)dt%3D%5Cint_%7B-%5Cinfty%7D%5Ex0dt" alt="\int_{-\infty}^xf\left(t\right)dt=\int_{-\infty}^x0dt"/></div> Kun 0≤x≤4</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-%5Cinfty%7D%5Exf%5Cleft(t%5Cright)dt%3D%5Cint_%7B-%5Cinfty%7D%5E0f%5Cleft(t%5Cright)dt%2B%5Cint_0%5Exf%5Cleft(t%5Cright)dt" alt="\int_{-\infty}^xf\left(t\right)dt=\int_{-\infty}^0f\left(t\right)dt+\int_0^xf\left(t\right)dt"/></div> </div> <div> </div> <br/> <div>336</div> <div>338</div> </div> </div> </div> 2019-12-05T11:03:13+02:00 https://peda.net/id/8daa0cfa166 2019-12-13T00:25:48+02:00 <div>305<br/> Lasketaan eri puolien pinta-alojen raja-arvot <div>Koska funktio ei ole määritelty kohdassa x ≤ 0, pinta-ala saadaan seuraavasti</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bt%5Crightarrow0%7D%5Cint_t%5E1%5Cfrac%7B1%7D%7B4%5Csqrt%5B%5D%7Bx%7D%7Ddx%3D%5Clim_%7Bt%5Crightarrow0%7D%5Cint_t%5E1%5Cfrac%7B1%7D%7Bx%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%7Ddx%3D%5Clim_%7Bt%5Crightarrow0%7D%5Cint_t%5E1x%5E%7B-%5Cfrac%7B1%7D%7B4%7D%7Ddx%3D%5Clim_%7Bt%5Crightarrow0%7D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!t%7D%5E1%5Cfrac%7B4%7D%7B3%7Dx%5E%7B%5Cfrac%7B3%7D%7B4%7D%7D%3D%5Clim_%7Bt%5Crightarrow0%7D%5Cleft(%5Cfrac%7B4%7D%7B3%7D-%5Cfrac%7B4%7D%7B3%7Dt%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%5Cright)%3D%5Cfrac%7B4%7D%7B3%7D-0%3D%5Cfrac%7B4%7D%7B3%7D" alt="\lim_{t\rightarrow0}\int_t^1\frac{1}{4\sqrt[]{x}}dx=\lim_{t\rightarrow0}\int_t^1\frac{1}{x^{\frac{1}{4}}}dx=\lim_{t\rightarrow0}\int_t^1x^{-\frac{1}{4}}dx=\lim_{t\rightarrow0}\bigg/_{\!\!\!\!\!t}^1\frac{4}{3}x^{\frac{3}{4}}=\lim_{t\rightarrow0}\left(\frac{4}{3}-\frac{4}{3}t^{\frac{4}{3}}\right)=\frac{4}{3}-0=\frac{4}{3}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bi%5Crightarrow%5Cinfty%7D%5Cint_1%5Ei%5Cfrac%7B1%7D%7B%5Csqrt%5B4%5D%7Bx%7D%7Ddx%3D%5Clim_%7Bi%5Crightarrow%5Cinfty%7D%5Cint_1%5Ei%5Cfrac%7B1%7D%7B%5E%7Bx%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%7D%7Ddx%3D%5Clim_%7Bi%5Crightarrow%5Cinfty%7D%5Cint_1%5Eix%5E%7B-%5Cfrac%7B1%7D%7B4%7D%7Ddx%3D%5Clim_%7Bi%5Crightarrow%5Cinfty%7D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!1%7D%5Ei%5Cfrac%7B4%7D%7B3%7Dx%5E%7B%5Cfrac%7B3%7D%7B4%7D%7D%3D%5Clim_%7Bi%5Crightarrow%5Cinfty%7D%5Cleft(%5C%20%5Cfrac%7B4%7D%7B3%7Di%5E%7B%5Cfrac%7B3%7D%7B4%7D%7D-%5Cfrac%7B4%7D%7B3%7D%5Cright)%3D%5Cinfty" alt="\lim_{i\rightarrow\infty}\int_1^i\frac{1}{\sqrt[4]{x}}dx=\lim_{i\rightarrow\infty}\int_1^i\frac{1}{^{x^{\frac{1}{4}}}}dx=\lim_{i\rightarrow\infty}\int_1^ix^{-\frac{1}{4}}dx=\lim_{i\rightarrow\infty}\bigg/_{\!\!\!\!\!1}^i\frac{4}{3}x^{\frac{3}{4}}=\lim_{i\rightarrow\infty}\left(\ \frac{4}{3}i^{\frac{3}{4}}-\frac{4}{3}\right)=\infty"/>  <div><span>Ensimmäinen pinta-ala on äärellinen, toinen on ääretön </span></div> </div> <div> <div><br/> 306<br/> <div>a)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cpi%5Cint_0%5E%7B%5C%20t%7D%5Cleft(%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7Be%5Ex%7D%7D%5Cright)%5E2dx%3D%5Cpi%5Cint_0%5E%7B%5C%20t%7D%5Cfrac%7B1%7D%7Be%5Ex%7Ddx%3D%5Cpi%5Cint_0%5E%7B%5C%20t%7D%5Cfrac%7B1%7D%7Be%5Ex%7Ddx%3D%5Cpi%5Cint_0%5E%7B%5C%20t%7De%5E%7B-x%7Ddx%3D%5Cpi%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5Et-e%5E%7B-x%7D" alt="V=\pi\int_0^{\ t}\left(\frac{1}{\sqrt[]{e^x}}\right)^2dx=\pi\int_0^{\ t}\frac{1}{e^x}dx=\pi\int_0^{\ t}\frac{1}{e^x}dx=\pi\int_0^{\ t}e^{-x}dx=\pi\bigg/_{\!\!\!\!\!0}^t-e^{-x}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cpi%5Cleft(-e%5E%7B-t%7D-%5Cleft(-e%5E0%5Cright)%5Cright)%3D%5Cpi%5Cleft(1-%5Cfrac%7B1%7D%7Be%5Et%7D%5Cright)" alt="=\pi\left(-e^{-t}-\left(-e^0\right)\right)=\pi\left(1-\frac{1}{e^t}\right)"/><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cpi%5Cleft(1-%5Cfrac%7B1%7D%7Be%5Et%7D%5Cright)%5Crightarrow%5Cpi%5Cleft(1-0%5Cright)%3D%5Cpi" alt="\pi\left(1-\frac{1}{e^t}\right)\rightarrow\pi\left(1-0\right)=\pi"/></div> </div> <div>310<br/> <div>a)</div> <div>Kuvaaja on x-akselin alapuolella, joten pinta-ala on integraalin vastaluku</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cint_0%5E1%5Cfrac%7Bx%5E2-1%7D%7B%5Csqrt%5B%5D%7Bx%7D%7Ddx%3D%5Clim_%7Bt%5Crightarrow0%2B%7D%5Cint_t%5E1%5Cfrac%7Bx%5E2-1%7D%7B%5Csqrt%5B%5D%7Bx%7D%7Ddx" alt="-\int_0^1\frac{x^2-1}{\sqrt[]{x}}dx=\lim_{t\rightarrow0+}\int_t^1\frac{x^2-1}{\sqrt[]{x}}dx"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Clim_%7Bt%5Crightarrow0%2B%7D%5Cint_t%5E1%5Cfrac%7Bx%5E2%7D%7B%5Csqrt%5B%5D%7Bx%7D%7D-%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7Bx%7D%7Ddx" alt="=-\lim_{t\rightarrow0+}\int_t^1\frac{x^2}{\sqrt[]{x}}-\frac{1}{\sqrt[]{x}}dx"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Clim_%7Bt%5Crightarrow0%2B%7D%5Cint_t%5E1%5Cleft(x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D-x%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D%5Cright)dx" alt="=-\lim_{t\rightarrow0+}\int_t^1\left(x^{\frac{3}{2}}-x^{-\frac{1}{2}}\right)dx"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Clim_%7Bt%5Crightarrow0%2B%7D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!t%7D%5E1%5Cleft(%5Cfrac%7B2%7D%7B5%7Dx%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D-2x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%5Cright)dx" alt="=-\lim_{t\rightarrow0+}\bigg/_{\!\!\!\!\!t}^1\left(\frac{2}{5}x^{\frac{5}{2}}-2x^{\frac{1}{2}}\right)dx"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Clim_%7Bt%5Crightarrow0%2B%7D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!t%7D%5E1%5Cleft(%5Cfrac%7B2%7D%7B5%7Dx%5E2%5Csqrt%5B%5D%7Bx%7D-2%5Csqrt%5B%5D%7Bx%7D%5Cright)dx" alt="=-\lim_{t\rightarrow0+}\bigg/_{\!\!\!\!\!t}^1\left(\frac{2}{5}x^2\sqrt[]{x}-2\sqrt[]{x}\right)dx"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Clim_%7Bt%5Crightarrow0%2B%7D%5Cleft(%5Cfrac%7B2%7D%7B5%7D%5Ccdot1-2%5Ccdot1-%5Cleft(%5Cfrac%7B2%7D%7B5%7Dt%5E2%5Csqrt%5B%5D%7Bt%7D-2%5Csqrt%5B%5D%7Bt%7D%5Cright)%5Cright)" alt="=-\lim_{t\rightarrow0+}\left(\frac{2}{5}\cdot1-2\cdot1-\left(\frac{2}{5}t^2\sqrt[]{t}-2\sqrt[]{t}\right)\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Cleft(%5Cfrac%7B2%7D%7B5%7D-2-0%5Cright)" alt="=-\left(\frac{2}{5}-2-0\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D1%5Cfrac%7B3%7D%7B5%7D" alt="=1\frac{3}{5}"/><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-1%7D%5E1%5Cfrac%7B1%7D%7B%5Csqrt%5B3%5D%7Bx%5E2%7D%7Ddx%3D%5Cint_%7B-1%7D%5E0%5Cfrac%7B1%7D%7B%5Csqrt%5B3%5D%7Bx%5E2%7D%7Ddx%2B%5Cint_0%5E1%5Cfrac%7B1%7D%7B%5Csqrt%5B3%5D%7Bx%5E2%7D%7Ddx" alt="\int_{-1}^1\frac{1}{\sqrt[3]{x^2}}dx=\int_{-1}^0\frac{1}{\sqrt[3]{x^2}}dx+\int_0^1\frac{1}{\sqrt[3]{x^2}}dx"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Clim_%7Bt%5Crightarrow0-%7D%5Cint_%7B-1%7D%5Etx%5E%7B-%5Cfrac%7B2%7D%7B3%7D%7Ddx%2B%5Clim_%7Bt%5Crightarrow0%2B%7D%5Cint_t%5E1x%5E%7B-%5Cfrac%7B2%7D%7B3%7D%7Ddx" alt="=\lim_{t\rightarrow0-}\int_{-1}^tx^{-\frac{2}{3}}dx+\lim_{t\rightarrow0+}\int_t^1x^{-\frac{2}{3}}dx"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Clim_%7Bt%5Crightarrow0-%7D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5Et3x%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%2B%5Clim_%7Bt%5Crightarrow0%2B%7D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!t%7D%5E13x%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="=\lim_{t\rightarrow0-}\bigg/_{\!\!\!\!\!{-1}}^t3x^{\frac{1}{3}}+\lim_{t\rightarrow0+}\bigg/_{\!\!\!\!\!t}^13x^{\frac{1}{3}}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Clim_%7Bt%5Crightarrow0-%7D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5Et3%5Csqrt%5B3%5D%7Bx%7D%2B%5Clim_%7Bt%5Crightarrow0%2B%7D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!t%7D%5E13%5Csqrt%5B3%5D%7Bx%7D" alt="=\lim_{t\rightarrow0-}\bigg/_{\!\!\!\!\!{-1}}^t3\sqrt[3]{x}+\lim_{t\rightarrow0+}\bigg/_{\!\!\!\!\!t}^13\sqrt[3]{x}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Clim_%7Bt%5Crightarrow0-%7D%5Cleft(3%5Csqrt%5B3%5D%7Bt%7D-3%5Csqrt%5B3%5D%7B1%7D%5Cright)%2B%5Clim_%7Bt%5Crightarrow0%2B%7D%5Cleft(3%5Csqrt%5B3%5D%7B1%7D-3%5Csqrt%5B3%5D%7Bt%7D%5Cright)dx" alt="=\lim_{t\rightarrow0-}\left(3\sqrt[3]{t}-3\sqrt[3]{1}\right)+\lim_{t\rightarrow0+}\left(3\sqrt[3]{1}-3\sqrt[3]{t}\right)dx"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D3%5Ccdot0-3%5Ccdot%5Cleft(-1%5Cright)%2B3%5Ccdot1-3%5Ccdot0" alt="=3\cdot0-3\cdot\left(-1\right)+3\cdot1-3\cdot0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D6" alt="=6"/></div> <div>313</div> <div>316</div> <div>319</div> </div> 2019-12-04T09:06:11+02:00 https://peda.net/id/f82036ce10e 2019-12-09T21:35:24+02:00 <div> <div>265</div> </div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7B12%7D%7B48%7D%3D%5Cfrac%7B1%7D%7B4%7D" alt="q=\frac{12}{48}=\frac{1}{4}"/></div> <div>|q|=0,25&lt;1, lukujono suppenee.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csum_%7Bk%3D1%7D%5E%7B%5Cinfty%7Da_k%3D%5Cfrac%7Ba_1%7D%7B1-q%7D%3D%5Cfrac%7B48%7D%7B1-%5Cfrac%7B1%7D%7B4%7D%7D%3D%5Cfrac%7B48%7D%7B%5Cfrac%7B3%7D%7B4%7D%7D%3D64" alt="\sum_{k=1}^{\infty}a_k=\frac{a_1}{1-q}=\frac{48}{1-\frac{1}{4}}=\frac{48}{\frac{3}{4}}=64"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=q%3D-%5Cfrac%7B3%7D%7B9%7D%3D-%5Cfrac%7B1%7D%7B3%7D" alt="q=-\frac{3}{9}=-\frac{1}{3}"/></div> <div>|q|=1/3&lt;1, lukujono suppenee. </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csum_%7Bk%3D1%7D%5E%7B%5Cinfty%7Da_k%3D%5Cfrac%7Ba_1%7D%7B1-q%7D%3D%5Cfrac%7B9%7D%7B1-%5Cleft(-%5Cfrac%7B1%7D%7B3%7D%5Cright)%7D%3D%5Cfrac%7B27%7D%7B4%7D" alt="\sum_{k=1}^{\infty}a_k=\frac{a_1}{1-q}=\frac{9}{1-\left(-\frac{1}{3}\right)}=\frac{27}{4}"/></div> c) <div><img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7B6%7D%7B2%7D%3D3" alt="q=\frac{6}{2}=3"/><br/> |q|=3&gt;1, lukujono hajaantuu.<br/> <br/> <div> <div>266</div> </div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_1%3D%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E%7B1-1%7D%3D1" alt="a_1=\left(\frac{1}{2}\right)^{1-1}=1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_2%3D%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E%7B2-1%7D%3D%5Cfrac%7B1%7D%7B2%7D" alt="a_2=\left(\frac{1}{2}\right)^{2-1}=\frac{1}{2}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=a_3%3D%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E%7B3-1%7D%3D%5Cfrac%7B1%7D%7B4%7D" alt="a_3=\left(\frac{1}{2}\right)^{3-1}=\frac{1}{4}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7B%5Cfrac%7B1%7D%7B2%7D%7D%7B1%7D%3D%5Cfrac%7B1%7D%7B2%7D" alt="q=\frac{\frac{1}{2}}{1}=\frac{1}{2}"/></div> <div>|q|=1/2&lt;1, lukujono suppenee. </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7B1%7D%7B1-%5Cfrac%7B1%7D%7B2%7D%7D%3D2" alt="S=\frac{1}{1-\frac{1}{2}}=2"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_1%3D4%5Ccdot%5Cleft(%5Cfrac%7B3%7D%7B2%7D%5Cright)%5E0%3D4%5Ccdot1%3D4" alt="a_1=4\cdot\left(\frac{3}{2}\right)^0=4\cdot1=4"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_2%3D4%5Ccdot%5Cleft(%5Cfrac%7B3%7D%7B2%7D%5Cright)%5E1%3D4%5Ccdot%5Cfrac%7B3%7D%7B2%7D%3D6" alt="a_2=4\cdot\left(\frac{3}{2}\right)^1=4\cdot\frac{3}{2}=6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a_3%3D4%5Ccdot%5Cleft(%5Cfrac%7B3%7D%7B2%7D%5Cright)%5E2%3D4%5Ccdot%5Cfrac%7B9%7D%7B4%7D%3D9" alt="a_3=4\cdot\left(\frac{3}{2}\right)^2=4\cdot\frac{9}{4}=9"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7B6%7D%7B4%7D%3D%5Cfrac%7B3%7D%7B2%7D" alt="q=\frac{6}{4}=\frac{3}{2}"/></div> <div>|q|=3/2&gt;1, lukujono ei suppene. </div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_1%3D0%7B%2C%7D3%5E%7B1-1%7D%3D1" alt="a_1=0{,}3^{1-1}=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=a_2%3D0%7B%2C%7D3%5E%7B2-1%7D%3D0%7B%2C%7D3" alt="a_2=0{,}3^{2-1}=0{,}3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a_3%3D0%7B%2C%7D3%5E%7B3-1%7D%3D0%7B%2C%7D09" alt="a_3=0{,}3^{3-1}=0{,}09"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7B0%7B%2C%7D3%7D%7B1%7D%3D0%7B%2C%7D3" alt="q=\frac{0{,}3}{1}=0{,}3"/></div> <div>|q|=0,3&lt;1, lukujono suppenee.  </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7B1%7D%7B1-0%7B%2C%7D3%7D%3D%5Cfrac%7B1%7D%7B0%7B%2C%7D7%7D%3D%5Cfrac%7B10%7D%7B7%7D" alt="S=\frac{1}{1-0{,}3}=\frac{1}{0{,}7}=\frac{10}{7}"/></div> <div> <div><br/> 267</div> </div> <div>a) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1%7B%2C%7D555...%3D1%2B0%7B%2C%7D555%3D1%2B5%5Ccdot0%7B%2C%7D1%2B5%5Ccdot0%7B%2C%7D01%2B5%5Ccdot0%7B%2C%7D001%2B..." alt="1{,}555...=1+0{,}555=1+5\cdot0{,}1+5\cdot0{,}01+5\cdot0{,}001+..."/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D1%2B5%5Ccdot0%7B%2C%7D1%5E1%2B5%5Ccdot0%7B%2C%7D1%5E2%2B5%5Ccdot0%7B%2C%7D1%5E3%2B..." alt="=1+5\cdot0{,}1^1+5\cdot0{,}1^2+5\cdot0{,}1^3+..."/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7B0%7B%2C%7D5%7D%7B1-0%7B%2C%7D1%7D%3D%5Cfrac%7B5%7D%7B9%7D" alt="S=\frac{0{,}5}{1-0{,}1}=\frac{5}{9}"/></div> <div>Siis</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1%7B%2C%7D555...%3D1%2B0%7B%2C%7D5%2B0%7B%2C%7D55%2B0%7B%2C%7D555...%3D1%2B%5Cfrac%7B5%7D%7B9%7D%3D%5Cfrac%7B14%7D%7B9%7D" alt="1{,}555...=1+0{,}5+0{,}55+0{,}555...=1+\frac{5}{9}=\frac{14}{9}"/><br/> <div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3%7B%2C%7D2121...%5C%20%3D3%2B0%7B%2C%7D2121%3D3%2B0%7B%2C%7D21%2B0%7B%2C%7D0021%2B..." alt="3{,}2121...\ =3+0{,}2121=3+0{,}21+0{,}0021+..."/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D3%2B21%5Ccdot0%7B%2C%7D01%2B21%5Ccdot0%7B%2C%7D0001%2B..." alt="=3+21\cdot0{,}01+21\cdot0{,}0001+..."/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D3%2B21%5Ccdot0%7B%2C%7D01%2B21%5Ccdot0%7B%2C%7D01%5E2%2B..." alt="=3+21\cdot0{,}01+21\cdot0{,}01^2+..."/></div> <div>Sarja <img src="https://math-demo.abitti.fi/math.svg?latex=21%5Ccdot0%7B%2C%7D01%2B21%5Ccdot0%7B%2C%7D01%5E2%2B..." alt="21\cdot0{,}01+21\cdot0{,}01^2+..."/>on geometrinen, jossa</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_1%3D0%7B%2C%7D21%5C%20ja%5C%20q%3D0%7B%2C%7D01" alt="a_1=0{,}21\ ja\ q=0{,}01"/></div> <div>Koska -1 &lt; q &lt; 1, sarja suppenee. Sarjan summa on</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7Ba_1%7D%7B1-q%7D%3D%5Cfrac%7B0%7B%2C%7D21%7D%7B1-0%7B%2C%7D01%7D%3D%5Cfrac%7B0%7B%2C%7D21%7D%7B0%7B%2C%7D99%7D%3D%5Cfrac%7B21%7D%7B99%7D" alt="S=\frac{a_1}{1-q}=\frac{0{,}21}{1-0{,}01}=\frac{0{,}21}{0{,}99}=\frac{21}{99}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3%7B%2C%7D2121...%3D3%2B%5Cfrac%7B21%7D%7B99%7D%3D%5Cfrac%7B318%7D%7B99%7D%5E%7B%5Ctext%7B(%7D3%7D%3D%5Cfrac%7B106%7D%7B33%7D" alt="3{,}2121...=3+\frac{21}{99}=\frac{318}{99}^{\text{(}3}=\frac{106}{33}"/><br/> <span><span><br/> 269</span></span> <div>a) Sarja on geometrinen sarja, jonka suhdeluku q=x. Sarja suppenee, kun -1&lt;x&lt;1.</div> <div>b) </div> <div>Jotta sarja suppenee, tulee olla −1 &lt; x &lt; 1. Tällöin </div> <span><span><img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7B1%7D%7B1-x%7D" alt="S=\frac{1}{1-x}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B1-x%7D%3Dx%2B3" alt="\frac{1}{1-x}=x+3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%3D%5Cleft(x%2B3%5Cright)%5Cleft(1-x%5Cright)" alt="1=\left(x+3\right)\left(1-x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%3D-x%5E2%2Bx%2B3-3x%3D-x%5E2-2x%2B3" alt="1=-x^2+x+3-3x=-x^2-2x+3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2B2x-2%3D0" alt="x^2+2x-2=0"/></span></span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%2B%5Csqrt%5B%5D%7B3%7D%5Capprox0%7B%2C%7D73%5C%20tai%5C%20x%3D-1-%5Csqrt%5B%5D%7B3%7D%3C-1" alt="x=-1+\sqrt[]{3}\approx0{,}73\ tai\ x=-1-\sqrt[]{3}&lt;-1"/>(Laskin)</div> <div>Vain ratkaisu<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%2B%5Csqrt%5B%5D%7B3%7D" alt="x=-1+\sqrt[]{3}"/>kelpaa yhtälön ratkaisuksi.<br/> <br/> <span>271</span> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7Ba_1%7D%7B1-q%7D" alt="S=\frac{a_1}{1-q}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B-%5Cfrac%7B2%7D%7B5%7D%7D%7B1%7D%3D-%5Cfrac%7B2%7D%7B5%7D" alt="q=\frac{a_{n+1}}{a_n}=\frac{-\frac{2}{5}}{1}=-\frac{2}{5}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7B1%7D%7B1-%5Cleft(-%5Cfrac%7B2%7D%7B5%7D%5Cright)%7D%3D%5Cfrac%7B5%7D%7B7%7D" alt="S=\frac{1}{1-\left(-\frac{2}{5}\right)}=\frac{5}{7}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7Ba_1%7D%7B1-q%7D" alt="S=\frac{a_1}{1-q}"/> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B-1%7D%7B-4%7D%3D%5Cfrac%7B1%7D%7B4%7D" alt="q=\frac{a_{n+1}}{a_n}=\frac{-1}{-4}=\frac{1}{4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7B-4%7D%7B1-%5Cfrac%7B1%7D%7B4%7D%7D%3D-%5Cfrac%7B16%7D%7B3%7D" alt="S=\frac{-4}{1-\frac{1}{4}}=-\frac{16}{3}"/> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7Ba_1%7D%7B1-q%7D" alt="S=\frac{a_1}{1-q}"/> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B%5Cfrac%7B2%7D%7B9%7D%7D%7B%5Cfrac%7B1%7D%7B6%7D%7D%3D%5Cfrac%7B4%7D%7B3%7D" alt="q=\frac{a_{n+1}}{a_n}=\frac{\frac{2}{9}}{\frac{1}{6}}=\frac{4}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7B%5Cfrac%7B1%7D%7B6%7D%7D%7B1-%5Cfrac%7B4%7D%7B3%7D%7D%3D-%5Cfrac%7B1%7D%7B2%7D" alt="S=\frac{\frac{1}{6}}{1-\frac{4}{3}}=-\frac{1}{2}"/><br/> <br/> <div>273</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_1%3D1" alt="a_1=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=q%3D1-x" alt="q=1-x"/><br/> <div>Jotta sarjalla olisi summa, sen tulee olla suppeneva, eli −1 &lt; q &lt; 1.<span> </span></div> <div>-1 &lt; 1-x &lt;1</div> <div>-2 &lt; -x &lt; 0</div> <div> 0 &lt; x &lt; 2</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7B1%7D%7B1-%5Cleft(1-x%5Cright)%7D%3D%5Cfrac%7B1%7D%7Bx%7D" alt="S=\frac{1}{1-\left(1-x\right)}=\frac{1}{x}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7Bx%7D%3D2x%2B1" alt="\frac{1}{x}=2x+1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=1%3D2x%5E2%2Bx" alt="1=2x^2+x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%2Bx-1%3D0" alt="2x^2+x-1=0"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%5C%20tai%5C%20x%3D%5Cfrac%7B1%7D%7B2%7D" alt="x=-1\ tai\ x=\frac{1}{2}"/></div> <div>Yhtälön ratkaisu x = 1/2 täyttää suppenemisehdon.  </div> <br/> <div>275</div> <div> <div>Pallon kulkema matka on:</div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1%2B2%5Ccdot0%7B%2C%7D75%5Ccdot1%2B2%5Ccdot0%7B%2C%7D75%5Ccdot0%7B%2C%7D75%5Ccdot1%2B..." alt="1+2\cdot0{,}75\cdot1+2\cdot0{,}75\cdot0{,}75\cdot1+..."/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D1%2B2%5Ccdot0%7B%2C%7D75%2B2%5Ccdot0%7B%2C%7D75%5E2%2B..." alt="=1+2\cdot0{,}75+2\cdot0{,}75^2+..."/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_1%3D2%5Ccdot0%7B%2C%7D75%3D1%7B%2C%7D5m" alt="a_1=2\cdot0{,}75=1{,}5m"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=q%3D0%7B%2C%7D75" alt="q=0{,}75"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7B1%7B%2C%7D5%7D%7B1-q%7D%3D%5Cfrac%7B1%7B%2C%7D5m%7D%7B1-0%7B%2C%7D75%7D%3D6%7B%2C%7D0m" alt="S=\frac{1{,}5}{1-q}=\frac{1{,}5m}{1-0{,}75}=6{,}0m"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1%7B%2C%7D0m%2B6%7B%2C%7D0m%3D7%7B%2C%7D0m" alt="1{,}0m+6{,}0m=7{,}0m"/></div> </div> <br/> <div>276</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%3D%5Cfrac%7B2%7D%7B3%5En%7D%3D%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D2%5Ccdot%5Cfrac%7B1%7D%7B3%5En%7D%3D%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D2%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Cright)%5En" alt="\sum_{n=1}^{\infty}=\frac{2}{3^n}=\sum_{n=1}^{\infty}2\cdot\frac{1}{3^n}=\sum_{n=1}^{\infty}2\cdot\left(\frac{1}{3}\right)^n"/></div> <div>Sarja on geometerinen sarja, missä<img src="https://math-demo.abitti.fi/math.svg?latex=a_1%3D%5Cfrac%7B2%7D%7B3%7D" alt="a_1=\frac{2}{3}"/>ja <img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7B1%7D%7B3%7D" alt="q=\frac{1}{3}"/>.</div> <div>Sarja suppenee, kun -1 &lt; q &lt; 1. Joten summaksi saadaan</div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S%3D%5Cfrac%7Ba_1%7D%7B1-q%7D%3D%5Cfrac%7B%5Cfrac%7B2%7D%7B3%7D%7D%7B1-%5Cfrac%7B1%7D%7B3%7D%7D%3D%5Cfrac%7B%5Cfrac%7B2%7D%7B3%7D%7D%7B%5Cfrac%7B2%7D%7B3%7D%7D%3D1" alt="S=\frac{a_1}{1-q}=\frac{\frac{2}{3}}{1-\frac{1}{3}}=\frac{\frac{2}{3}}{\frac{2}{3}}=1"/></div> <div>Lasketaan sarjan n ensimmäisen jäsenen geometrinen summa.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_n%3Da_1%5Ccdot%5Cfrac%7B1%5Ccdot%20q%5En%7D%7B1-q%7D%3D%5Cfrac%7B2%7D%7B3%7D%5Ccdot%5Cfrac%7B1-%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Cright)%5En%7D%7B1-%5Cfrac%7B1%7D%7B3%7D%7D%3D1-%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Cright)%5En" alt="S_n=a_1\cdot\frac{1\cdot q^n}{1-q}=\frac{2}{3}\cdot\frac{1-\left(\frac{1}{3}\right)^n}{1-\frac{1}{3}}=1-\left(\frac{1}{3}\right)^n"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1-%5Cleft(1-%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Cright)%5En%5Cright)%5Cle0%7B%2C%7D001" alt="1-\left(1-\left(\frac{1}{3}\right)^n\right)\le0{,}001"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1-1%2B%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Cright)%5En%5Cle0%7B%2C%7D001" alt="1-1+\left(\frac{1}{3}\right)^n\le0{,}001"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Cright)%5En%5Cle%5Cfrac%7B1%7D%7B1000%7D" alt="\left(\frac{1}{3}\right)^n\le\frac{1}{1000}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_%7B%5Cfrac%7B1%7D%7B3%7D%7D%5Cfrac%7B1%7D%7B1000%7D%3Dn" alt="\log_{\frac{1}{3}}\frac{1}{1000}=n"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3D6.2877098228685%5Capprox7" alt="n=6.2877098228685\approx7"/></div> <div>Pitää laskea vähinttäin 7 jäsentä yhteen.</div> <div> </div> <div>278</div> <div>Suhdeluku ei ole 2/3 vaan<img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7Ba_2%7D%7Ba_1%7D%3D%5Cfrac%7B-%5Cfrac%7B1%7D%7B2%7D%7D%7B-%5Cfrac%7B1%7D%7B3%7D%7D%3D%5Cfrac%7B3%7D%7B2%7D" alt="q=\frac{a_2}{a_1}=\frac{-\frac{1}{2}}{-\frac{1}{3}}=\frac{3}{2}"/>. Koska suhdeluku on suurempi kuin yksi(3/2&gt;1), sarja ei suppene, vaan hajaantuu. Sarjalle ei voi laskea summaa, kuten ratkaisuyrityksessä on tehty.<br/> <br/> <div>280</div> <div><span>a)</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a_1%3D%5Cfrac%7B1%7D%7B1%2B1%7D-%5Cfrac%7B1%7D%7B1%2B2%7D%3D%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7B3%7D%3D%5Cfrac%7B1%7D%7B6%7D" alt="a_1=\frac{1}{1+1}-\frac{1}{1+2}=\frac{1}{2}-\frac{1}{3}=\frac{1}{6}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_2%3D%5Cfrac%7B1%7D%7B2%2B1%7D-%5Cfrac%7B1%7D%7B2%2B2%7D%3D%5Cfrac%7B1%7D%7B3%7D-%5Cfrac%7B1%7D%7B4%7D%3D%5Cfrac%7B1%7D%7B12%7D" alt="a_2=\frac{1}{2+1}-\frac{1}{2+2}=\frac{1}{3}-\frac{1}{4}=\frac{1}{12}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_3%3D%5Cfrac%7B1%7D%7B3%2B1%7D-%5Cfrac%7B1%7D%7B3%2B2%7D%3D%5Cfrac%7B1%7D%7B4%7D-%5Cfrac%7B1%7D%7B5%7D%3D%5Cfrac%7B1%7D%7B20%7D" alt="a_3=\frac{1}{3+1}-\frac{1}{3+2}=\frac{1}{4}-\frac{1}{5}=\frac{1}{20}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a_4%3D%5Cfrac%7B1%7D%7B4%2B1%7D-%5Cfrac%7B1%7D%7B4%2B2%7D%3D%5Cfrac%7B1%7D%7B5%7D-%5Cfrac%7B1%7D%7B6%7D%3D%5Cfrac%7B1%7D%7B30%7D" alt="a_4=\frac{1}{4+1}-\frac{1}{4+2}=\frac{1}{5}-\frac{1}{6}=\frac{1}{30}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=a_5%3D%5Cfrac%7B1%7D%7B5%2B1%7D-%5Cfrac%7B1%7D%7B5%2B2%7D%3D%5Cfrac%7B1%7D%7B6%7D-%5Cfrac%7B1%7D%7B7%7D%3D%5Cfrac%7B1%7D%7B42%7D" alt="a_5=\frac{1}{5+1}-\frac{1}{5+2}=\frac{1}{6}-\frac{1}{7}=\frac{1}{42}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_5%3Da_1%2Ba_2%2Ba_3%2Ba_4%2Ba_5%3D%5Cfrac%7B5%7D%7B14%7D" alt="S_5=a_1+a_2+a_3+a_4+a_5=\frac{5}{14}"/></div> </div> <div> <div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_n%3D%5Cleft(%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7B3%7D%5Cright)%2B%5Cleft(%5Cfrac%7B1%7D%7B3%7D-%5Cfrac%7B1%7D%7B4%7D%5Cright)%2B...%2B%5Cleft(%5Cfrac%7B1%7D%7Bn%7D-%5Cfrac%7B1%7D%7Bn%2B1%7D%5Cright)%2B%5Cleft(%5Cfrac%7B1%7D%7Bn%2B1%7D-%5Cfrac%7B1%7D%7Bn%2B2%7D%5Cright)" alt="S_n=\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+...+\left(\frac{1}{n}-\frac{1}{n+1}\right)+\left(\frac{1}{n+1}-\frac{1}{n+2}\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7Bn%2B2%7D%3D%5Cfrac%7Bn%2B2%7D%7B2%5Cleft(n%2B2%5Cright)%7D-%5Cfrac%7B2%7D%7B2%5Cleft(n%2B2%5Cright)%7D%3D%5Cfrac%7Bn%7D%7B2n%2B4%7D" alt="=\frac{1}{2}-\frac{1}{n+2}=\frac{n+2}{2\left(n+2\right)}-\frac{2}{2\left(n+2\right)}=\frac{n}{2n+4}"/><br/> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_n%3D%5Cfrac%7Bn%7D%7B2n%2B4%7D%3D%5Cfrac%7B1%7D%7B2%2B%5Cfrac%7B4%7D%7Bn%7D%7D%5Crightarrow%5Cfrac%7B1%7D%7B2%2B0%7D%3D%5Cfrac%7B1%7D%7B2%7D%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="S_n=\frac{n}{2n+4}=\frac{1}{2+\frac{4}{n}}\rightarrow\frac{1}{2+0}=\frac{1}{2}{,}\ kun\ n\rightarrow\infty"/><br/> <br/> </div> </div> <div> <div>281<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7Bn%7D%7B2n%2B1%7D%3D%5Cfrac%7B1%7D%7B2%2B%5Cfrac%7B1%7D%7Bn%7D%7D%5Crightarrow%5Cfrac%7B1%7D%7B2%2B0%7D%3D%5Cfrac%7B1%7D%7B2%7D" alt="a_n=\frac{n}{2n+1}=\frac{1}{2+\frac{1}{n}}\rightarrow\frac{1}{2+0}=\frac{1}{2}"/></div> <div>Koska yleisen termin raja-arvo ei ole 0, sarja ei suppene.</div> </div> <div>282<br/> a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_n%3D%5Cfrac%7B1%7D%7Bn%7D-%5Cfrac%7B1%7D%7Bn%2B2%7D%3D%5Cfrac%7Bn%2B2%7D%7Bn%5Cleft(n%2B2%5Cright)%7D-%5Cfrac%7Bn%7D%7Bn%5Cleft(n%2B2%5Cright)%7D%3D%5Cfrac%7B2%7D%7Bn%5Cleft(n%2B2%5Cright)%7D%3D%5Cfrac%7B%5Cfrac%7B2%7D%7Bn%5E2%7D%7D%7Bn%5E2%5Cleft(1%2B%5Cfrac%7B2%7D%7Bn%7D%5Cright)%7D%5Crightarrow%5Cfrac%7B0%7D%7B1%2B0%7D%3D0%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="S_n=\frac{1}{n}-\frac{1}{n+2}=\frac{n+2}{n\left(n+2\right)}-\frac{n}{n\left(n+2\right)}=\frac{2}{n\left(n+2\right)}=\frac{\frac{2}{n^2}}{n^2\left(1+\frac{2}{n}\right)}\rightarrow\frac{0}{1+0}=0{,}\ kun\ n\rightarrow\infty"/></div> <span>Sarja voi olla suppeneva, määitetään n:n ensimmäisen jäsenen summa</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_n%3D%5Cleft(1-%5Cfrac%7B1%7D%7B3%7D%5Cright)%2B%5Cleft(%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7B4%7D%5Cright)%2B%5Cleft(%5Cfrac%7B1%7D%7B3%7D-%5Cfrac%7B1%7D%7B5%7D%5Cright)%2B%5Cleft(%5Cfrac%7B1%7D%7B4%7D-%5Cfrac%7B1%7D%7B6%7D%5Cright)%2B...%5Cleft(%5Cfrac%7B1%7D%7Bn-1%7D-%5Cfrac%7B1%7D%7Bn%2B1%7D%5Cright)%2B%5Cleft(%5Cfrac%7B1%7D%7Bn%7D-%5Cfrac%7B1%7D%7Bn%2B2%7D%5Cright)" alt="S_n=\left(1-\frac{1}{3}\right)+\left(\frac{1}{2}-\frac{1}{4}\right)+\left(\frac{1}{3}-\frac{1}{5}\right)+\left(\frac{1}{4}-\frac{1}{6}\right)+...\left(\frac{1}{n-1}-\frac{1}{n+1}\right)+\left(\frac{1}{n}-\frac{1}{n+2}\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1%2B%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7Bn%2B1%7D-%5Cfrac%7B1%7D%7Bn%2B2%7D%3D%5Cfrac%7B2%5Cleft(n%2B1%5Cright)%5Cleft(n%2B2%5Cright)%2B%5Cleft(n%2B1%5Cright)%5Cleft(n%2B2%5Cright)-2%5Cleft(n%2B2%5Cright)-2%5Cleft(n%2B1%5Cright)%7D%7B2%5Cleft(n%2B1%5Cright)%5Cleft(n%2B2%5Cright)%7D" alt="1+\frac{1}{2}-\frac{1}{n+1}-\frac{1}{n+2}=\frac{2\left(n+1\right)\left(n+2\right)+\left(n+1\right)\left(n+2\right)-2\left(n+2\right)-2\left(n+1\right)}{2\left(n+1\right)\left(n+2\right)}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B2n%5E2%2B6n%2B4%2Bn%5E2%2B3n%2B2-2n-4-2n-2%7D%7B2%5Cleft(n%2B1%5Cright)%5Cleft(n%2B2%5Cright)%7D" alt="=\frac{2n^2+6n+4+n^2+3n+2-2n-4-2n-2}{2\left(n+1\right)\left(n+2\right)}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B3n%5E2%2B5n%7D%7B2n%5E2%2B6n%2B4%7D" alt="=\frac{3n^2+5n}{2n^2+6n+4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B3%2B%5Cfrac%7B5%7D%7Bn%7D%7D%7B2%2B%5Cfrac%7B6%7D%7Bn%7D%2B%5Cfrac%7B4%7D%7Bn%5E2%7D%7D%5Crightarrow%5Cfrac%7B3%2B0%7D%7B2%2B0%2B0%7D%3D%5Cfrac%7B3%7D%7B2%7D%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="=\frac{3+\frac{5}{n}}{2+\frac{6}{n}+\frac{4}{n^2}}\rightarrow\frac{3+0}{2+0+0}=\frac{3}{2}{,}\ kun\ n\rightarrow\infty"/><br/> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_n%3D%5Cleft(-1%5Cright)%5En%5Cfrac%7B3n%2B1%7D%7B2n%7D" alt="S_n=\left(-1\right)^n\frac{3n+1}{2n}"/></div> <div>Takastellaan yhteenlaskettavien termien osaa <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3n%2B1%7D%7B2n%7D" alt="\frac{3n+1}{2n}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3n%2B1%7D%7B2n%7D%3D%5Cfrac%7B3%2B%5Cfrac%7B1%7D%7Bn%7D%7D%7B2%7D%5Crightarrow%5Cfrac%7B3%2B0%7D%7B2%7D%3D%5Cfrac%7B3%7D%7B2%7D%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="\frac{3n+1}{2n}=\frac{3+\frac{1}{n}}{2}\rightarrow\frac{3+0}{2}=\frac{3}{2}{,}\ kun\ n\rightarrow\infty"/></div> <div>Yhteenlaskettavat luvut <img src="https://math-demo.abitti.fi/math.svg?latex=a_n" alt="a_n"/> eivät lähesty nollaa, joten sarja ei suppene.<br/> <span>c)</span> <div>Määritetään yhteenlaskettavan <img src="https://math-demo.abitti.fi/math.svg?latex=a_n" alt="a_n"/>raja-arvo<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5Cfrac%7Bn%2B1%7D%7Bn%7D%3D%5Cln%5Cleft(1%2B%5Cfrac%7B1%7D%7Bn%7D%5Cright)%5Crightarrow%5Cln%5Cleft(1%2B0%5Cright)%3D%5Cln1%3D0" alt="\ln\frac{n+1}{n}=\ln\left(1+\frac{1}{n}\right)\rightarrow\ln\left(1+0\right)=\ln1=0"/></div> Yleisen termin raja-arvo on 0, joten sarja voi supeta</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=S_n%3D%5Cln%5Cfrac%7Bn%2B1%7D%7Bn%7D%3D%5Cln%5Cleft(n%2B1%5Cright)-%5Cln%20n" alt="S_n=\ln\frac{n+1}{n}=\ln\left(n+1\right)-\ln n"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Cln2-%5Cln1%5Cright)%2B%5Cleft(%5Cln3-%5Cln2%5Cright)%2B%5Cleft(%5Cln4-%5Cln3%5Cright)" alt="=\left(\ln2-\ln1\right)+\left(\ln3-\ln2\right)+\left(\ln4-\ln3\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Cln1%2B%5Cln%5Cleft(n%2B1%5Cright)%5Crightarrow%5Cln%5Cleft(n%2B1%5Cright)%3D%5Cinfty" alt="=-\ln1+\ln\left(n+1\right)\rightarrow\ln\left(n+1\right)=\infty"/></div> <div>Sarja ei suppene</div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> 2019-11-27T09:53:15+02:00 https://peda.net/id/fd9fba340f5 2019-11-27T00:36:59+02:00 <div> <div>243</div> </div> <div>a) On geometrinen lukujono, jossa <img src="https://math-demo.abitti.fi/math.svg?latex=a_1%3D-2%7B%2C%7D%5C%20q%3D3" alt="a_1=-2{,}\ q=3"/>, lukujono hajaantuu, koska q=3&gt;1</div> <div>b) On geometrinen, koska jonon n:s jäsen on muotoa <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3Dbq%5En%7B%2C%7D%5C%20miss%C3%A4%5C%20b%3D5%7B%2C%7D%5C%20q%3D-0%7B%2C%7D65" alt="a_n=bq^n{,}\ missä\ b=5{,}\ q=-0{,}65"/>, lukujono suppenee, kun q=-0,65, q∈]-1,1].</div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B%5Cleft(%5Cfrac%7B5%7D%7B4%7D%5Cright)%5E%7B%5Cleft(n%2B1%5Cright)-1%7D%7D%7B%5Cleft(%5Cfrac%7B5%7D%7B4%7D%5Cright)%5E%7Bn-1%7D%7D%3D%5Cfrac%7B%5Cleft(%5Cfrac%7B5%7D%7B4%7D%5Cright)%5En%7D%7B%5Cleft(%5Cfrac%7B5%7D%7B4%7D%5Cright)%5E%7Bn-1%7D%7D%3D%5Cleft(%5Cfrac%7B5%7D%7B4%7D%5Cright)%5E%7Bn-n-1%7D%3D%5Cfrac%7B5%7D%7B4%7D" alt="\frac{a_{n+1}}{a_n}=\frac{\left(\frac{5}{4}\right)^{\left(n+1\right)-1}}{\left(\frac{5}{4}\right)^{n-1}}=\frac{\left(\frac{5}{4}\right)^n}{\left(\frac{5}{4}\right)^{n-1}}=\left(\frac{5}{4}\right)^{n-n-1}=\frac{5}{4}"/></div> <div>Lukujonon peräkkäisten jäsenten suhde on vakio, joten lukujono on geometrinen. Suhdeluku q = 5/4. Lukujono ei suppene eli lukujono hajaantuu, koska q &gt; 1. <br/> <br/> <div>245</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B16%7D%7B15%7D%3D1%7B%2C%7D0666...%5Capprox1%7B%2C%7D07" alt="\frac{16}{15}=1{,}0666...\approx1{,}07"/> Suhdeluku |q|=1,07&gt;1, lukujono hajaantuu.</div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B-%5Cfrac%7B1%7D%7B2%7Da_n%7D%7Ba_n%7D%3D-%5Cfrac%7B1%7D%7B2%7D" alt="q=\frac{a_{n+1}}{a_n}=\frac{-\frac{1}{2}a_n}{a_n}=-\frac{1}{2}"/> Suhdeluku |q|= 1/2&lt;1, lukujono suppenee.</div> <span>c)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=q%3D-1%7B%2C%7D5" alt="q=-1{,}5"/> Suhdeluku |q|=1,5&gt;1, lukujono hajaantuu.</div> <div> <div><br/> 246<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B%5Cleft(-0%7B%2C%7D75%5Cright)%5E%7Bn%2B1%7D%7D%7B%5Cleft(-0%7B%2C%7D75%5Cright)%5En%7D%3D%5Cleft(-0%7B%2C%7D75%5Cright)%5E%7Bn%2B1-n%7D%3D-0%7B%2C%7D75%3C1" alt="q=\frac{a_{n+1}}{a_n}=\frac{\left(-0{,}75\right)^{n+1}}{\left(-0{,}75\right)^n}=\left(-0{,}75\right)^{n+1-n}=-0{,}75&lt;1"/><br/> Suhdeluku |q|=0,75&lt;1, lukujono suppenee.<span> </span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7Da_n%3D0" alt="\lim_{n\rightarrow\infty}a_n=0"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B3%5Ccdot0%7B%2C%7D5%5E%7B%5Cleft(n%2B1%5Cright)-1%7D%7D%7B3%5Ccdot0%7B%2C%7D5%5E%7Bn-1%7D%7D%3D%5Cfrac%7B0%7B%2C%7D5%5En%7D%7B0%7B%2C%7D5%5E%7Bn-1%7D%7D%3D0%7B%2C%7D5%5E%7Bn-%5Cleft(n-1%5Cright)%7D%3D0%7B%2C%7D5%3C1" alt="q=\frac{a_{n+1}}{a_n}=\frac{3\cdot0{,}5^{\left(n+1\right)-1}}{3\cdot0{,}5^{n-1}}=\frac{0{,}5^n}{0{,}5^{n-1}}=0{,}5^{n-\left(n-1\right)}=0{,}5&lt;1"/><br/> Suhdeluku |q|=0,5&lt;1, lukujono suppenee.<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7Da_n%3D0" alt="\lim_{n\rightarrow\infty}a_n=0"/></div> <span>c)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B-5%5Ccdot2%7B%2C%7D5%5E%7Bn%2B1%7D%7D%7B-5%5Ccdot2%7B%2C%7D5%5En%7D%3D2%7B%2C%7D5%5E%7B%5Cleft(n%2B1%5Cright)-n%7D%3D2%7B%2C%7D5%3E1" alt="q=\frac{a_{n+1}}{a_n}=\frac{-5\cdot2{,}5^{n+1}}{-5\cdot2{,}5^n}=2{,}5^{\left(n+1\right)-n}=2{,}5&gt;1"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <span>Suhdeluku |q|=2,5&gt;1, lukujono hajaantuu, joten sillä ei ole raja-arvoa.</span> <div>ei ole raja-arvoa <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B%5Cleft(-0%7B%2C%7D25%5Cright)%5E%7Bn%2B2%2B1%7D%7D%7B%5Cleft(-0%7B%2C%7D25%5Cright)%5E%7Bn%2B2%7D%7D%3D%5Cfrac%7B%5Cleft(-0%7B%2C%7D25%5Cright)%5E%7Bn%2B3%7D%7D%7B%5Cleft(-0%7B%2C%7D25%5Cright)%5E%7Bn%2B2%7D%7D%3D%5Cleft(-0%7B%2C%7D25%5Cright)%5E%7Bn-n%2B3-2%7D%3D-0%7B%2C%7D25%3C1" alt="q=\frac{a_{n+1}}{a_n}=\frac{\left(-0{,}25\right)^{n+2+1}}{\left(-0{,}25\right)^{n+2}}=\frac{\left(-0{,}25\right)^{n+3}}{\left(-0{,}25\right)^{n+2}}=\left(-0{,}25\right)^{n-n+3-2}=-0{,}25&lt;1"/><br/> Suhdeluku |q|=0,25&lt;1, lukujono suppenee.<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7Da_n%3D0" alt="\lim_{n\rightarrow\infty}a_n=0"/> </div> </div> <div>e)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=q%3D%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B4%5Ccdot%5Cleft(-1%5Cright)%5E%7Bn%2B1%2B1%7D%7D%7B4%5Ccdot%5Cleft(-1%5Cright)%5E%7Bn%2B1%7D%7D%3D%5Cfrac%7B%5Cleft(-1%5Cright)%5E%7Bn%2B2%7D%7D%7B%5Cleft(-1%5Cright)%5E%7Bn%2B1%7D%7D%3D%5Cleft(-1%5Cright)%5E%7Bn-n%2B2-1%7D%3D-1" alt="q=\frac{a_{n+1}}{a_n}=\frac{4\cdot\left(-1\right)^{n+1+1}}{4\cdot\left(-1\right)^{n+1}}=\frac{\left(-1\right)^{n+2}}{\left(-1\right)^{n+1}}=\left(-1\right)^{n-n+2-1}=-1"/></div> <div>Suhdeluku |-1|=1=1</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7Da_n%3Da_1%3D4%5Ccdot%5Cleft(-1%5Cright)%5E%7B1%2B1%7D%3D4%5Ccdot%5Cleft(-1%5Cright)%5E2%3D4" alt="\lim_{n\rightarrow\infty}a_n=a_1=4\cdot\left(-1\right)^{1+1}=4\cdot\left(-1\right)^2=4"/></div> <span>Lukujono on 4, −4, 4, −4, 4, …, ei raja-arvoa  </span><br/> <div>f)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%5Crightarrow%5Cinfty" alt="a_n\rightarrow\infty"/></div> <div> <div>Ei raja-arvoa<br/> <br/> </div> </div> </div> <div> <div> <div>248</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B2n%2B1%2B1%7D%7B2n%2B1%7D%3D%5Cfrac%7B2n%2B2%7D%7B2n%2B1%7D%3D%5Cfrac%7B2%7D%7B1%7D%3D2" alt="\frac{a_{n+1}}{a_n}=\frac{2n+1+1}{2n+1}=\frac{2n+2}{2n+1}=\frac{2}{1}=2"/></div> <div>Suhdeluku |q|=2&gt;1, lukujono hajaantuu.</div> <div>Lukujono on geomterinen</div> <div>ei ole raja-arvoa </div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B5%5Ccdot%5Cleft(%5Cfrac%7Be%7D%7B%5Cpi%7D%5Cright)%5E%7Bn%2B1-1%7D%7D%7B5%5Ccdot%5Cleft(%5Cfrac%7Be%7D%7B%5C%20%5Cpi%7D%5Cright)%5E%7Bn-1%7D%7D%3D%5Cfrac%7B5%5Ccdot%5Cleft(%5Cfrac%7Be%7D%7B%5Cpi%7D%5Cright)%5E%7Bn-1%7D-%5Cleft(%5Cfrac%7Be%7D%7B%5Cpi%7D%5Cright)%5E1%7D%7B%5Cleft(%5Cfrac%7Be%7D%7B%5Cpi%7D%5Cright)%5E%7Bn-1%7D%7D%3D%5Cfrac%7Be%7D%7B%5Cpi%7D" alt="\frac{a_{n+1}}{a_n}=\frac{5\cdot\left(\frac{e}{\pi}\right)^{n+1-1}}{5\cdot\left(\frac{e}{\ \pi}\right)^{n-1}}=\frac{5\cdot\left(\frac{e}{\pi}\right)^{n-1}-\left(\frac{e}{\pi}\right)^1}{\left(\frac{e}{\pi}\right)^{n-1}}=\frac{e}{\pi}"/></div> <div>Suhdeluku |q|≈087&lt;1, lukujono suppenee. </div> <span>Lukujono on geomterinen</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7Da_n%3D0" alt="\lim_{n\rightarrow\infty}a_n=0"/><span> </span><br/> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B%5Cleft(%5Csqrt%5B%5D%7B2%7D%5C%20%5Cright)%5E%7Bn%2B1-1%7D%7D%7B%5Cleft(%5Csqrt%5B%5D%7B2%7D%5Cright)%5E%7Bn-1%7D%7D%3D%5Cfrac%7B%5Cleft(%5Csqrt%5B%5D%7B2%7D%5Cright)%5En%7D%7B%5Cleft(%5Csqrt%5B%5D%7B2%7D%5Cright)%5E%7Bn-1%7D%7D%3D%5Cleft(%5Csqrt%5B%5D%7B2%7D%5Cright)%5E%7Bn-n-1%7D%3D%5Cleft(%5Csqrt%5B%5D%7B2%7D%5Cright)%5E%7B-1%7D%3D%5Cfrac%7B%5Csqrt%5B%5D%7B2%7D%7D%7B2%7D" alt="\frac{a_{n+1}}{a_n}=\frac{\left(\sqrt[]{2}\ \right)^{n+1-1}}{\left(\sqrt[]{2}\right)^{n-1}}=\frac{\left(\sqrt[]{2}\right)^n}{\left(\sqrt[]{2}\right)^{n-1}}=\left(\sqrt[]{2}\right)^{n-n-1}=\left(\sqrt[]{2}\right)^{-1}=\frac{\sqrt[]{2}}{2}"/></div> <div>Suhdeluku |q|≈0,71&lt;1, lukujono suppenee. </div> <div>Lukujono on geomterinen<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bn%5Crightarrow%5Cinfty%7Da_n%3D0" alt="\lim_{n\rightarrow\infty}a_n=0"/> </div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7B2n-1%7D%7Bn%2B1%7D" alt="a_n=\frac{2n-1}{n+1}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_1%3D%5Cfrac%7B2n-1%7D%7Bn%2B1%7D%7B%2C%7D%5C%20a_2%3D%5Cfrac%7B3%7D%7B3%7D%3D1%7B%2C%7D%5C%20a_3%3D%5Cfrac%7B5%7D%7B4%7D" alt="a_1=\frac{2n-1}{n+1}{,}\ a_2=\frac{3}{3}=1{,}\ a_3=\frac{5}{4}"/></div> <div> Lukujono ei ole geometrinen, koska peräkkäisten jäsenten suhde ei ole vakio. </div> <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7B2n-1%7D%7Bn%2B1%7D%3D%5Cfrac%7Bn%5Cleft(2-%5Cfrac%7B1%7D%7Bn%7D%5Cright)%7D%7Bn%5Cleft(1%2B%5Cfrac%7B1%7D%7Bn%7D%5Cright)%7D%5Crightarrow%5Cfrac%7B2-0%7D%7B1%2B0%7D%3D%5Cfrac%7B2%7D%7B1%7D%3D2" alt="a_n=\frac{2n-1}{n+1}=\frac{n\left(2-\frac{1}{n}\right)}{n\left(1+\frac{1}{n}\right)}\rightarrow\frac{2-0}{1+0}=\frac{2}{1}=2"/> <div>Lukujonon raja-arvo on 2.<br/> <div> </div> </div> </div> <div> <div>249</div> <div>a)</div> <div>Harrastelija voi tietää todeksi väitteen: Lukujono ei ole monotoninen. Lukujonossa a1 &gt; a2, mutta a2 &lt; a3, joten lukujono ei voi olla monotoninen. Kahta muuta ei voi laskujen perusteella tietää todeksi.  </div> <div>b) </div> <div>Osoitetaan, että lukujono on geometrinen, laskemalla lukujonon peräkkäisten termien suhde.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B%5Cfrac%7B12%7D%7B%5Cleft(-2%5Cright)%5E%7Bn%2B1-1%7D%7D%7D%7B%5Cfrac%7B12%7D%7B%5Cleft(-2%5Cright)%5E%7Bn-1%7D%7D%7D%3D%5Cfrac%7B12%7D%7B%5Cleft(-2%5Cright)%5En%7D%5Ccdot%5Cfrac%7B%5Cleft(-2%5Cright)%5E%7Bn-1%7D%7D%7B12%7D%3D%5Cleft(-2%5Cright)%5E%7Bn-1-n%7D%3D%5Cleft(-2%5Cright)%5E%7B-1%7D%3D-%5Cfrac%7B1%7D%7B2%7D" alt="\frac{a_{n+1}}{a_n}=\frac{\frac{12}{\left(-2\right)^{n+1-1}}}{\frac{12}{\left(-2\right)^{n-1}}}=\frac{12}{\left(-2\right)^n}\cdot\frac{\left(-2\right)^{n-1}}{12}=\left(-2\right)^{n-1-n}=\left(-2\right)^{-1}=-\frac{1}{2}"/></div> <span> Lukujonon peräkkäisten termien suhde on vakio, joten lukujono on geometrinen.  </span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7B12%7D%7B%5Cleft(-2%5Cright)%5E%7Bn-1%7D%7D%3D12%5Ccdot%5Cfrac%7B1%7D%7B%5Cleft(-2%5Cright)%5E%7Bn-1%7D%7D%3D12%5Ccdot%5Cfrac%7B1%5E%7Bn-1%7D%7D%7B%5Cleft(-2%5Cright)%5E%7Bn-1%7D%7D%5Crightarrow12%5Ccdot%5Cleft(-%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E%7Bn-1%7D%3D12%5Ccdot0%3D0%7B%2C%7D%5C%20%5C%20kun%5C%20n%5Crightarrow%5Cinfty%5C%20" alt="a_n=\frac{12}{\left(-2\right)^{n-1}}=12\cdot\frac{1}{\left(-2\right)^{n-1}}=12\cdot\frac{1^{n-1}}{\left(-2\right)^{n-1}}\rightarrow12\cdot\left(-\frac{1}{2}\right)^{n-1}=12\cdot0=0{,}\ \ kun\ n\rightarrow\infty\ "/><br/> <div> <div>Raja-arvo on 0<br/> <br/> </div> </div> </div> </div> <div>250 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D9%5E%7Bn%2B1%7D%3D0%7B%2C%7D9%5En%5Ccdot0%7B%2C%7D9" alt="0{,}9^{n+1}=0{,}9^n\cdot0{,}9"/></div> <div>Kun positiivinen luku kerrotaan lukua 1 pienemmällä positiivisella luvulla, sen arvo pienenee, eli se on lähempänä nollaa.</div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1%7B%2C%7D1%5E%7Bn%2B1%7D%3D1%7B%2C%7D1%5En%5Ccdot1%7B%2C%7D1" alt="1{,}1^{n+1}=1{,}1^n\cdot1{,}1"/></div> <span>Kun positiivinen luku kerrotaan lukua 1 suuremmalla positiivisella luvulla, sen arvo kasvaa, eli se on etäämpänä nollasta.  </span><br/> <span>c)</span> <div>a- ja b-kohtien perusteella, kun positiivista lukua kerrotaan lukua 1 pienemmällä positiivisella luvulla, sen arvo pienenee ja raja-arvo on 0. Vastaavasti kun lukua kerrotaan lukua 1 suuremmalla positiivisella luvulla, sen arvo kasvaa rajatta. </div> </div> <div> <div>251</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B9%7D%7B6%7D%3D%5Cfrac%7B3%7D%7B2%7D" alt="\frac{9}{6}=\frac{3}{2}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D9%5Ccdot%5Cleft(%5Cfrac%7B3%7D%7B2%7D%5Cright)%5E%7Bn-1%7D" alt="a_n=9\cdot\left(\frac{3}{2}\right)^{n-1}"/></div> <div>Lukujono on vähenevä, koska −1 &lt; q &lt; 1, eli lukujono on monotoninen.  </div> <br/> <div> <div> <div> <div>252</div> </div> </div> </div> </div> <div> </div> </div> </div> 2019-11-25T10:10:06+02:00 https://peda.net/id/b1e4eb240c4 2019-11-24T15:10:27+02:00 <span>224</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3Dn-n%5E2" alt="a_n=n-n^2"/></div> <div>Lukujono on </div> <div>1. kasvava, jos <img src="https://math-demo.abitti.fi/math.svg?latex=a_%7Bn%2B1%7D%3Ea_n" alt="a_{n+1}&gt;a_n"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"-->kaikialla n=1,2,3, ...</div> <div>2. vähenevä, jos <img src="https://math-demo.abitti.fi/math.svg?latex=a_%7Bn%2B1%7D%3Ca_n" alt="a_{n+1}&lt;a_n"/>kaikialla n=1,2,3,...</div> <div>Koska kyseessä olevassa funktiossa yritetään vähentämään n² n:stä, ja n² on melkein aina suurempi kuin n(paitsi lukujonon ensimmäinen jäsen 1, silloin n²=1²=1)</div> <div>Näin ollen funktio on vähenevä kaikialla, ja se on monotoninen.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=n-n%5E2%3Dn%5E2%5Cleft(%5Cfrac%7B1%7D%7Bn%7D-1%5Cright)%5Crightarrow%5Cinfty%5Cleft(0-1%5Cright)%3D-%5Cinfty%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="n-n^2=n^2\left(\frac{1}{n}-1\right)\rightarrow\infty\left(0-1\right)=-\infty{,}\ kun\ n\rightarrow\infty"/></div> <div>Funktio ei suppenee, se hajantuu.<br/> <br/> <div> <div>225</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7Bn%7D%7B2n%2B1%7D" alt="a_n=\frac{n}{2n+1}"/></div> a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_%7Bn%2B1%7D-a_n" alt="a_{n+1}-a_n"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7Bn%2B1%7D%7B2%5Cleft(n%2B1%5Cright)%2B1%7D-%5Cfrac%7Bn%7D%7B2n%2B1%7D" alt="=\frac{n+1}{2\left(n+1\right)+1}-\frac{n}{2n+1}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7Bn%2B1%7D%7B2n%2B2%2B1%7D-%5Cfrac%7Bn%7D%7B2n%2B1%7D" alt="=\frac{n+1}{2n+2+1}-\frac{n}{2n+1}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7Bn%2B1%7D%7B2n%2B3%7D-%5Cfrac%7Bn%7D%7B2n%2B1%7D" alt="=\frac{n+1}{2n+3}-\frac{n}{2n+1}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B%5Cleft(2n%2B1%5Cright)%5Cleft(n%2B1%5Cright)-%5Cleft(2n%2B3%5Cright)n%7D%7B%5Cleft(2n%2B3%5Cright)%5Cleft(2n%2B1%5Cright)%7D%3D%5Cfrac%7B2n%5E2%2Bn%2B2n%2B1-2n%5E2-3n%7D%7B%5Cleft(2n%2B3%5Cright)%5Cleft(2n%2B1%5Cright)%7D%3D%5Cfrac%7B1%7D%7B%5Cleft(2n%2B3%5Cright)%5Cleft(2n%2B1%5Cright)%7D" alt="=\frac{\left(2n+1\right)\left(n+1\right)-\left(2n+3\right)n}{\left(2n+3\right)\left(2n+1\right)}=\frac{2n^2+n+2n+1-2n^2-3n}{\left(2n+3\right)\left(2n+1\right)}=\frac{1}{\left(2n+3\right)\left(2n+1\right)}"/><br/> <div>Koska n≥1, 2n+3&gt;0 ja 2n+1&gt;0, joten<img src="https://math-demo.abitti.fi/math.svg?latex=a_%7Bn%2B1%7D-a_n%3E0" alt="a_{n+1}-a_n&gt;0"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--> ja lukujono on kasvava.</div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D" alt="\frac{a_{n+1}}{a_n}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B%5Cfrac%7Bn%2B1%7D%7B2%5Cleft(n%2B1%5Cright)%2B1%7D%7D%7B%5Cfrac%7Bn%7D%7B2n%2B1%7D%7D" alt="=\frac{\frac{n+1}{2\left(n+1\right)+1}}{\frac{n}{2n+1}}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7Bn%2B1%7D%7B2n%2B3%7D%5Ccdot%5Cfrac%7B2n%2B1%7D%7Bn%7D" alt="=\frac{n+1}{2n+3}\cdot\frac{2n+1}{n}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B2n%5E2%2B3n%2B1%7D%7B2n%5E2%2B3n%7D%3D1%2B%5Cfrac%7B1%7D%7B2n%5E2%2B3n%7D" alt="=\frac{2n^2+3n+1}{2n^2+3n}=1+\frac{1}{2n^2+3n}"/></div> <div>Koska n≥1, 2n+3&gt;0 ja 2n+1&gt;0, joten <img src="https://math-demo.abitti.fi/math.svg?latex=a_%7Bn%2B1%7D-a_n%3E0" alt="a_{n+1}-a_n&gt;0"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"-->ja lukujono on kasvava.</div> <span>c) </span><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bn%7D%7B2n%2B1%7D%3D%5Cfrac%7Bn%5Ccdot1%7D%7Bn%5Cleft(2%2B%5Cfrac%7B1%7D%7Bn%7D%5Cright)%7D%5Crightarrow%5Cfrac%7B1%7D%7B2%2B%5Cfrac%7B1%7D%7Bn%7D%7D%3D%5Cfrac%7B1%7D%7B2%2B0%7D%3D%5Cfrac%7B1%7D%7B2%7D%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="\frac{n}{2n+1}=\frac{n\cdot1}{n\left(2+\frac{1}{n}\right)}\rightarrow\frac{1}{2+\frac{1}{n}}=\frac{1}{2+0}=\frac{1}{2}{,}\ kun\ n\rightarrow\infty"/><br/> <div> </div> <div>226</div> <div>A–I, B–ei kumpikaan, C–II, D–ei kumpikaan, E–I  </div> <br/> <div> <div>228</div> <div>a)</div> <div>Tarkaistellaan lukujonon tyyppiä laskemalla kahden peräkkäisten jäsennen erotusta</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_%7Bn%2B1%7D-a_n" alt="a_{n+1}-a_n"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(1-2%5Cleft(n%2B1%5Cright)%5Cright)-%5Cleft(1-2n%5Cright)" alt="=\left(1-2\left(n+1\right)\right)-\left(1-2n\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(1-2n%2B2%5Cright)-%5Cleft(1-2n%5Cright)" alt="=\left(1-2n+2\right)-\left(1-2n\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D1-2n-2-1%2B2n%3D-2" alt="=1-2n-2-1+2n=-2"/><br/> Lukujono on aritmeettien</div> <div>Koska peräkkäisten jäsenten erotus on negatiivinen, lukujono on vähenevä, eli se on monotoninen.  </div> <div>Kun n→∞, <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D1-2n%5Crightarrow-%5Cinfty" alt="a_n=1-2n\rightarrow-\infty"/>. Lukujono hajaantuu.</div> <div> <div>b)</div> <div>Tarkaistellaan lukujonon tyyppiä laskemalla kahden peräkkäisten jäsennen erotusta<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a_%7Bn%2B1%7D-a_n" alt="a_{n+1}-a_n"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D2%2B%5Cleft(-1%5Cright)%5E%7Bn%2B1%7D-%5Cleft(2%2B%5Cleft(-1%5Cright)%5En%5Cright)" alt="=2+\left(-1\right)^{n+1}-\left(2+\left(-1\right)^n\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(-1%5Cright)%5En%5Cleft(-1%5Cright)-%5Cleft(-1%5Cright)%5En%3D-%5Cleft(-1%5Cright)%5En%5Ccdot1-%5Cleft(-1%5Cright)%5En" alt="=\left(-1\right)^n\left(-1\right)-\left(-1\right)^n=-\left(-1\right)^n\cdot1-\left(-1\right)^n"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Cleft(-1%5Cright)%5En-%5Cleft(-1%5Cright)%5En" alt="=-\left(-1\right)^n-\left(-1\right)^n"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-2%5Cleft(-1%5Cright)%5En" alt="=-2\left(-1\right)^n"/> <div>Lukujono ei ole aritmeettinen, koska kahden peräkkäisten jäsenten erotus ei ole vakio.</div> <div>Lukujonon peräkkäisten jäsenten erotuksen merkki riippuu siitä, onko n parillinen vai pariton. Lukujono ei ole monotoninen.  </div> <div>Lukujono on 2, −2, 2, −2, 2, …, joten lukujono ei suppene.  </div> <div>c)</div> <div>Tarkaistellaan lukujonon tyyppiä laskemalla kahden peräkkäisten jäsennen erotusta<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a_%7Bn%2B1%7D-a_n" alt="a_{n+1}-a_n"/> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Cleft(n%2B1%5Cright)-%5Cfrac%7B1%7D%7Bn%2B1%7D%5Cright)-%5Cleft(n-%5Cfrac%7B1%7D%7Bn%7D%5Cright)" alt="=\left(\left(n+1\right)-\frac{1}{n+1}\right)-\left(n-\frac{1}{n}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3Dn%2B1-%5Cfrac%7B1%7D%7Bn%2B1%7D-n%2B%5Cfrac%7B1%7D%7Bn%7D%3D1-%5Cfrac%7B1%7D%7Bn%2B1%7D%2B%5Cfrac%7B1%7D%7Bn%7D" alt="=n+1-\frac{1}{n+1}-n+\frac{1}{n}=1-\frac{1}{n+1}+\frac{1}{n}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D1-%5Cfrac%7Bn%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%2B%5Cfrac%7Bn%2B1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D" alt="=1-\frac{n}{n\left(n+1\right)}+\frac{n+1}{n\left(n+1\right)}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D1%2B%5Cfrac%7B-n%2Bn%2B1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%3D1%2B%5Cfrac%7B1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D" alt="=1+\frac{-n+n+1}{n\left(n+1\right)}=1+\frac{1}{n\left(n+1\right)}"/></div> <div>Peräkkäisten jäsenten erotus ei ole vakio, joten lukujono ei ole aritmeettinen. Koska n ≥ 1, peräkkäisten jäsenten erotus on aina positiivinen. Lukujono on kasvava, eli se on monotoninen. Kun n→∞,<img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3Dn-%5Cfrac%7B1%7D%7Bn%7D%5Crightarrow%5Cinfty" alt="a_n=n-\frac{1}{n}\rightarrow\infty"/>.<span> </span>Lukujono hajaantuu.  </div> </div> <br/> <div> <div><span>230</span><br/> <span>a)</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3De%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D%5Crightarrow%20e%5E0%3D1%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="a_n=e^{\frac{1}{n}}\rightarrow e^0=1{,}\ kun\ n\rightarrow\infty"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D" alt="\frac{a_{n+1}}{a_n}"/><span> </span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7Be%5E%7B%5Cfrac%7B1%7D%7Bn%2B1%7D%7D%7D%7Be%5En%7D%3De%5E%7B%5Cfrac%7B1%7D%7Bn%2B1%7D-%5Cfrac%7B1%7D%7Bn%7D%7D%3De%5E%7B%5Cfrac%7Bn-%5Cleft(n%2B1%5Cright)%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%7D%3De%5E%7B-%5Cfrac%7B1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%7D%3D%5Cfrac%7B1%7D%7Be%5E%7B%5Cfrac%7B1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%7D%7D" alt="=\frac{e^{\frac{1}{n+1}}}{e^n}=e^{\frac{1}{n+1}-\frac{1}{n}}=e^{\frac{n-\left(n+1\right)}{n\left(n+1\right)}}=e^{-\frac{1}{n\left(n+1\right)}}=\frac{1}{e^{\frac{1}{n\left(n+1\right)}}}"/><br/> <span>Koska n ≥ 1, </span><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%3E0%7B%2C%7D%5C%20jolloin%5C%20e%5E%7B%5Cfrac%7B1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%7D%3E1%5C%20ja%5C%20%5Cfrac%7B1%7D%7Be%5E%7B%5Cfrac%7B1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%7D%7D%3C1." alt="\frac{1}{n\left(n+1\right)}&gt;0{,}\ jolloin\ e^{\frac{1}{n\left(n+1\right)}}&gt;1\ ja\ \frac{1}{e^{\frac{1}{n\left(n+1\right)}}}&lt;1."/><span>Lukujono on vähenevä eli monotoninen.</span> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Csqrt%5Bn%5D%7B2%5E%7Bn%2B1%7D%7D%3D2%5E%7B%5Cfrac%7Bn%2B1%7D%7Bn%7D%7D%5Crightarrow2%5E%7B1%2B%5Cfrac%7B1%7D%7Bn%7D%7D%3D2%5E%7B1%2B0%7D%3D2%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="a_n=\sqrt[n]{2^{n+1}}=2^{\frac{n+1}{n}}\rightarrow2^{1+\frac{1}{n}}=2^{1+0}=2{,}\ kun\ n\rightarrow\infty"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%3D%5Cfrac%7B2%5E%7B%5Cfrac%7Bn%2B2%7D%7Bn%2B1%7D%7D%7D%7B2%5E%7B%5Cfrac%7Bn%2B1%7D%7Bn%7D%7D%7D%3D2%5E%7B%5Cfrac%7Bn%2B2%7D%7Bn%2B1%7D-%5Cfrac%7Bn%2B1%7D%7Bn%7D%7D%3D2%5E%7B%5Cfrac%7Bn%5Cleft(n%2B2%5Cright)-%5Cleft(n%2B1%5Cright)%5E2%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%7D%3D2%5E%7B%5Cfrac%7Bn%5E2%2B2n-n%5E2-2n-1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%7D%3D2%5E%7B-%5Cfrac%7B1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%7D%3D%5Cfrac%7B1%7D%7B2%5E%7B%5Cfrac%7B1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%7D%7D" alt="\frac{a_{n+1}}{a_n}=\frac{2^{\frac{n+2}{n+1}}}{2^{\frac{n+1}{n}}}=2^{\frac{n+2}{n+1}-\frac{n+1}{n}}=2^{\frac{n\left(n+2\right)-\left(n+1\right)^2}{n\left(n+1\right)}}=2^{\frac{n^2+2n-n^2-2n-1}{n\left(n+1\right)}}=2^{-\frac{1}{n\left(n+1\right)}}=\frac{1}{2^{\frac{1}{n\left(n+1\right)}}}"/><br/> Koska n ≥ 1, <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%3E0%7B%2C%7D%5C%20jolloin%5C%202%5E%7B%5Cfrac%7B1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%7D%3E1%5C%20ja%5C%20%5Cfrac%7B1%7D%7B2%5E%7B%5Cfrac%7B1%7D%7Bn%5Cleft(n%2B1%5Cright)%7D%7D%7D%3C1." alt="\frac{1}{n\left(n+1\right)}&gt;0{,}\ jolloin\ 2^{\frac{1}{n\left(n+1\right)}}&gt;1\ ja\ \frac{1}{2^{\frac{1}{n\left(n+1\right)}}}&lt;1."/></div> <div>Lukujono on vähenevä eli monotoninen.<br/> <br/> </div> </div> </div> <div> <div>233 <div> <div>Tarkastellaan funktiota </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Csqrt%5B%5D%7Bx%5E2%2B1%7D-x%7B%2C%7D%5C%20x%5Cge1" alt="f\left(x\right)=\sqrt[]{x^2+1}-x{,}\ x\ge1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Cfrac%7Bx%7D%7B%5Csqrt%5B%5D%7Bx%5E2%2B1%7D%7D-1%7B%2C%7D%5C%20x%3E1" alt="f'\left(x\right)=\frac{x}{\sqrt[]{x^2+1}}-1{,}\ x&gt;1"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--> <div>Ratkaistaan derivaatan nollakodat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%7D%7B%5Csqrt%5B%5D%7Bx%5E2%2B1%7D%7D-1%3D0" alt="\frac{x}{\sqrt[]{x^2+1}}-1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%7D%7B%5Csqrt%5B%5D%7Bx%5E2%2B1%7D%7D%3D1" alt="\frac{x}{\sqrt[]{x^2+1}}=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Csqrt%5B%5D%7Bx%5E2%2B1%7D" alt="x=\sqrt[]{x^2+1}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3Dx%5E2%2B1" alt="x^2=x^2+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=0%3D1" alt="0=1"/><br/> <div>Deivaatalla ei ole nollakohtia. Derivaatan merkki ei vaihdu määrittelyjoukossa x&gt;1</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(2%5Cright)%3D%5Cfrac%7B2%7D%7B%5C%20%5Csqrt%5B%5D%7B5%7D%7D-1%3C0" alt="f'\left(2\right)=\frac{2}{\ \sqrt[]{5}}-1&lt;0"/></div> <div>Derivaatan merkki on negatiivinen, joten funktio f on vähenevä</div> <div>Vastaava lukujono on myös vähenevä, joten lukujonon ensimmäinen säsen on suurin</div> <div>Nyt</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_1%3D%5Csqrt%5B%5D%7B1%5E2%2B1%7D-1%3D%5Csqrt%5B%5D%7B2%7D-1" alt="a_1=\sqrt[]{1^2+1}-1=\sqrt[]{2}-1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Csqrt%5B%5D%7Bn%5E2%2B1%7D-n%3D%5Csqrt%5B%5D%7Bn%5E2%5Cleft(1%2B%5Cfrac%7B1%7D%7Bn%5E2%7D%5Cright)%7D-n%3Dn%5Cleft(%5Csqrt%5B%5D%7B1%2B%5Cfrac%7B1%7D%7Bn%5E2%7D%7D-1%5Cright)%5Crightarrow0%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="a_n=\sqrt[]{n^2+1}-n=\sqrt[]{n^2\left(1+\frac{1}{n^2}\right)}-n=n\left(\sqrt[]{1+\frac{1}{n^2}}-1\right)\rightarrow0{,}\ kun\ n\rightarrow\infty"/></div> Koska n ≥ 1, niin <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7B1%2B%5Cfrac%7B1%7D%7Bn%5E2%7D%7D%3E1%5C%20" alt="\sqrt[]{1+\frac{1}{n^2}}&gt;1\ "/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"-->ja <img src="https://math-demo.abitti.fi/math.svg?latex=%5C%20%5Csqrt%5B%5D%7B1%2B%5Cfrac%7B1%7D%7Bn%5E2%7D%7D-1%3E0" alt="\ \sqrt[]{1+\frac{1}{n^2}}-1&gt;0"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"-->. Lukujonon kaikki arvot ovat positiivisia. Lukujonon jäsenet ovat hyvin lähellä nollaa ja positiivisia.<br/> <div> </div> </div> </div> <div>236<br/> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7Bn%2B10%7D%7Bn!%7D" alt="a_n=\frac{n+10}{n!}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_%7Bn%2B1%7D-a_n%3D%5Cfrac%7B%5Cleft(n%2B1%5Cright)%2B10%7D%7B%5Cleft(n%2B1%5Cright)!%7D-%5Cfrac%7Bn%2B10%7D%7Bn!%7D" alt="a_{n+1}-a_n=\frac{\left(n+1\right)+10}{\left(n+1\right)!}-\frac{n+10}{n!}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7Bn%2B11%7D%7B%5Cleft(n%2B1%5Cright)n!%7D-%5Cfrac%7Bn%2B10%7D%7Bn!%7D" alt="=\frac{n+11}{\left(n+1\right)n!}-\frac{n+10}{n!}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7Bn%2B11-%5Cleft(n%2B1%5Cright)%5Cleft(n%2B10%5Cright)%7D%7B%5Cleft(n%2B1%5Cright)n!%7D" alt="=\frac{n+11-\left(n+1\right)\left(n+10\right)}{\left(n+1\right)n!}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7Bn%2B11-n%5E2-11n-10%7D%7B%5Cleft(n%2B1%5Cright)!%7D" alt="=\frac{n+11-n^2-11n-10}{\left(n+1\right)!}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B-n%5E2-10n%2B1%7D%7B%5Cleft(n%2B1%5Cright)!%7D" alt="=\frac{-n^2-10n+1}{\left(n+1\right)!}"/></div> <span>Koska n ≥ 1, (n+1)!&gt;0 </span> <div>Tutkitaan osoittajan -n²-10n+1 merkkiä</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-n%5E2-10n%2B1%3D0" alt="-n^2-10n+1=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=n%5Capprox-10%7B%2C%7D1%5C%20tai%5C%20n%5Capprox0%7B%2C%7D1" alt="n\approx-10{,}1\ tai\ n\approx0{,}1"/><br/> <div>Kuvaaja on alaspäin aukeava paraabeli. Kun n ≥ 1, lausekkeen -n²-10n+1 merkki on negatiivinen.</div> <div>Lukujonon kahden peräkkäisen termin erotus on siis negatiivinen, joten lukujono on vähenevä.</div> <div> </div> </div> </div> <div>241<br/>  <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7Bn%5E4%7D%7B2%5En%7D%7B%2C%7D%5C%20n%3D1%7B%2C%7D%5C%202%7B%2C%7D%5C%203%7B%2C%7D..." alt="a_n=\frac{n^4}{2^n}{,}\ n=1{,}\ 2{,}\ 3{,}..."/><br/> <div>Tarkstellaan funktiota </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cfrac%7Bx%5E4%7D%7B2%5Ex%7D" alt="f\left(x\right)=\frac{x^4}{2^x}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Cfrac%7B4x%5E3%5Ccdot2%5Ex-x%5E4%5Ccdot2%5Ex%5Ccdot%5Cln2%7D%7B%5Cleft(2%5Ex%5Cright)%5E2%7D%3D%5Cfrac%7B2%5Ex%5Ccdot%20x%5E3%5Cleft(4-x%5Cln2%5Cright)%7D%7B2%5E%7B2x%7D%7D%3D%5Cfrac%7Bx%5E3%5Cleft(4-x%5Cln2%5Cright)%7D%7B2%5Ex%7D" alt="f'\left(x\right)=\frac{4x^3\cdot2^x-x^4\cdot2^x\cdot\ln2}{\left(2^x\right)^2}=\frac{2^x\cdot x^3\left(4-x\ln2\right)}{2^{2x}}=\frac{x^3\left(4-x\ln2\right)}{2^x}"/><br/> <div>Koska x ≥1, niin x³ &gt; 0 ja<img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ex%3E0" alt="2^x&gt;0"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"-->, joten derivaatan merkki riippuu vain lausekkeen 4-xln2 merkistä.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4-x%5Cln2%3D0" alt="4-x\ln2=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cln2%3D4" alt="x\ln2=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B4%7D%7B%5Cln2%7D%5Capprox5%7B%2C%7D77" alt="x=\frac{4}{\ln2}\approx5{,}77"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(5%5Cright)%5Capprox2%7B%2C%7D1%3E0" alt="f'\left(5\right)\approx2{,}1&gt;0"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(6%5Cright)%5Capprox-0%7B%2C%7D5%3C0" alt="f'\left(6\right)\approx-0{,}5&lt;0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%261%26%26%5Cfrac%7B4%7D%7B%5Cln2%7D%26%5C%5C%0A%5Chline%0Af%27%5Cleft(x%5Cright)%26%26%2B%26%26-%5C%5C%0Af%5Cleft(x%5Cright)%26%26%5Cnearrow%26%26%5Csearrow%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;1&amp;&amp;\frac{4}{\ln2}&amp;\\&#10;\hline&#10;f'\left(x\right)&amp;&amp;+&amp;&amp;-\\&#10;f\left(x\right)&amp;&amp;\nearrow&amp;&amp;\searrow&#10;\end{array}"/> <div>Funktio f saa suurimman arvonsa derivaatan nollakohdassa <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B4%7D%7B%5Cln2%7D%5Capprox5%7B%2C%7D77" alt="x=\frac{4}{\ln2}\approx5{,}77"/>. Lukujonon suurin jäsen on siten<img src="https://math-demo.abitti.fi/math.svg?latex=a_5" alt="a_5"/>tai <img src="https://math-demo.abitti.fi/math.svg?latex=a_6" alt="a_6"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_5%5Capprox19%7B%2C%7D5" alt="a_5\approx19{,}5"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=a_6%5Capprox20%7B%2C%7D25" alt="a_6\approx20{,}25"/> <div>Lukujopnon suurin jäsen on<img src="https://math-demo.abitti.fi/math.svg?latex=a_6" alt="a_6"/><br/> <div> </div> </div> <br/> <div> </div> </div> </div> </div> 2019-11-21T11:45:47+02:00 https://peda.net/id/6f4081860b6 2019-11-20T21:40:04+02:00 <div>205</div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D%5Cfrac%7Bn-2%7D%7Bn%7D" alt="a=\frac{n-2}{n}"/><br/> <div>a) Koordinaatiston piirretään pisteitä, jotka esittävät lukujonon jäseniä. Pisteen x-koordinaatti n lukuhonon jäsenen järejestysnumero n ja y.koordinaatti on jäsen <img src="https://math-demo.abitti.fi/math.svg?latex=a_n" alt="a_n"/>.</div> <div>Pisteet ovat muotoa <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(n%7B%2C%7Da_n%5Cright)" alt="\left(n{,}a_n\right)"/></div> <div>Kuvaajan mukaan lukujonon raja-arvo on kohdassa x=1</div> <div>b) </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7Bn-2%7D%7Bn%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cleft(%5Cfrac%7Bn%7D%7Bn%7D-%5Cfrac%7B2%7D%7Bn%7D%5Cright)%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cleft(1-%5Cfrac%7B2%7D%7Bn%7D%5Cright)%3D1-0%3D1" alt="\lim_{x\rightarrow\infty}\frac{n-2}{n}=\lim_{x\rightarrow\infty}\left(\frac{n}{n}-\frac{2}{n}\right)=\lim_{x\rightarrow\infty}\left(1-\frac{2}{n}\right)=1-0=1"/><br/> <div>Lukujen a ja b etäisyys on <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7Ca-b%5Cright%7C" alt="\left|a-b\right|"/></div> <div>Määritetään monnesta jäsenestä alkaen lukujonon jäsenen etäisyys raja-arvosta 1 on alle 0,01. Ratkaistaan n epäyhtälöstä</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Cfrac%7Bn-2%7D%7Bn%7D-1%5Cright%7C%3C0%7B%2C%7D01" alt="\left|\frac{n-2}{n}-1\right|&lt;0{,}01"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Cfrac%7Bn-2%7D%7Bn%7D-%5Cfrac%7Bn%7D%7Bn%7D%5Cright%7C%3C0%7B%2C%7D01" alt="\left|\frac{n-2}{n}-\frac{n}{n}\right|&lt;0{,}01"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Cfrac%7Bn-2-n%7D%7Bn%7D%5Cright%7C%3C0%7B%2C%7D01" alt="\left|\frac{n-2-n}{n}\right|&lt;0{,}01"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C-%5Cfrac%7B2%7D%7Bn%7D%5Cright%7C%3C0%7B%2C%7D01" alt="\left|-\frac{2}{n}\right|&lt;0{,}01"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cleft(-%5Cfrac%7B2%7D%7Bn%7D%5Cright)%3C0%7B%2C%7D01" alt="-\left(-\frac{2}{n}\right)&lt;0{,}01"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%7D%7Bn%7D%3C0%7B%2C%7D01" alt="\frac{2}{n}&lt;0{,}01"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D01n%3E2" alt="0{,}01n&gt;2"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff; color: #333333; font-family: 'Times New Roman'; font-size: 17px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3E200" alt="n&gt;200"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff; color: #333333; font-family: 'Times New Roman'; font-size: 17px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"--><br/> <div>201. jäsenestä alkaen<br/> <br/> <span>206</span> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7B3n%5E3%2Bn-2%7D%7Bn%5E2-1%7D%3D%5Cfrac%7Bn%5E2%5Cleft(3n%2B%5Cfrac%7B1%7D%7Bn%7D-%5Cfrac%7B2%7D%7Bn%5E2%7D%5Cright)%7D%7Bn%5E2%5Cleft(1-%5Cfrac%7B1%7D%7Bn%5E2%7D%5Cright)%7D%5Crightarrow%5Cinfty%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="a_n=\frac{3n^3+n-2}{n^2-1}=\frac{n^2\left(3n+\frac{1}{n}-\frac{2}{n^2}\right)}{n^2\left(1-\frac{1}{n^2}\right)}\rightarrow\infty{,}\ kun\ n\rightarrow\infty"/></div> <div>Raja-arvoa ei ole.</div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7B2n%5E3%2Bn-2%7D%7B6n%5E3-n%5E2%7D%3D%5Cfrac%7Bn%5E3%5Cleft(2%2B%5Cfrac%7B1%7D%7Bn%5E2%7D-%5Cfrac%7B2%7D%7Bn%5E3%7D%5Cright)%7D%7Bn%5E3%5Cleft(6-%5Cfrac%7B1%7D%7Bn%7D%5Cright)%7D%5Crightarrow%5Cfrac%7B2%7D%7B6%7D%3D%5Cfrac%7B1%7D%7B3%7D%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="a_n=\frac{2n^3+n-2}{6n^3-n^2}=\frac{n^3\left(2+\frac{1}{n^2}-\frac{2}{n^3}\right)}{n^3\left(6-\frac{1}{n}\right)}\rightarrow\frac{2}{6}=\frac{1}{3}{,}\ kun\ n\rightarrow\infty"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7B4n%5E3%2B1%7D%7Bn%5E4%2B1%7D%3D%5Cfrac%7Bn%5E4%5Cleft(%5Cfrac%7B4%7D%7Bn%7D%2B%5Cfrac%7B1%7D%7Bn%5E4%7D%5Cright)%7D%7Bn%5E4%5Cleft(1%2B%5Cfrac%7B1%7D%7Bn%5E4%7D%5Cright)%7D%5Crightarrow%5Cfrac%7B0%7D%7B1%7D%3D0%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="a_n=\frac{4n^3+1}{n^4+1}=\frac{n^4\left(\frac{4}{n}+\frac{1}{n^4}\right)}{n^4\left(1+\frac{1}{n^4}\right)}\rightarrow\frac{0}{1}=0{,}\ kun\ n\rightarrow\infty"/><br/> <br/> </div> <div>207</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3Dn%5E2-100n%5Crightarrow%20n%5E2%5Cleft(1-%5Cfrac%7B100%7D%7Bn%5E2%7D%5Cright)%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="a_n=n^2-100n\rightarrow n^2\left(1-\frac{100}{n^2}\right){,}\ kun\ n\rightarrow\infty"/></div> <div>Lukujonon<img src="https://math-demo.abitti.fi/math.svg?latex=a_n" alt="a_n"/>jäsenet kasvavat rajatta, joten lukujono hajaantuu.</div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3Dn%5E2-n%5E3%3D%5Crightarrow%20n%5E3%5Cleft(%5Cfrac%7B1%7D%7Bn%7D-1%5Cright)%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow-%5Cinfty" alt="a_n=n^2-n^3=\rightarrow n^3\left(\frac{1}{n}-1\right){,}\ kun\ n\rightarrow-\infty"/></div> <span> Lukujonon</span><img src="https://math-demo.abitti.fi/math.svg?latex=a_n" alt="a_n"/><span>jäsenet vähenevät rajatta, joten lukujono hajaantuu. </span><br/> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(-1%5Cright)%5En" alt="\left(-1\right)^n"/>on −1 parittomilla eksponenteilla ja 1 parillisilla eksponenteilla. Lukujonon<img src="https://math-demo.abitti.fi/math.svg?latex=a_n" alt="a_n"/> jäsenet ovat vuorotellen 1 − 1 = 0 ja 1 + 1 = 2, joten lukujono hajaantuu.  </div> <br/> <div>208</div> <div>A III ja V</div> <div>B IV, </div> <div>C I ja V <br/> <br/> <div>210</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Ccos%5Cleft(%5Cfrac%7B%5Cpi%7D%7B3%7D%2Bn%5Ccdot2%5Cpi%5Cright)%3D%5Cfrac%7B1%7D%7B2%7D%2Bn%5Ccdot1%3D%5Cfrac%7B1%7D%7B2%7D%2Bn" alt="a_n=\cos\left(\frac{\pi}{3}+n\cdot2\pi\right)=\frac{1}{2}+n\cdot1=\frac{1}{2}+n"/></div> <div>Koska kosinin jakso on 2π, ja cos2π=1/2, näin ollen lukujonon kaikki jäsenet ovat 1/2 , joten lukujono raja-arvo on 1/2.</div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Ccos%5Cleft(n%5Ccdot%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%3D0" alt="a_n=\cos\left(n\cdot\frac{\pi}{2}\right)=0"/></div> <div>Lukujonossa toistuvat jäsenet 0, −1, 0 ja 1 tässä järjestyksessä. Lukujonolla ei ole raja-arvoa.  </div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Ccos%5Cfrac%7B%5Cpi%7D%7Bn%7D" alt="a_n=\cos\frac{\pi}{n}"/></div> <div>Kun n kasvaa,<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cpi%7D%7Bn%7D" alt="\frac{\pi}{n}"/>lähestyy nollaa, jolloin <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cfrac%7B%5Cpi%7D%7Bn%7D" alt="\cos\frac{\pi}{n}"/> lähestyy arvoa 1. </div> </div> <div><br/> 213<br/> <div>Lukujonon jäsenten nimittäjä on yhtä suurempi kuin jäsenen järjestysluku ja osoittajat muodostavat aritmeettisen lukujonon, jossa peräkkäisten jäsenten erotus on 3. Nimittäjä on siis muotoa n + 1 ja osoittaja</div> <img src="https://math-demo.abitti.fi/math.svg?latex=1%2B%5Cleft(n-1%5Cright)%5Ccdot3%3D3n-2" alt="1+\left(n-1\right)\cdot3=3n-2"/> <div>Lukujonon n:s jäsen onm <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7B3n-2%7D%7Bn%2B1%7D" alt="a_n=\frac{3n-2}{n+1}"/></div> <div>Lukujonon raja-arvo:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7B3n-2%7D%7Bn%2B1%7D%3D%5Cfrac%7B3-%5Cfrac%7B2%7D%7Bn%7D%7D%7B1%2B%5Cfrac%7B1%7D%7Bn%7D%7D%5Crightarrow-%5Cfrac%7B3-0%7D%7B1%2B0%7D%3D3%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="a_n=\frac{3n-2}{n+1}=\frac{3-\frac{2}{n}}{1+\frac{1}{n}}\rightarrow-\frac{3-0}{1+0}=3{,}\ kun\ n\rightarrow\infty"/></div> <div>Luku jonon jäsenet ovat pienempiä kuin raja-arvo.</div> <div>Ratkaistaan epäyhtälö <img src="https://math-demo.abitti.fi/math.svg?latex=3-a_n%3C0%7B%2C%7D001" alt="3-a_n&lt;0{,}001"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3-%5Cleft(%5Cfrac%7B3n-2%7D%7Bn%2B1%7D%5Cright)%3C%5Cfrac%7B1%7D%7B1000%7D" alt="3-\left(\frac{3n-2}{n+1}\right)&lt;\frac{1}{1000}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3n%2B3-3n%2B2%7D%7Bn%2B1%7D%3C%5Cfrac%7B1%7D%7B1000%7D" alt="\frac{3n+3-3n+2}{n+1}&lt;\frac{1}{1000}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B5%7D%7Bn%2B1%7D%3C%5Cfrac%7B1%7D%7B1000%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Ccdot1000" alt="\frac{5}{n+1}&lt;\frac{1}{1000}\ \ \ \ \ \left|\right|\cdot1000"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=5000%3Cn%2B1" alt="5000&lt;n+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3E4999" alt="n&gt;4999"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <div>Lukujonon jäsenen poikkeama raja-arvosta on itseisarvoltaan pienempi kuin 0,001 lukujonon 5000. jäsenestä alkaen.  <br/> <div> </div> </div> </div> <div>216<br/> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Csqrt%5B%5D%7Bn%2B1%7D-%5Csqrt%5B%5D%7Bn%7D%3D%5Cfrac%7B%5Cleft(%5Csqrt%5B%5D%7Bn%2B1%7D-%5Csqrt%5B%5D%7Bn%7D%5Cright)%5Cleft(%5Csqrt%5B%5D%7Bn%2B1%7D%2B%5Csqrt%5B%5D%7Bn%7D%5Cright)%7D%7B%5Csqrt%5B%5D%7Bn%2B1%7D%2B%5Csqrt%5B%5D%7Bn%7D%7D%3D%5Cfrac%7B%5Cleft(n%2B1%5Cright)-n%7D%7B%5Csqrt%5B%5D%7Bn%2B1%7D%2B%5Csqrt%5B%5D%7Bn%7D%7D%3D%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7Bn%2B1%7D%2B%5Csqrt%5B%5D%7Bn%7D%7D%5Crightarrow0%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="a_n=\sqrt[]{n+1}-\sqrt[]{n}=\frac{\left(\sqrt[]{n+1}-\sqrt[]{n}\right)\left(\sqrt[]{n+1}+\sqrt[]{n}\right)}{\sqrt[]{n+1}+\sqrt[]{n}}=\frac{\left(n+1\right)-n}{\sqrt[]{n+1}+\sqrt[]{n}}=\frac{1}{\sqrt[]{n+1}+\sqrt[]{n}}\rightarrow0{,}\ kun\ n\rightarrow\infty"/></div> <span>b)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7B2n-1%7D%7B%5Csqrt%5B%5D%7Bn%5E2%2B4%7D%7D%3D%5Cfrac%7Bn%5Cleft(2-%5Cfrac%7B1%7D%7Bn%7D%5Cright)%7D%7Bn%5Csqrt%5B%5D%7B1%2B%5Cfrac%7B4%7D%7Bn%5E2%7D%7D%7D%3D%5Cfrac%7B2-%5Cfrac%7B1%7D%7Bn%7D%7D%7B%5Csqrt%5B%5D%7B1%2B%5Cfrac%7B4%7D%7Bn%5E2%7D%7D%7D%3D%5Cfrac%7B2-0%7D%7B%5Csqrt%5B%5D%7B1%2B0%7D%7D%5Crightarrow2%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="a_n=\frac{2n-1}{\sqrt[]{n^2+4}}=\frac{n\left(2-\frac{1}{n}\right)}{n\sqrt[]{1+\frac{4}{n^2}}}=\frac{2-\frac{1}{n}}{\sqrt[]{1+\frac{4}{n^2}}}=\frac{2-0}{\sqrt[]{1+0}}\rightarrow2{,}\ kun\ n\rightarrow\infty"/><br/> c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3Dn-%5Csqrt%5B%5D%7Bn%5E2%2B3n%2B1%7D%3D%5Cfrac%7B%5Cleft(n%2B%5Csqrt%5B%5D%7Bn%5E2%2B3n%2B1%7D%5Cright)%5Cleft(n-%5Csqrt%5B%5D%7Bn%5E2%2B3n%2B1%7D%5Cright)%7D%7Bn%2B%5Csqrt%5B%5D%7Bn%5E2%2B3n%2B1%7D%7D%3D%5Cfrac%7Bn%5E2-%5Cleft(n%5E2%2B3n%2B1%5Cright)%7D%7Bn%2B%5Csqrt%5B%5D%7Bn%5E2%2B3n%2B1%7D%7D%3D%5Cfrac%7B-3n-1%7D%7Bn%2Bn%5Csqrt%5B%5D%7B1%2B%5Cfrac%7B3%7D%7Bn%7D%2B%5Cfrac%7B1%7D%7Bn%5E2%7D%7D%7D%3D%5Cfrac%7B-3-%5Cfrac%7B1%7D%7Bn%7D%7D%7B1%2B%5Csqrt%5B%5D%7B1%2B%5Cfrac%7B3%7D%7Bn%7D%2B%5Cfrac%7B1%7D%7Bn%5E2%7D%7D%7D%3D%5Cfrac%7B-3-0%7D%7B1%2B%5Csqrt%5B%5D%7B1%2B0%2B0%7D%7D%5Crightarrow-%5Cfrac%7B3%7D%7B2%7D%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="a_n=n-\sqrt[]{n^2+3n+1}=\frac{\left(n+\sqrt[]{n^2+3n+1}\right)\left(n-\sqrt[]{n^2+3n+1}\right)}{n+\sqrt[]{n^2+3n+1}}=\frac{n^2-\left(n^2+3n+1\right)}{n+\sqrt[]{n^2+3n+1}}=\frac{-3n-1}{n+n\sqrt[]{1+\frac{3}{n}+\frac{1}{n^2}}}=\frac{-3-\frac{1}{n}}{1+\sqrt[]{1+\frac{3}{n}+\frac{1}{n^2}}}=\frac{-3-0}{1+\sqrt[]{1+0+0}}\rightarrow-\frac{3}{2}{,}\ kun\ n\rightarrow\infty"/><br/> <div> d)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7Be%5En-1%7D%7Be%5En%7D%3D1-%5Cfrac%7B1%7D%7Be%5En%7D%3D1-0%5Crightarrow1%7B%2C%7D%5C%20kun%5C%20n%5Crightarrow%5Cinfty" alt="a_n=\frac{e^n-1}{e^n}=1-\frac{1}{e^n}=1-0\rightarrow1{,}\ kun\ n\rightarrow\infty"/></div> </div> </div> 2019-11-20T09:14:47+02:00 https://peda.net/id/edfa56f4071 2019-11-19T21:30:30+02:00 <div> <div> <div> <div>150</div> </div> </div> </div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B3%7D%7B2x%2B1%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7Bx%5Cleft(%5Cfrac%7B3%7D%7Bx%7D%5Cright)%7D%7Bx%5Cleft(2%2B%5Cfrac%7B1%7D%7Bx%7D%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cleft(%5Cfrac%7B3%7D%7B%5Cfrac%7Bx%7D%7B2%2B%5Cfrac%7B1%7D%7Bx%7D%7D%7D%5Cright)%3D%5Cfrac%7B0%7D%7B2%2B0%7D%3D%3D%5Cfrac%7B0%7D%7B2%7D%3D0" alt="\lim_{x\rightarrow\infty}\frac{3}{2x+1}=\lim_{x\rightarrow\infty}\frac{x\left(\frac{3}{x}\right)}{x\left(2+\frac{1}{x}\right)}=\lim_{x\rightarrow\infty}\left(\frac{3}{\frac{x}{2+\frac{1}{x}}}\right)=\frac{0}{2+0}==\frac{0}{2}=0"/></div> <span>b)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-%5Cinfty%7D%5Cfrac%7Bx%7D%7B1-2x%7D%3D%5Clim_%7Bx%5Crightarrow-%5Cinfty%7D%5Cfrac%7Bx%5Ccdot1%7D%7Bx%5Cleft(%5Cfrac%7B1%7D%7Bx%7D-2%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow-%5Cinfty%7D%5Cfrac%7B1%7D%7B%5Cfrac%7B1%7D%7Bx%7D-2%7D%3D%5Cfrac%7B1%7D%7B0-2%7D%3D-%5Cfrac%7B1%7D%7B2%7D" alt="\lim_{x\rightarrow-\infty}\frac{x}{1-2x}=\lim_{x\rightarrow-\infty}\frac{x\cdot1}{x\left(\frac{1}{x}-2\right)}=\lim_{x\rightarrow-\infty}\frac{1}{\frac{1}{x}-2}=\frac{1}{0-2}=-\frac{1}{2}"/></div> <div>c)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7Bx%5E2-5x%7D%7B2x%5E2%2Bx%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7Bx%5E2%5Cleft(1-%5Cfrac%7B5%7D%7Bx%7D%5Cright)%7D%7Bx%5E2%5Cleft(2%2B%5Cfrac%7B1%7D%7Bx%7D%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B1-5%5Ccdot%5Cfrac%7B1%7D%7Bx%7D%7D%7B2%2B%5Cfrac%7B1%7D%7Bx%7D%7D%3D%5Cfrac%7B1-0%7D%7B2%2B0%7D%3D%5Cfrac%7B1%7D%7B2%7D" alt="\lim_{x\rightarrow\infty}\frac{x^2-5x}{2x^2+x}=\lim_{x\rightarrow\infty}\frac{x^2\left(1-\frac{5}{x}\right)}{x^2\left(2+\frac{1}{x}\right)}=\lim_{x\rightarrow\infty}\frac{1-5\cdot\frac{1}{x}}{2+\frac{1}{x}}=\frac{1-0}{2+0}=\frac{1}{2}"/><br/> <div><br/> <div> <div>152<br/> <span>A–II, B–I, C–I, D–II, E–II, F–II  <br/> </span></div> <div>153<br/> <div><span>a)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B-3x%2B1%7D%7B2x%2B5%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7Bx%5Cleft(-3%2B%5Cfrac%7B1%7D%7Bx%7D%5Cright)%7D%7Bx%5Cleft(2%2B%5Cfrac%7B5%7D%7Bx%7D%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B-3%2B%5Cfrac%7B1%7D%7Bx%7D%7D%7B2%2B5%5Ccdot%5Cfrac%7B1%7D%7Bx%7D%7D%3D%5Cfrac%7B-3%2B0%7D%7B2%2B0%7D%3D-%5Cfrac%7B3%7D%7B2%7D" alt="\lim_{x\rightarrow\infty}\frac{-3x+1}{2x+5}=\lim_{x\rightarrow\infty}\frac{x\left(-3+\frac{1}{x}\right)}{x\left(2+\frac{5}{x}\right)}=\lim_{x\rightarrow\infty}\frac{-3+\frac{1}{x}}{2+5\cdot\frac{1}{x}}=\frac{-3+0}{2+0}=-\frac{3}{2}"/></div> <span>b)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7Bx%5E3-3x%5E2-1%7D%7Bx%5E3%2B1%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7Bx%5E3%5Cleft(1-%5Cfrac%7B3%7D%7Bx%7D-%5Cfrac%7B1%7D%7Bx%5E3%7D%5Cright)%7D%7Bx%5E3%5Cleft(1%2B%5Cfrac%7B1%7D%7Bx%5E3%7D%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B1-%5Cfrac%7B3%7D%7Bx%7D-%5Cfrac%7B1%7D%7Bx%5E3%7D%7D%7B1%2B%5Cfrac%7B1%7D%7Bx%5E3%7D%7D%3D%5Cfrac%7B1-0-0%7D%7B1%2B0%7D%3D%5Cfrac%7B1%7D%7B1%7D%3D1" alt="\lim_{x\rightarrow\infty}\frac{x^3-3x^2-1}{x^3+1}=\lim_{x\rightarrow\infty}\frac{x^3\left(1-\frac{3}{x}-\frac{1}{x^3}\right)}{x^3\left(1+\frac{1}{x^3}\right)}=\lim_{x\rightarrow\infty}\frac{1-\frac{3}{x}-\frac{1}{x^3}}{1+\frac{1}{x^3}}=\frac{1-0-0}{1+0}=\frac{1}{1}=1"/></div> <span>c)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B-x%5E2%2B5x%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7Bx%5E2%5Cleft(-1%2B%5Cfrac%7B5%7D%7Bx%7D%5Cright)%7D%7Bx%5E2%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%5Cright)%7D%3D%5Cfrac%7B-1%7D%7B0%7D%3D%5Cinfty" alt="\lim_{x\rightarrow\infty}\frac{-x^2+5x}{x+1}=\lim_{x\rightarrow\infty}\frac{x^2\left(-1+\frac{5}{x}\right)}{x^2\left(\frac{1}{x}+\frac{1}{x^2}\right)}=\frac{-1}{0}=\infty"/></div> </div> </div> <div>154<br/> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cleft(x%5E2-x%5E3%5Cright)%3Dx%5E3%5Cleft(%5Cfrac%7B1%7D%7Bx%7D-1%5Cright)%3D%5Cinfty%5Cleft(0-1%5Cright)%3D%5Cinfty" alt="\lim_{x\rightarrow\infty}\left(x^2-x^3\right)=x^3\left(\frac{1}{x}-1\right)=\infty\left(0-1\right)=\infty"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow2%7D%5Cfrac%7Bx%5E2-4%7D%7Bx-2%7D%3D%5Cfrac%7B%5Cleft(x-2%5Cright)%5Cleft(x%2B2%5Cright)%7D%7B%5Cleft(x-2%5Cright)%7D%3Dx%2B2%3D2%2B2%3D4" alt="\lim_{x\rightarrow2}\frac{x^2-4}{x-2}=\frac{\left(x-2\right)\left(x+2\right)}{\left(x-2\right)}=x+2=2+2=4"/></div> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow2%7D%5Cfrac%7Bx%5E2%2B4%7D%7Bx-2%7D%3D%5Cfrac%7B2%5E2%2B4%7D%7B2-2%7D%3D%5Cfrac%7B8%7D%7B0%7D" alt="\lim_{x\rightarrow2}\frac{x^2+4}{x-2}=\frac{2^2+4}{2-2}=\frac{8}{0}"/><span>eli raja-arvoa ei eole olemassa. Tutkitaan onko funktiola epäoleellinen raja.arvo laskemalla toispuliset raja-arvot.</span><br/> <div>Lasketaan funtkion <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cfrac%7Bx%5E2%2B4%7D%7Bx-2%7D" alt="f\left(x\right)=\frac{x^2+4}{x-2}"/>arvoja, kun x→2- ja x→2+</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow2-%7D%5Cfrac%7Bx%5E2%2B4%7D%7Bx-2%7D%3D-%5Cinfty" alt="\lim_{x\rightarrow2-}\frac{x^2+4}{x-2}=-\infty"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow2%2B%7D%5Cfrac%7Bx%5E2%2B4%7D%7Bx-2%7D%3D%5Cinfty" alt="\lim_{x\rightarrow2+}\frac{x^2+4}{x-2}=\infty"/></div> <div>Toispuoleiset raja-arvot ovat eri suuret, joten funktiolla ei ole epäoleellista raja-arvoa kohdassa x=2</div> <br/> 155 <div>a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B1%7D%7Be%5Ex%7D%3D0" alt="\lim_{x\rightarrow\infty}\frac{1}{e^x}=0"/></div> <div>b) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-%5Cinfty%7D%5Cfrac%7B1%7D%7Be%5E%7B-x%7D%7D%3D%5Clim_%7Bx%5Crightarrow-%5Cinfty%7D%5Cfrac%7B1%7D%7B%5Cfrac%7B1%7D%7Be%5Ex%7D%7D%3D%5Clim_%7Bx%5Crightarrow-%5Cinfty%7D1%5Ccdot%5Cfrac%7Be%5Ex%7D%7B1%7D%3De%5Ex%3D0" alt="\lim_{x\rightarrow-\infty}\frac{1}{e^{-x}}=\lim_{x\rightarrow-\infty}\frac{1}{\frac{1}{e^x}}=\lim_{x\rightarrow-\infty}1\cdot\frac{e^x}{1}=e^x=0"/></div> <div><span>c)</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-%5Cinfty%7D%5Cfrac%7Be%5Ex%2B1%7D%7Be%5Ex%7D%3D%5Cfrac%7Be%5Ex%7D%7Be%5Ex%7D%2B%5Cfrac%7B1%7D%7Be%5Ex%7D%3D1%2B%5Cleft(%5Cfrac%7B1%7D%7Be%7D%5Cright)%5Ex%3D%5Cinfty" alt="\lim_{x\rightarrow-\infty}\frac{e^x+1}{e^x}=\frac{e^x}{e^x}+\frac{1}{e^x}=1+\left(\frac{1}{e}\right)^x=\infty"/><br/> <br/> <span>159</span> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7DR%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B9x%5E2-1%7D%7B3x%5E2-5x-2%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B9-%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%7B3-%5Cfrac%7B5%7D%7Bx%7D-%5Cfrac%7B2%7D%7Bx%5E2%7D%7D%3D%5Cfrac%7B9-0%7D%7B3-0-0%7D%3D3" alt="\lim_{x\rightarrow\infty}R\left(x\right)=\lim_{x\rightarrow\infty}\frac{9x^2-1}{3x^2-5x-2}=\lim_{x\rightarrow\infty}\frac{9-\frac{1}{x^2}}{3-\frac{5}{x}-\frac{2}{x^2}}=\frac{9-0}{3-0-0}=3"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-%5Cfrac%7B1%7D%7B3%7D%7DR%5Cleft(x%5Cright)%3D%5Cfrac%7B%5Cleft(3x-1%5Cright)%5Cleft(3x%2B1%5Cright)%7D%7B%5Cleft(3x%2B1%5Cright)%5Cleft(x-2%5Cright)%7D%3D%5Cfrac%7B3x-1%7D%7Bx-2%7D%3D%5Cfrac%7B3%5Ccdot%5Cleft(-%5Cfrac%7B1%7D%7B3%7D%5Cright)-1%7D%7B-%5Cfrac%7B1%7D%7B3%7D-2%7D%3D%5Cfrac%7B-1-1%7D%7B-%5Cfrac%7B7%7D%7B3%7D%7D%3D%5Cfrac%7B6%7D%7B7%7D" alt="\lim_{x\rightarrow-\frac{1}{3}}R\left(x\right)=\frac{\left(3x-1\right)\left(3x+1\right)}{\left(3x+1\right)\left(x-2\right)}=\frac{3x-1}{x-2}=\frac{3\cdot\left(-\frac{1}{3}\right)-1}{-\frac{1}{3}-2}=\frac{-1-1}{-\frac{7}{3}}=\frac{6}{7}"/><br/> <br/> </div> </div> <div> <div> <div>160</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Cfrac%7Be%5Ex%5Cleft(1%2Be%5Ex%5Cright)-e%5Ex%5Ccdot%20e%5Ex%7D%7B%5Cleft(1%2Be%5Ex%5Cright)%5E2%7D%3D%5Cfrac%7Be%5Ex%2Be%5E%7B2x%7D-e%5E%7B2x%7D%7D%7B%5Cleft(1%2Be%5Ex%5Cright)%5E2%7D%3D%5Cfrac%7Be%5Ex%7D%7B%5Cleft(1%2Be%5Ex%5Cright)%5E2%7D" alt="f'\left(x\right)=\frac{e^x\left(1+e^x\right)-e^x\cdot e^x}{\left(1+e^x\right)^2}=\frac{e^x+e^{2x}-e^{2x}}{\left(1+e^x\right)^2}=\frac{e^x}{\left(1+e^x\right)^2}"/></div> <div>funktion f derivaatta on kaikkialla positiviinen, joten f on aidosti kasvava</div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7Be%5Ex%7D%7B1%2Be%5Ex%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B1%7D%7B%5Cfrac%7B1%7D%7Be%5Ex%7D%2B1%7D%3D%5Cfrac%7B1%7D%7B0%2B1%7D%3D1" alt="\lim_{x\rightarrow\infty}\frac{e^x}{1+e^x}=\lim_{x\rightarrow\infty}\frac{1}{\frac{1}{e^x}+1}=\frac{1}{0+1}=1"/></div> <span>c)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(10%5Cright)%3D%5Cfrac%7Be%5E%7B10%7D%7D%7B1%2Be%5E%7B10%7D%7D%3D0%7B%2C%7D999954...%3E0%7B%2C%7D99" alt="f\left(10\right)=\frac{e^{10}}{1+e^{10}}=0{,}999954...&gt;0{,}99"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <div>Koska funktio f on aidosti kasvava, kaikille x ≥ 10, f(x) ≥ f(10). Väite pätee.<br/> <br/> </div> </div> <div> <div>161</div> <div>a) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(t%5Cright)%3D%5Cfrac%7B2e%5E%7B0%7B%2C%7D05t%7D%7D%7B%5Cleft(e%5E%7B0%7B%2C%7D05t%7D%2B4%5Cright)%5E2%7D" alt="f'\left(t\right)=\frac{2e^{0{,}05t}}{\left(e^{0{,}05t}+4\right)^2}"/></div> <span>f''(t) on kaikialla t:n arvoilla positiivinen, joten fuynktio f kasvaa loputtomasti.</span> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bt%5Crightarrow%5Cinfty%7Df%5Cleft(t%5Cright)%3D%5Clim_%7Bt%5Crightarrow%5Cinfty%7D%5Cfrac%7B10e%5E%7B0%7B%2C%7D05t%7D%7D%7Be%5E%7B0%7B%2C%7D05t%7D%2B4%7D%3D%5Clim_%7Bt%5Crightarrow%5Cinfty%7D%5Cfrac%7B10%7D%7B1%2B%5Cfrac%7B4%7D%7Be%5E%7B0%7B%2C%7D05t%7D%7D%7D%3D%5Cfrac%7B10%7D%7B1%2B0%7D%3D10" alt="\lim_{t\rightarrow\infty}f\left(t\right)=\lim_{t\rightarrow\infty}\frac{10e^{0{,}05t}}{e^{0{,}05t}+4}=\lim_{t\rightarrow\infty}\frac{10}{1+\frac{4}{e^{0{,}05t}}}=\frac{10}{1+0}=10"/></div> <div>Yläraja on 10 000 yksilöä.<br/> <br/> </div> </div> <div>163<br/> a) <div>Takastellaan funktion jatkuvuutta kohdassa x=1. Jotta funktio olisi jatkuva, <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-1%5Cright)%3D%5Clim_%7Bx%5Crightarrow-1%7Df%5Cleft(x%5Cright)" alt="f\left(-1\right)=\lim_{x\rightarrow-1}f\left(x\right)"/><br/> <br/> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-1%5Cright)%3Da%5Ccdot%5Cleft(-1%5Cright)%5E2%3Da" alt="f\left(-1\right)=a\cdot\left(-1\right)^2=a"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-1-%7Dax%5E2%3Da%5Ccdot%5Cleft(-1%5Cright)%5E2%3Da" alt="\lim_{x\rightarrow-1-}ax^2=a\cdot\left(-1\right)^2=a"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7Bx%5E2%7D%7B1%2Bx%5E2%7D%3D%5Cfrac%7B%5Cleft(-1%5Cright)%5E2%7D%7B1%2B%5Cleft(-1%5Cright)%5E2%7D%3D%5Cfrac%7B1%7D%7B2%7D" alt="\lim_{x\rightarrow-1+}\frac{x^2}{1+x^2}=\frac{\left(-1\right)^2}{1+\left(-1\right)^2}=\frac{1}{2}"/></div> <div>Jotta funktio olisi jatkuva, a on oltava 1/2.</div> <div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=kun%5C%20a%3D%5Cfrac%7B1%7D%7B2%7D%7B%2C%7D%5C%20f%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0A%5Cfrac%7B1%7D%7B2%7Dx%5E2%7B%2C%7D%26x%5Cle-1%5C%5C%0A%5Cfrac%7Bx%5E2%7D%7B1%2Bx%5E2%7D%7B%2C%7D%26x%3E-1%0A%5Cend%7Bcases%7D" alt="kun\ a=\frac{1}{2}{,}\ f\left(x\right)=\begin{cases}&#10;\frac{1}{2}x^2{,}&amp;x\le-1\\&#10;\frac{x^2}{1+x^2}{,}&amp;x&gt;-1&#10;\end{cases}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <div>Tarkastellaan funktion dervoituvuutta kohdassa x = -1</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7Bf%5Cleft(x%5Cright)-f%5Cleft(-1%5Cright)%7D%7Bx-%5Cleft(-1%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7B%5Cfrac%7B1%7D%7B2%7Dx%5E2-%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cleft(-1%5Cright)%5E2%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7B%5Cfrac%7B1%7D%7B2%7Dx%5E2-%5Cfrac%7B1%7D%7B2%7D%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7B%5Cfrac%7B1%7D%7B2%7D%5Cleft(x%2B1%5Cright)%5Cleft(x-1%5Cright)%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7B1%7D%7B2%7D%5Cleft(x-1%5Cright)%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft(-1-1%5Cright)%3D-1" alt="\lim_{x\rightarrow-1-}\frac{f\left(x\right)-f\left(-1\right)}{x-\left(-1\right)}=\lim_{x\rightarrow-1-}\frac{\frac{1}{2}x^2-\frac{1}{2}\cdot\left(-1\right)^2}{x+1}=\lim_{x\rightarrow-1-}\frac{\frac{1}{2}x^2-\frac{1}{2}}{x+1}=\lim_{x\rightarrow-1-}\frac{\frac{1}{2}\left(x+1\right)\left(x-1\right)}{x+1}=\lim_{x\rightarrow-1-}\frac{1}{2}\left(x-1\right)=\frac{1}{2}\left(-1-1\right)=-1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7Bf%5Cleft(x%5Cright)-f%5Cleft(-1%5Cright)%7D%7Bx-%5Cleft(-1%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7B%5Cfrac%7Bx%5E2%7D%7B1%2Bx%5E2%7D-%5Cfrac%7B%5Cleft(-1%5Cright)%5E2%7D%7B1%2B%5Cleft(-1%5Cright)%5E2%7D%7D%7Bx-%5Cleft(-1%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7B%5Cfrac%7Bx%5E2%7D%7B1%2Bx%5E2%7D-%5Cfrac%7B1%7D%7B2%7D%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7B%5Cfrac%7B2x%5E2%7D%7B2%5Cleft(1%2Bx%5E2%5Cright)%7D-%5Cfrac%7B1%2Bx%5E2%7D%7B2%5Cleft(1%2Bx%5E2%5Cright)%7D%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7Bx%5E2-1%7D%7B2%5Cleft(1%2Bx%5E2%5Cright)%5Cleft(x%2B1%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7B%5Cleft(x%2B1%5Cright)%5Cleft(x-1%5Cright)%7D%7B2%5Cleft(1%2Bx%5E2%5Cright)%5Cleft(x%2B1%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7B%5Cleft(x-1%5Cright)%7D%7B2%5Cleft(1%2Bx%5E2%5Cright)%7D%3D%5Cfrac%7B-1-1%7D%7B2%5Cleft(1%2B%5Cleft(-1%5Cright)%5E2%5Cright)%7D%3D%5Cfrac%7B-2%7D%7B4%7D%3D-%5Cfrac%7B1%7D%7B2%7D" alt="\lim_{x\rightarrow-1+}\frac{f\left(x\right)-f\left(-1\right)}{x-\left(-1\right)}=\lim_{x\rightarrow-1+}\frac{\frac{x^2}{1+x^2}-\frac{\left(-1\right)^2}{1+\left(-1\right)^2}}{x-\left(-1\right)}=\lim_{x\rightarrow-1+}\frac{\frac{x^2}{1+x^2}-\frac{1}{2}}{x+1}=\lim_{x\rightarrow-1+}\frac{\frac{2x^2}{2\left(1+x^2\right)}-\frac{1+x^2}{2\left(1+x^2\right)}}{x+1}=\lim_{x\rightarrow-1+}\frac{x^2-1}{2\left(1+x^2\right)\left(x+1\right)}=\lim_{x\rightarrow-1+}\frac{\left(x+1\right)\left(x-1\right)}{2\left(1+x^2\right)\left(x+1\right)}=\lim_{x\rightarrow-1+}\frac{\left(x-1\right)}{2\left(1+x^2\right)}=\frac{-1-1}{2\left(1+\left(-1\right)^2\right)}=\frac{-2}{4}=-\frac{1}{2}"/><br/> <br/> <div> <div>Funktio ei ole derivoituva kohdassa x=-a, koska sen erotusosamäärän toispuoleiset raja-arvot eivät ole samat. Funtkio f ei ole derfoituva kaikkialla.<br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7Bx%5E2%7D%7B1%2Bx%5E2%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7Bx%5E2%5Ccdot1%7D%7Bx%5E2%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7Bx%5E2%7D%2B1%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cfrac%7B1%7D%7B%5Cfrac%7B1%7D%7Bx%5E2%7D%2B1%7D%3D%5Cfrac%7B1%7D%7B0%2B1%7D%3D1" alt="\lim_{x\rightarrow\infty}\frac{x^2}{1+x^2}=\lim_{x\rightarrow\infty}\frac{x^2\cdot1}{x^2\cdot\left(\frac{1}{x^2}+1\right)}=\lim_{x\rightarrow\infty}\frac{1}{\frac{1}{x^2}+1}=\frac{1}{0+1}=1"/></div> </div> </div> </div> <div><br/> 164<br/> <div>Tarkastellaan funktion f kulkua derivaatan avulla.  </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Cfrac%7B%5Cleft(6x%2B1%5Cright)%5Cleft(x%5E2%2B1%5Cright)-%5Cleft(3x%5E2%2Bx%2B3%5Cright)%5Ccdot2x%7D%7B%5Cleft(x%5E2%2B1%5Cright)%5E2%7D%3D%5Cfrac%7B-x%5E2%2B1%7D%7B%5Cleft(x%5E2%2B1%5Cright)%5E2%7D" alt="f'\left(x\right)=\frac{\left(6x+1\right)\left(x^2+1\right)-\left(3x^2+x+3\right)\cdot2x}{\left(x^2+1\right)^2}=\frac{-x^2+1}{\left(x^2+1\right)^2}"/></div> <div>Derivaatan lausekkeen nimittäjä on positiivinen kaikilla x:n arvoilla, joten derivaatan merkkiin vaikuttaa vain osoittajan merkki.  </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-x%5E2%2B1%3D0" alt="-x^2+1=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%5C%20tai%5C%20x%3D1" alt="x=-1\ tai\ x=1"/><br/> <div>Funktio f on kasvava välillä x ≤ −1, vähenevä välillä −1 ≤ x ≤ 1 ja kasvava välillä x ≥ 1.  </div> <div>Funktiolla on paikallinen minimiarvo  </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-1%5Cright)%3D2%5Cfrac%7B1%7D%7B2%7D" alt="f\left(-1\right)=2\frac{1}{2}"/>ja paikallinen maksimiarvo <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)%3D3%5Cfrac%7B1%7D%7B2%7D" alt="f\left(1\right)=3\frac{1}{2}"/></div> <div>Funktiolla f on nollakohta osoittajan nollakohdissa.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E2%2Bx%2B3%3D0" alt="3x^2+x+3=0"/></div> <div>ei ratkaisuja</div> <div>Funktiolal ei ole nollakohtia</div> <div>Tarkastellaan vielä, mitä arvoja funktio saa, kun muuttuja x kasvaa tai pienenee rajatta.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cpm%5Cinfty%7Df%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow%5Cpm%5Cinfty%7D%5Cfrac%7B3x%5E2%2Bx%2B3%7D%7Bx%5E2%2B1%7D%3D%5Clim_%7Bx%5Crightarrow%5Cpm%5Cinfty%7D%5Cfrac%7Bx%5E2%5Cleft(3%2B%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B3%7D%7Bx%5E2%7D%5Cright)%7D%7Bx%5E2%5Cleft(1%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow%5Cpm%5Cinfty%7D%5Cfrac%7B3%2B%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B3%7D%7Bx%5E2%7D%7D%7B1%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%3D%5Cfrac%7B3%2B0%2B0%7D%7B1%2B0%7D%3D3" alt="\lim_{x\rightarrow\pm\infty}f\left(x\right)=\lim_{x\rightarrow\pm\infty}\frac{3x^2+x+3}{x^2+1}=\lim_{x\rightarrow\pm\infty}\frac{x^2\left(3+\frac{1}{x}+\frac{3}{x^2}\right)}{x^2\left(1+\frac{1}{x^2}\right)}=\lim_{x\rightarrow\pm\infty}\frac{3+\frac{1}{x}+\frac{3}{x^2}}{1+\frac{1}{x^2}}=\frac{3+0+0}{1+0}=3"/></div> <span>Funktion pienin arvo on </span><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B5%7D%7B2%7D%3D2%5Cfrac%7B1%7D%7B2%7D" alt="\frac{5}{2}=2\frac{1}{2}"/><span>ja suurin </span><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B7%7D%7B2%7D%3D3%5Cfrac%7B1%7D%7B2%7D" alt="\frac{7}{2}=3\frac{1}{2}"/><span>.</span><br/> <br/> </div> <div>167<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cleft(%5Cln%5Cleft(4x%2B3%5Cright)-%5Cln%5Cleft(3x%2B4%5Cright)%5Cright)%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cleft(%5Cln%5Cfrac%7B4x%2B3%7D%7B3x%2B4%7D%5Cright)%3D%5Clim_%7Bx%5Crightarrow%5Cinfty%7D%5Cleft(%5Cln%5Cfrac%7B4%2B%5Cfrac%7B3%7D%7Bx%7D%7D%7B3%2B%5Cfrac%7B4%7D%7Bx%7D%7D%5Cright)%3D%5Cln%5Cfrac%7B4%2B0%7D%7B3%2B0%7D%3D%5Cln%5Cfrac%7B4%7D%7B3%7D" alt="\lim_{x\rightarrow\infty}\left(\ln\left(4x+3\right)-\ln\left(3x+4\right)\right)=\lim_{x\rightarrow\infty}\left(\ln\frac{4x+3}{3x+4}\right)=\lim_{x\rightarrow\infty}\left(\ln\frac{4+\frac{3}{x}}{3+\frac{4}{x}}\right)=\ln\frac{4+0}{3+0}=\ln\frac{4}{3}"/></div> </div> </div> </div> </div> </div> 2019-11-14T21:35:35+02:00 https://peda.net/id/de72be5a05e 2019-11-13T22:21:08+02:00 123 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0A2x%2B3%7B%2C%7D%26kun%5C%20x%3C-1%5C%5C%0Ax%2B1%7B%2C%7D%26kun%5C%20x%5C%20%5Cge1%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\begin{cases}&#10;2x+3{,}&amp;kun\ x&lt;-1\\&#10;x+1{,}&amp;kun\ x\ \ge1&#10;\end{cases}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7Bf%5Cleft(x%5Cright)-f%5Cleft(-1%5Cright)%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7B2x%2B3-1%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7B2x%2B2%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7B2%5Cleft(x%2B1%5Cright)%7D%7Bx%2B1%7D%3D2" alt="f'\left(x\right)=\lim_{x\rightarrow-1-}\lim_{x\rightarrow-1-}\frac{f\left(x\right)-f\left(-1\right)}{x+1}=\lim_{x\rightarrow-1-}\frac{2x+3-1}{x+1}=\lim_{x\rightarrow-1-}\frac{2x+2}{x+1}=\lim_{x\rightarrow-1-}\frac{2\left(x+1\right)}{x+1}=2"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7Bf%5Cleft(x%5Cright)-f%5Cleft(-1%5Cright)%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7Bx%2B1%2B0%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7Bx%2B1%7D%7Bx%2B1%7D%3D1" alt="f'\left(x\right)=\lim_{x\rightarrow-1+}\frac{f\left(x\right)-f\left(-1\right)}{x+1}=\lim_{x\rightarrow-1+}\frac{x+1+0}{x+1}=\lim_{x\rightarrow-1+}\frac{x+1}{x+1}=1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cne1" alt="2\ne1"/>, funktio ei ole derivoituva.</div> <div>Määritetään funktion jatkuvuutta</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-1-%7D2x%2B3%3D2%5Ccdot-1%2B3%3D-2%2B3%3D1" alt="\lim_{x\rightarrow-1-}2x+3=2\cdot-1+3=-2+3=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-1%2B%7Dx%2B1%3D-1%2B1%3D0" alt="\lim_{x\rightarrow-1+}x+1=-1+1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%5Cne0" alt="1\ne0"/>, funktio ei ole jatkuva. <div><br/> <div>b)</div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0Ax%5E2%2B1%7B%2C%7D%26kun%5C%20x%3C1%5C%5C%0A-2x%7B%2C%7D%26kun%5C%20x%5Cge-1%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\begin{cases}&#10;x^2+1{,}&amp;kun\ x&lt;1\\&#10;-2x{,}&amp;kun\ x\ge-1&#10;\end{cases}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7Bf%5Cleft(x%5Cright)-f%5Cleft(-1%5Cright)%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7Bx%5E2%2B1-2%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7Bx%5E2-1%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1-%7D%3D%5Cfrac%7B%5Cleft(x%2B1%5Cright)%5Cleft(x-1%5Cright)%7D%7B%5Cleft(x%2B1%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow-1-%7Dx-1%3D-2" alt="f'\left(x\right)=\lim_{x\rightarrow-1-}\frac{f\left(x\right)-f\left(-1\right)}{x+1}=\lim_{x\rightarrow-1-}\frac{x^2+1-2}{x+1}=\lim_{x\rightarrow-1-}\frac{x^2-1}{x+1}=\lim_{x\rightarrow-1-}=\frac{\left(x+1\right)\left(x-1\right)}{\left(x+1\right)}=\lim_{x\rightarrow-1-}x-1=-2"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7Bf%5Cleft(x%5Cright)-f%5Cleft(-1%5Cright)%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7B-2x-2%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7B-2%5Cleft(x%2B1%5Cright)%7D%7Bx%2B1%7D%3D-2" alt="f'\left(x\right)=\lim_{x\rightarrow-1+}\frac{f\left(x\right)-f\left(-1\right)}{x+1}=\lim_{x\rightarrow-1+}\frac{-2x-2}{x+1}=\lim_{x\rightarrow-1+}\frac{-2\left(x+1\right)}{x+1}=-2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=-2%3D-2" alt="-2=-2"/>, funktio on derivoituva, joten se on myös jatkuva.<br/> <br/> <div> <div>124</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0Ax%5E2-1%7B%2C%7D%26kun%5C%20x%3C2%5C%5C%0Ax%2B1%7B%2C%7D%26kun%5C%20x%5Cge2%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\begin{cases}&#10;x^2-1{,}&amp;kun\ x&lt;2\\&#10;x+1{,}&amp;kun\ x\ge2&#10;\end{cases}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow2-%7D%5Cfrac%7Bx%5E2-1-3%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2-%7D%5Cfrac%7Bx%5E2-4%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2-%7D%5Cfrac%7B%5Cleft(x-2%5Cright)%5Cleft(x%2B2%5Cright)%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2-%7Dx%2B2%3D4" alt="f'\left(x\right)=\lim_{x\rightarrow2-}\frac{x^2-1-3}{x-2}=\lim_{x\rightarrow2-}\frac{x^2-4}{x-2}=\lim_{x\rightarrow2-}\frac{\left(x-2\right)\left(x+2\right)}{x-2}=\lim_{x\rightarrow2-}x+2=4"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow2%2B%7D%5Cfrac%7Bx%2B1-3%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2%2B%7D%5Cfrac%7Bx-2%7D%7Bx-2%7D%3D1" alt="f'\left(x\right)=\lim_{x\rightarrow2+}\frac{x+1-3}{x-2}=\lim_{x\rightarrow2+}\frac{x-2}{x-2}=1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4%5Cne1" alt="4\ne1"/>, funktio ei ole derivoituva</div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0Ax%5E2-1%7B%2C%7D%26kun%5C%20x%3C2%5C%5C%0A4x-5%7B%2C%7D%26kun%5C%20x%5Cge2%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\begin{cases}&#10;x^2-1{,}&amp;kun\ x&lt;2\\&#10;4x-5{,}&amp;kun\ x\ge2&#10;\end{cases}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow2-%7D%5Cfrac%7Bx%5E2-1-%5Cleft(4-1%5Cright)%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2-%7D%5Cfrac%7Bx%5E2-1-3%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2-%7D%5Cfrac%7Bx%5E2-4%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2-%7D%5Cfrac%7B%5Cleft(x-2%5Cright)%5Cleft(x%2B2%5Cright)%7D%7B%5Cleft(x-2%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow2-%7Dx%2B2%3D2%2B2%3D4" alt="f'\left(x\right)=\lim_{x\rightarrow2-}\frac{x^2-1-\left(4-1\right)}{x-2}=\lim_{x\rightarrow2-}\frac{x^2-1-3}{x-2}=\lim_{x\rightarrow2-}\frac{x^2-4}{x-2}=\lim_{x\rightarrow2-}\frac{\left(x-2\right)\left(x+2\right)}{\left(x-2\right)}=\lim_{x\rightarrow2-}x+2=2+2=4"/><br/> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow2%2B%7D%5Cfrac%7B4x-5-%5Cleft(8-5%5Cright)%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2%2B%7D%5Cfrac%7B4x-5-3%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2%2B%7D%5Cfrac%7B4x-8%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2%2B%7D%5Cfrac%7B4%5Cleft(x-2%5Cright)%7D%7Bx-2%7D%3D4" alt="f'\left(x\right)=\lim_{x\rightarrow2+}\frac{4x-5-\left(8-5\right)}{x-2}=\lim_{x\rightarrow2+}\frac{4x-5-3}{x-2}=\lim_{x\rightarrow2+}\frac{4x-8}{x-2}=\lim_{x\rightarrow2+}\frac{4\left(x-2\right)}{x-2}=4"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=4%3D4" alt="4=4"/>, funktio on derivoituva ja jatkuva</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(2%5Cright)%3D4" alt="f'\left(2\right)=4"/><br/> <br/> </div> </div> <div> <div>127</div> <div>a)</div> <div>Kirjoitetaan funktio f paloittain määriteltynä funktiona.  </div> <div>Määritetään funktion nollakohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1-x%5E2%3D0" alt="1-x^2=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%5C%20tai%5C%20x%3D1" alt="x=-1\ tai\ x=1"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1-x%5E2%3D-x%5E2%2B1" alt="1-x^2=-x^2+1"/></div> <div>Koska vakio x:n kerroin on negatiivinen, funktion kuvaaja on alaspäin aukeava, ja näin ollen</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26%26-1%26%261%26%5C%5C%0A%5Chline%0Af%5Cleft(x%5Cright)%26-%26%26%2B%26%26-%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;&amp;-1&amp;&amp;1&amp;\\&#10;\hline&#10;f\left(x\right)&amp;-&amp;&amp;+&amp;&amp;-&#10;\end{array}"/><br/> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cleft%7C1-x%5E2%5Cright%7C%3D%5Cbegin%7Bcases%7D%0A-%5Cleft(1-x%5E2%5Cright)%7B%2C%7D%26kun%5C%20x%3C-1%5C%5C%0A1-x%5E2%7B%2C%7D%26kun%5C%20-1%5Cle%20x%3C1%5C%5C%0A-%5Cleft(1-x%5E2%5Cright)%7B%2C%7D%26kun%5C%20x%5Cge1%0A%5Cend%7Bcases%7D%3D%5Cbegin%7Bcases%7D%0Ax%5E2-1%7B%2C%7D%26kun%5C%20x%3C-1%5C%5C%0A1-x%5E2%7B%2C%7D%26kun%5C%20-1%5Cle%20x%3C1%5C%5C%0Ax%5E2-1%7B%2C%7D%26kun%5C%20x%5Cge1%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\left|1-x^2\right|=\begin{cases}&#10;-\left(1-x^2\right){,}&amp;kun\ x&lt;-1\\&#10;1-x^2{,}&amp;kun\ -1\le x&lt;1\\&#10;-\left(1-x^2\right){,}&amp;kun\ x\ge1&#10;\end{cases}=\begin{cases}&#10;x^2-1{,}&amp;kun\ x&lt;-1\\&#10;1-x^2{,}&amp;kun\ -1\le x&lt;1\\&#10;x^2-1{,}&amp;kun\ x\ge1&#10;\end{cases}"/></div> Funktio f on polynomifunktiona jatkuva väleillä x &lt; −1, −1 &lt; x &lt; 1 ja x &gt; 1. Funktio f on jatkuva kohdissa x = −1 ja x = 1, jos <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1%7Df%5Cleft(x%5Cright)%3Df%5Cleft(-1%5Cright)" alt="\lim_{x\rightarrow1}f\left(x\right)=f\left(-1\right)"/>ja <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1%7Df%5Cleft(x%5Cright)%3Df%5Cleft(1%5Cright)" alt="\lim_{x\rightarrow1}f\left(x\right)=f\left(1\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1-%7Dx%5E2-1%3D1-1%3D0" alt="\lim_{x\rightarrow1-}x^2-1=1-1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1%2B%7D1-x%5E2%3D1-1%3D0" alt="\lim_{x\rightarrow1+}1-x^2=1-1=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-1%5Cright)%3D1-x%5E2%3D1-1%3D0" alt="f\left(-1\right)=1-x^2=1-1=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=0%3D0%3D0" alt="0=0=0"/></div> <div>Funktio f on jatkuva kohdassa x = −1.  </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1-%7D1-x%5E2%3D1-1%3D0" alt="\lim_{x\rightarrow1-}1-x^2=1-1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1%2B%7Dx%5E2-1%3D1-1%3D0" alt="\lim_{x\rightarrow1+}x^2-1=1-1=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)%3Dx%5E2-1%3D1-1%3D0" alt="f\left(1\right)=x^2-1=1-1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=0%3D0%3D0" alt="0=0=0"/></div> <div>Funktio f on jatkuva kohdassa x = 1.</div> <div>Funktio on kaikialla jatkuva.<br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7Bx%5E2-1-0%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7Bx%5E2-1%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1-%7D%5Cfrac%7B%5Cleft(x%2B1%5Cright)%5Cleft(x-1%5Cright)%7D%7B%5Cleft(x%2B1%5Cright)%7D%3Dx-1%3D-2" alt="f'\left(x\right)=\lim_{x\rightarrow-1-}\frac{x^2-1-0}{x+1}=\lim_{x\rightarrow-1-}\frac{x^2-1}{x+1}=\lim_{x\rightarrow-1-}\frac{\left(x+1\right)\left(x-1\right)}{\left(x+1\right)}=x-1=-2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7B1-x%5E2-%5Cleft(1-%5Cleft(-1%5Cright)%5E2%5Cright)%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7B1-x%5E2-0%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1%2B%7D%5Cfrac%7B%5Cleft(1-x%5Cright)%5Cleft(1%2Bx%5Cright)%7D%7B1%2Bx%7D%3D1-x%3D2" alt="f'\left(x\right)=\lim_{x\rightarrow-1+}\frac{1-x^2-\left(1-\left(-1\right)^2\right)}{x+1}=\lim_{x\rightarrow-1+}\frac{1-x^2-0}{x+1}=\lim_{x\rightarrow-1+}\frac{\left(1-x\right)\left(1+x\right)}{1+x}=1-x=2"/><br/> <span>Funktio f ei ole derivoituva kohdassa x=-1, joten se ei ole derivoituva kaikialla.<br/> <br/> </span> <div>128</div> <div>a) Funktio f on derivoituva kohdassa x = 2, koska erotusosamäärällä on raja-arvo kohdassa x = 2.  </div> <div>b) Funktio f on jatkuva kohdassa x = 2, koska se on derivoituva kohdassa x = 2.</div> <div>c)<img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow2%7Df%5Cleft(x%5Cright)%3D3" alt="\lim_{x\rightarrow2}f\left(x\right)=3"/>, koska funktio on jatkuva, sen raja-arvo kohdassa x = 2 on sama kuin sen arvo kohdassa 2.</div> <div><br/> 131</div> </div> </div> </div> <div><span>a) Epätosi. Jatkuva funktio ei välttämättä ole derivoituva. Esimerkiksi funktio f (x) = |x| ei ole derivoituva kohdassa x = 0, vaikka se on jatkuva tässä kohdassa.  </span> <div>b) Tosi. Jatkuvuus on derivoituvuuden välttämätön ehto, joten kaikki derivoituvat funktiot ovat jatkuvia.</div> <div>c) Tosi. Koska funktion erotusosamäärällä on raja-arvo kohdassa x = a, se on derivoituva ja siten myös jatkuva tässä kohdassa. Koska funktio f on jatkuva kohdassa x = a, sen raja-arvo ja arvo ovat yhtä suuret.<br/> <br/> </div> </div> <div> <div> <div>132</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0Ax%5E2-3x%7B%2C%7D%26kun%5C%20x%5Cle2%5C%5C%0Ax%2Ba%7B%2C%7D%26kun%5C%20x%3E2%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\begin{cases}&#10;x^2-3x{,}&amp;kun\ x\le2\\&#10;x+a{,}&amp;kun\ x&gt;2&#10;\end{cases}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <div>Lasketaa funktiolle raja-arvo</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow2-%7Dx%5E2-3x%3D2%5E2-3%5Ccdot2%3D4-6%3D-2" alt="\lim_{x\rightarrow2-}x^2-3x=2^2-3\cdot2=4-6=-2"/></div> <div>Koska funktion pitäisi olla jatkuva, joten</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%2Ba%3D-2" alt="2+a=-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D-4" alt="a=-4"/></div> <div>ja näin ollen</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow2%2B%7Dx-4%3D2-4%3D-2" alt="\lim_{x\rightarrow2+}x-4=2-4=-2"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0Ax%5E2-3x%7B%2C%7D%26kun%5C%20x%5Cle2%5C%5C%0Ax-4%7B%2C%7D%26kun%5C%20x%3E2%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\begin{cases}&#10;x^2-3x{,}&amp;kun\ x\le2\\&#10;x-4{,}&amp;kun\ x&gt;2&#10;\end{cases}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--> </div> <div>Tarkastellaan funktion derivoituvuutta</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5C%20%5Crightarrow2-%7D%5C%20%5Cfrac%7Bx%5E2-3x-%5Cleft(2%5E2-3%5Ccdot2%5Cright)%7D%7Bx-2%7D%3D%5Clim_%7Bx%5C%20%5Crightarrow2-%7D%5Cfrac%7Bx%5E2-3x%2B2%7D%7Bx-2%7D%3D%5Clim_%7Bx%5C%20%5Crightarrow2-%7D%5Cfrac%7B%5Cleft(x-2%5Cright)%5Cleft(x-1%5Cright)%7D%7Bx-2%7D%3D%5Clim_%7Bx%5C%20%5Crightarrow2-%7Dx-1%3D2-1%3D1" alt="f'\left(x\right)=\lim_{x\ \rightarrow2-}\ \frac{x^2-3x-\left(2^2-3\cdot2\right)}{x-2}=\lim_{x\ \rightarrow2-}\frac{x^2-3x+2}{x-2}=\lim_{x\ \rightarrow2-}\frac{\left(x-2\right)\left(x-1\right)}{x-2}=\lim_{x\ \rightarrow2-}x-1=2-1=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5C%20%5Crightarrow2%2B%7D%5Cfrac%7Bx-4-%5Cleft(2-4%5Cright)%7D%7Bx-2%7D%3D%5Clim_%7Bx%5C%20%5Crightarrow2%2B%7D%5Cfrac%7Bx-4%2B2%7D%7Bx-2%7D%3D%5Clim_%7Bx%5C%20%5Crightarrow2%2B%7D%5Cfrac%7Bx-2%7D%7Bx-2%7D%3D1" alt="f'\left(x\right)=\lim_{x\ \rightarrow2+}\frac{x-4-\left(2-4\right)}{x-2}=\lim_{x\ \rightarrow2+}\frac{x-4+2}{x-2}=\lim_{x\ \rightarrow2+}\frac{x-2}{x-2}=1"/></div> <div>Funktio on derivoituva kohdassa x=2.</div> </div> <div><br/> 133<br/> <span>A, I, II, IV</span> <div>B I</div> <div>C I, II, III</div> </div> <div>135<br/> <br/> </div> <div>137</div> <div> <div> <div> <div>l38</div> </div> </div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0A2x-3%7B%2C%7D%26kun%5C%20x%5Cle1%5C%5C%0Ax%5E2-2%7B%2C%7D%26kun%5C%20x%3E1%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\begin{cases}&#10;2x-3{,}&amp;kun\ x\le1\\&#10;x^2-2{,}&amp;kun\ x&gt;1&#10;\end{cases}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <div>Derivaattafunktio on 2, kun x &lt; 1 ja 2x, kun x &gt; 1</div> <div>Tarkastellaan funktio dervoituvuutta kohdassa x=1</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow1-%7D%5Cfrac%7B2x-3-%5Cleft(2%5Ccdot1-3%5Cright)%7D%7Bx-1%7D%3D%5Clim_%7Bx%5Crightarrow1-%7D%5Cfrac%7B2x-3%2B1%7D%7Bx-1%7D%3D%5Clim_%7Bx%5Crightarrow1-%7D%5Cfrac%7B2x-2%7D%7Bx-1%7D%3D%5Clim_%7Bx%5Crightarrow1-%7D%5Cfrac%7B2%5Cleft(x-1%5Cright)%7D%7Bx-1%7D%3D2" alt="f'\left(x\right)=\lim_{x\rightarrow1-}\frac{2x-3-\left(2\cdot1-3\right)}{x-1}=\lim_{x\rightarrow1-}\frac{2x-3+1}{x-1}=\lim_{x\rightarrow1-}\frac{2x-2}{x-1}=\lim_{x\rightarrow1-}\frac{2\left(x-1\right)}{x-1}=2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow1%2B%7D%5Cfrac%7Bx%5E2-2-%5Cleft(1%5E2-2%5Cright)%7D%7Bx-1%7D%3D%5Clim_%7Bx%5Crightarrow1%2B%7D%5Cfrac%7Bx%5E2-2%2B1%7D%7Bx-1%7D%3D%5Clim_%7Bx%5Crightarrow1%2B%7D%5Cfrac%7Bx%5E2-1%7D%7Bx-1%7D%3D%5Clim_%7Bx%5Crightarrow1%2B%7D%5Cfrac%7B%5Cleft(x%2B1%5Cright)%5Cleft(x-1%5Cright)%7D%7Bx-1%7D%3D%5Clim_%7Bx%5Crightarrow1%2B%7Dx%2B1%3D2" alt="f'\left(x\right)=\lim_{x\rightarrow1+}\frac{x^2-2-\left(1^2-2\right)}{x-1}=\lim_{x\rightarrow1+}\frac{x^2-2+1}{x-1}=\lim_{x\rightarrow1+}\frac{x^2-1}{x-1}=\lim_{x\rightarrow1+}\frac{\left(x+1\right)\left(x-1\right)}{x-1}=\lim_{x\rightarrow1+}x+1=2"/><br/> <div>Funktio f on derivoituva myös kohdassa x = 1. </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0A2%7B%2C%7D%26kun%5C%20x%5Cle1%5C%5C%0A2x%7B%2C%7D%26kun%5C%20x%3E1%0A%5Cend%7Bcases%7D" alt="f'\left(x\right)=\begin{cases}&#10;2{,}&amp;kun\ x\le1\\&#10;2x{,}&amp;kun\ x&gt;1&#10;\end{cases}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff; color: #333333; font-family: 'Times New Roman'; font-size: 17px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"--><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0Ax%5E2%7B%2C%7D%26kun%5C%20x%5Cle2%5C%5C%0A-x%5E2%2B8x-8%7B%2C%7D%26kun%5C%20x%3E2%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\begin{cases}&#10;x^2{,}&amp;kun\ x\le2\\&#10;-x^2+8x-8{,}&amp;kun\ x&gt;2&#10;\end{cases}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <div>Derivaattafunktio on 2x, kun x &lt; 2 ja -2x+8, kun x &gt; 2</div> <div>Tarkastellaan funktion derivoituvuutta kohdassa x = 2<br/> <div> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow2-%7D%5Cfrac%7Bx%5E2-2%5E2%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2-%7D%5Cfrac%7Bx%5E2-4%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2-%7D%5Cfrac%7B%5Cleft(x%2B2%5Cright)%5Cleft(x-2%5Cright)%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2-%7Dx%2B2%3D4" alt="f'\left(x\right)=\lim_{x\rightarrow2-}\frac{x^2-2^2}{x-2}=\lim_{x\rightarrow2-}\frac{x^2-4}{x-2}=\lim_{x\rightarrow2-}\frac{\left(x+2\right)\left(x-2\right)}{x-2}=\lim_{x\rightarrow2-}x+2=4"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow2%2B%7D%5Cfrac%7B-x%5E2%2B8x-8-%5Cleft(-2%5Ccdot2%2B8%5Cright)%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2%2B%7D%5Cfrac%7B-x%5E2%2B8x-8-4%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2%2B%7D%5Cfrac%7B-x%5E2%2B8x-12%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2%2B%7D%5Cfrac%7B%5Cleft(x-2%5Cright)%5Cleft(-x%2B6%5Cright)%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2%2B%7D-x%2B6%3D4" alt="f'\left(x\right)=\lim_{x\rightarrow2+}\frac{-x^2+8x-8-\left(-2\cdot2+8\right)}{x-2}=\lim_{x\rightarrow2+}\frac{-x^2+8x-8-4}{x-2}=\lim_{x\rightarrow2+}\frac{-x^2+8x-12}{x-2}=\lim_{x\rightarrow2+}\frac{\left(x-2\right)\left(-x+6\right)}{x-2}=\lim_{x\rightarrow2+}-x+6=4"/></div> <div>Funktio f on derivoituva myös kohdassa x = 2</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0A2x%7B%2C%7D%26kun%5C%20x%5Cle2%5C%5C%0A-2x%2B8%7B%2C%7D%26kun%5C%20x%3E2%0A%5Cend%7Bcases%7D" alt="f'\left(x\right)=\begin{cases}&#10;2x{,}&amp;kun\ x\le2\\&#10;-2x+8{,}&amp;kun\ x&gt;2&#10;\end{cases}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> </div> </div> </div> </div> 2019-11-13T09:19:02+02:00 https://peda.net/id/752c35d4049 2019-11-11T22:46:22+02:00 101 <div>a) 4</div> <div>b) raja-arvoa ei ole</div> <div>c) 3</div> <div>d) 1</div> <div>e) 1</div> <div>f) 2</div> <div> </div> <div>102</div> <div>A III</div> <div>B I</div> <div>C II</div> <div>D IV</div> <div> </div> <div> <div><span>106</span><br/> <span>a) </span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1%7D%5Cfrac%7B1%7D%7B%5Cleft(x-1%5Cright)%5E2%7D%3D%5Cfrac%7B1%7D%7B0%7D%5C%20" alt="\lim_{x\rightarrow1}\frac{1}{\left(x-1\right)^2}=\frac{1}{0}\ "/><span>, joten voidaan oleta, että funktiolla ei ole tavallista raja-arvoa.</span> <div>Kun x lähestyy arvoa 1 oikealta tai vasemmalta, lähestyy (x − 1)² arvoa 0 positiiviselta puolelta. Funktiolla on epäoleellinen raja-arvo ∞ kohdassa x = 1.  </div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1-x%5E2%7D%7Bx-1%7D%3D%5Cfrac%7B1-1%7D%7B1-1%7D%3D%5Cfrac%7B0%7D%7B0%7D" alt="\frac{1-x^2}{x-1}=\frac{1-1}{1-1}=\frac{0}{0}"/>, funktiota täytyy sieventä</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1%7D%5Cfrac%7B1-x%5E2%7D%7Bx-1%7D%3D%5Cfrac%7B-%5Cleft(x-1%5Cright)%5Cleft(1%2Bx%5Cright)%7D%7Bx-1%7D%3D-1%2Bx%5Crightarrow-1-1%3D-2" alt="\lim_{x\rightarrow1}\frac{1-x^2}{x-1}=\frac{-\left(x-1\right)\left(1+x\right)}{x-1}=-1+x\rightarrow-1-1=-2"/>, kun x→1</div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1%7D%5Cfrac%7Bx-2%7D%7B1-x%7D%3D%5Cfrac%7B1-2%7D%7B1-1%7D%3D%5Cfrac%7B-1%7D%7B0%7D" alt="\lim_{x\rightarrow1}\frac{x-2}{1-x}=\frac{1-2}{1-1}=\frac{-1}{0}"/>:</div> <div> Kun x lähestyy arvoa 1, osoittaja lähestyy arvoa 1 − 2 = −1. Kun x lähestyy arvoa 1 oikealta, lähestyy nimittäjä arvoa 0 negatiiviselta puolelta ja kun x lähestyy arvoa 1 vasemmalta, lähestyy nimittäjä arvoa 0 positiiviselta puolelta. </div> <span> Funktiolla f on epäoleellinen oikeanpuoleinen raja-arvo ∞ ja epäoleellinen vasemmanpuoleinen raja-arvo −∞ kohdassa x = 1. Funktiolla f ei ole raja-arvoa kohdassa x = 1.  </span><br/> <span>d)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1%7D%5Cfrac%7Bx-1%7D%7Bx%7D%3D%5Cfrac%7B1-1%7D%7B1%7D%3D%5Cfrac%7B0%7D%7B1%7D%3D0" alt="\lim_{x\rightarrow1}\frac{x-1}{x}=\frac{1-1}{1}=\frac{0}{1}=0"/>, kun x→1<br/> <br/> </div> </div> <div> <div> <div>108</div> <div>Funktiolla f on raja-arvo kohdassa x = 3, jos sen toispuoliset raja-arvot ovat samat.  </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow3-%7D2%5Ex%2Ba%3D2%5E3%2Ba%3D8%2Ba" alt="\lim_{x\rightarrow3-}2^x+a=2^3+a=8+a"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow3%2B%7D%5Csqrt%5B%5D%7B4x-8%7D%3D%5Csqrt%5B%5D%7B12-8%7D%3D%5Csqrt%5B%5D%7B4%7D%3D2" alt="\lim_{x\rightarrow3+}\sqrt[]{4x-8}=\sqrt[]{12-8}=\sqrt[]{4}=2"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=8%2Ba%3D2" alt="8+a=2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D2-8" alt="a=2-8"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D-6" alt="a=-6"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(3%5Cright)%3D%5Csqrt%5B%5D%7B4%5Ccdot3-8%7D%3D2" alt="f\left(3\right)=\sqrt[]{4\cdot3-8}=2"/></div> <div>Funktio on polynomifunktiona jatkuva välillä x &lt; 3 ja juurifunktion ja polynomifunktio yhdistettynä funktiona jatkuva välillä x &gt; 3. Kun a = -6, funktiolla on raja-arvo 2 kohdassa x=3 ja f(3)=2. Funktio on jatkuva kohdassa x = 3, joten funktio f on<span> jatkuva.</span></div> </div> <div> </div> <div> <div>109</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cleft%7Cx%5Cright%7C%3D%5Cbegin%7Bcases%7D%0A-x%7B%2C%7D%26kun%5C%20x%3C0%5C%5C%0Ax%7B%2C%7D%26kun%5C%20x%5Cge0%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\left|x\right|=\begin{cases}&#10;-x{,}&amp;kun\ x&lt;0\\&#10;x{,}&amp;kun\ x\ge0&#10;\end{cases}"/></div>  Kun x lähesty kohtaa nolla, myös |x| lähestyy arvoa 0. Funktiolla f on raja-arvo 0 kohdassa x = 0.  <br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cleft%7Cx%5Cright%7C%2Bx%3D%5Cbegin%7Bcases%7D%0A-x%2Bx%7B%2C%7D%26kun%5C%20x%3C0%5C%5C%0Ax%2Bx%7B%2C%7D%26kun%5Cge0%0A%5Cend%7Bcases%7D%3D%5Cbegin%7Bcases%7D%0A0%7B%2C%7D%26kun%5C%20x%3C0%5C%5C%0A2x%7B%2C%7D%5C%20%26kun%5C%20x%5Cge0%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\left|x\right|+x=\begin{cases}&#10;-x+x{,}&amp;kun\ x&lt;0\\&#10;x+x{,}&amp;kun\ge0&#10;\end{cases}=\begin{cases}&#10;0{,}&amp;kun\ x&lt;0\\&#10;2x{,}\ &amp;kun\ x\ge0&#10;\end{cases}"/></div> <div> Kun x lähesty kohtaa nolla vasemmalta, funktion f arvo on 0 ja kun x lähestyy kohtaa nolla oikealta, funktion arvot lähestyvät nollaa. Funktiolla f on raja-arvo 0 kohdassa x = 0. </div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cfrac%7B%5Cleft%7Cx%5Cright%7C%7D%7Bx%7D%3D%5Cbegin%7Bcases%7D%0A%5Cfrac%7B-x%7D%7Bx%7D%7B%2C%7D%26kun%5C%20x%3C0%5C%5C%0A%5Cfrac%7Bx%7D%7Bx%7D%7B%2C%7D%26kun%5Cge0%0A%5Cend%7Bcases%7D%3D%5Cbegin%7Bcases%7D%0A-1%7B%2C%7D%26kun%5C%20x%3C0%5C%5C%0A1%7B%2C%7D%5C%20%26kun%5C%20x%5Cge0%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\frac{\left|x\right|}{x}=\begin{cases}&#10;\frac{-x}{x}{,}&amp;kun\ x&lt;0\\&#10;\frac{x}{x}{,}&amp;kun\ge0&#10;\end{cases}=\begin{cases}&#10;-1{,}&amp;kun\ x&lt;0\\&#10;1{,}\ &amp;kun\ x\ge0&#10;\end{cases}"/></div> <div> Funktion f vasemmanpuoleinen raja-arvo kohdassa x = 0 on −1 ja oikeanpuoleinen on 1. Funktiolla f ei ole raja-arvoa kohdassa x = 0.<span> <br/> </span></div> <div> <div> <div><br/> 112</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow0%7D%5Cfrac%7Be%5E%7B2x%7D-1%7D%7Be%5Ex-1%7D%3D%5Cfrac%7B%5Cleft(e%5Ex%5Cright)%5E2-1%7D%7Be%5Ex-1%7D%3D%5Cfrac%7B%5Cleft(e%5Ex-1%5Cright)%5Cleft(e%5Ex%2B1%5Cright)%7D%7Be%5Ex-1%7D%3De%5E0%2B1%3D2" alt="\lim_{x\rightarrow0}\frac{e^{2x}-1}{e^x-1}=\frac{\left(e^x\right)^2-1}{e^x-1}=\frac{\left(e^x-1\right)\left(e^x+1\right)}{e^x-1}=e^0+1=2"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1%7D%5Cfrac%7B%5Cln%20x%5E3%7D%7B%5Cln%20x%7D%3D%5Cfrac%7B3%5Cln%20x%7D%7B%5Cln%20x%7D%3D%5Clim_%7Bx%5Crightarrow1%7D3%3D3" alt="\lim_{x\rightarrow1}\frac{\ln x^3}{\ln x}=\frac{3\ln x}{\ln x}=\lim_{x\rightarrow1}3=3"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow3%7D%5Cfrac%7B%5Cleft%7C2x-6%5Cright%7C%7D%7Bx-3%7D%3D%5Clim_%7Bx%5Crightarrow3%7D%5Cfrac%7B2%5Cleft%7Cx-3%5Cright%7C%7D%7Bx-3%7D%3D%5Clim_%7Bx%5Crightarrow3%7D%5Cfrac%7B2%5Cleft(x-3%5Cright)%7D%7Bx-3%7D%3D2" alt="\lim_{x\rightarrow3}\frac{\left|2x-6\right|}{x-3}=\lim_{x\rightarrow3}\frac{2\left|x-3\right|}{x-3}=\lim_{x\rightarrow3}\frac{2\left(x-3\right)}{x-3}=2"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow3%7D%5Cfrac%7B%5Cleft%7C2x-6%5Cright%7C%7D%7Bx-3%7D%3D%5Clim_%7Bx%5Crightarrow3%7D%5Cfrac%7B-2%5Cleft%7Cx-3%5Cright%7C%7D%7Bx-3%7D%3D%5Clim_%7Bx%5Crightarrow3%7D%5Cfrac%7B-2%5Cleft(x-3%5Cright)%7D%7Bx-3%7D%3D-2" alt="\lim_{x\rightarrow3}\frac{\left|2x-6\right|}{x-3}=\lim_{x\rightarrow3}\frac{-2\left|x-3\right|}{x-3}=\lim_{x\rightarrow3}\frac{-2\left(x-3\right)}{x-3}=-2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=-2%5Cne2" alt="-2\ne2"/>, raja-arvoa ei ole</div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow3%7D%5Cfrac%7Bx%5E2-3%7D%7Bx-3%7D" alt="\lim_{x\rightarrow3}\frac{x^2-3}{x-3}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow3%2B%7D%5Cfrac%7Bx%5E2-3%7D%7Bx-3%7D%3D%5Cfrac%7B6%7D%7B0%7D%3D%5Cinfty" alt="\lim_{x\rightarrow3+}\frac{x^2-3}{x-3}=\frac{6}{0}=\infty"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow3-%7D%5Cfrac%7Bx%5E2-3%7D%7Bx-3%7D%3D%5Cfrac%7B6%7D%7B0%7D%3D-%5Cinfty" alt="\lim_{x\rightarrow3-}\frac{x^2-3}{x-3}=\frac{6}{0}=-\infty"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cinfty%5Cne%5Cinfty" alt="-\infty\ne\infty"/>, raja-arvoa ei ole</div> </div> </div> <div> </div> <div> <div> <div>115</div> <div>a)</div> <div>Suorakulmion kanta on x ja korkeus y=1/x+1. Suorakulmion pinta-ala on<br/>  <img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(x%5Cright)%3Dx%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%2B1%5Cright)%3D1%2Bx" alt="A\left(x\right)=x\left(\frac{1}{x}+1\right)=1+x"/></div> <div>Kun x lähestyy nollaa, pinta-ala lähestyy arvoa 1</div> <div>b)<br/> Suorakulmion kanta on x ja korkeus y=2/x+1/x². Suorakulmion pinta-ala on  </div> <img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(x%5Cright)%3Dx%5Cleft(%5Cfrac%7B2%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%5Cright)%3D2%2B%5Cfrac%7B1%7D%7Bx%7D" alt="A\left(x\right)=x\left(\frac{2}{x}+\frac{1}{x^2}\right)=2+\frac{1}{x}"/></div> <div>Kun x lähestyy nollaa, pinta-ala kasvaa äärettömän suureksi.</div> </div> </div> 2019-11-11T17:18:04+02:00