https://peda.net/id/2eb9273417c 2019-01-14T09:29:52+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/17470630347 2019-02-19T21:03:49+02:00 <span>Esim. Sievennä</span> <div>a) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5E0%3D1" alt="e^0=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%20e%3D%5Clog_ee%3D1" alt="\ln e=\log_ee=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln1%3D0" alt="\ln1=0"/><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cfrac%7B1%7D%7B4a%5E2%7D%5Cright)%5E%7B-3%7D%3D%5Cleft(4a%5E2%5Cright)%5E3%3D4%5E3a%5E6%3D64a%5E6" alt="\left(\frac{1}{4a^2}\right)^{-3}=\left(4a^2\right)^3=4^3a^6=64a^6"/></div> <span>c) Muuta murtopotenssiksi</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%7D%7B%5Csqrt%5B%5D%7Bx%7D%7D%3D%5Cfrac%7Bx%7D%7Bx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%3Dx%5E%7B1-%5Cfrac%7B1%7D%7B2%7D%7D%3Dx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D" alt="\frac{x}{\sqrt[]{x}}=\frac{x}{x^{\frac{1}{2}}}=x^{1-\frac{1}{2}}=x^{\frac{1}{2}}"/></div> <span>d) Muuta juurimuotoon</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D%3D%5Csqrt%5B%5D%7Bx%5E5%7D%3D%5Csqrt%5B%5D%7Bx%5E2%5Ccdot%20x%5E2%5Ccdot%20x%5E1%7D%3Dx%5E2%5Csqrt%5B%5D%7Bx%7D" alt="x^{\frac{5}{2}}=\sqrt[]{x^5}=\sqrt[]{x^2\cdot x^2\cdot x^1}=x^2\sqrt[]{x}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D%3Dx%5E%7B2%5Cfrac%7B1%7D%7B2%7D%7D%3Dx%5E%7B2%2B%5Cfrac%7B1%7D%7B2%7D%7D%3D%5Ctext%7Bx%7D%5E2%5Ccdot%20x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%3Dx%5E2%5Csqrt%5B%5D%7Bx%7D" alt="x^{\frac{5}{2}}=x^{2\frac{1}{2}}=x^{2+\frac{1}{2}}=\text{x}^2\cdot x^{\frac{1}{2}}=x^2\sqrt[]{x}"/><br/> <div>Esim. Derivoi<img src="https://math-demo.abitti.fi/math.svg?latex=3xe%5E%7B4x%7D" alt="3xe^{4x}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D3xe%5E%7B4x%7D%3D3%5Ccdot%20e%5E%7B4x%7D%2B3x%5Ccdot%20e%5E%7B4x%7D%5Ccdot4%3D3e%5E%7B4x%7D%2B12xe%5E%7B4x%7D" alt="D3xe^{4x}=3\cdot e^{4x}+3x\cdot e^{4x}\cdot4=3e^{4x}+12xe^{4x}"/></div> <br/> <span>- Juurifunktio </span><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dn%5Csqrt%5B%5D%7Bx%7D" alt="f\left(x\right)=n\sqrt[]{x}"/><br/> <div>* jatkuva ja kasvava</div> <div>* n parillinen: Määrittelyehto x≥0, arvojoukko ℝ+</div> <div>* n pariton: Määritelty kaikilla x, arvojoukko koko ℝ</div> <div>- Derivoidaan käyttämällä murtopotenssia</div> <div>Esim. </div> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Csqrt%5B4%5D%7Bx%7D%3Dx%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D%5Cfrac%7B1%7D%7B4%7Dx%5E%7B%5Cfrac%7B1%7D%7B4%7D-1%7D%3D%5Cfrac%7B1%7D%7B4%7Dx%5E%7B-%5Cfrac%7B3%7D%7B4%7D%7D%3D%5Cfrac%7B1%7D%7B4%7D%5Ccdot%5Cfrac%7B1%7D%7Bx%5E%7B%5Cfrac%7B3%7D%7B4%7D%7D%7D%3D%5Cfrac%7B1%7D%7B4%5Csqrt%5B4%5D%7Bx%5E3%7D%7D" alt="D\sqrt[4]{x}=x^{\frac{1}{4}}=\frac{1}{4}x^{\frac{1}{4}-1}=\frac{1}{4}x^{-\frac{3}{4}}=\frac{1}{4}\cdot\frac{1}{x^{\frac{3}{4}}}=\frac{1}{4\sqrt[4]{x^3}}"/> <div>- Parillisen juuriyhtälön ratkaisussa muistettava neliöönkorotusehto</div> <div> </div> <div>Eim. Ratkaise yhtälö <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7Bx%5E2-1%7D%3Dx%2B1" alt="\sqrt[]{x^2-1}=x+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-1%5Cge0" alt="x^2-1\ge0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-1%3D0" alt="x^2-1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D1" alt="x^2=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D%5Cpm1" alt="x^2=\pm1"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cle-1%5C%20tai%5C%20x%5Cge1" alt="x\le-1\ tai\ x\ge1"/><br/> <div>Neiöön korotusehto: </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%2B1%5Cge0" alt="x+1\ge0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cge1" alt="x\ge1"/><br/> <div>Molemmat ehdot toteutuvat, kun <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%5C%20tai%5C%20x%5Cge1" alt="x=-1\ tai\ x\ge1"/></div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7Bx%5E2-1%7D%3Dx%2B1%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Cleft(%5Cright)%5E2" alt="\sqrt[]{x^2-1}=x+1\ \ \ \ \ \left|\right|\left(\right)^2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-1%3D%5Cleft(x%2B1%5Cright)%5E2" alt="x^2-1=\left(x+1\right)^2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-1%3Dx%5E2%2B2x%2B1" alt="x^2-1=x^2+2x+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%2B2%3D0" alt="2x+2=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%3D-2" alt="2x=-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1" alt="x=-1"/><br/> <div>- Verrannollisuus</div> <div>Suuret x&gt;0 ja y&lt;0 ovat suoraan verrannolliset, jos </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7By%7D%7Bx%7D%3Dk%5C%20eli%5C%20y%3Dkx" alt="\frac{y}{x}=k\ eli\ y=kx"/> <div>- Kääntäen verrannolliset, jos <img src="https://math-demo.abitti.fi/math.svg?latex=yx%3Dk%5C%20eli%5C%20y%3D%5Cfrac%7Bk%7D%7Bx%7D" alt="yx=k\ eli\ y=\frac{k}{x}"/></div> <div>K=verrannollisuuskerroin</div> <div> </div> <div> </div> <div> </div> 2019-02-19T21:03:49+02:00 https://peda.net/id/6dd2b65a2dc 2019-02-11T09:19:54+02:00 <div>Lause</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cln%20x%3D%5Cfrac%7B1%7D%7Bx%7D%7B%2C%7D%5C%20kun%5C%20x%3E0" alt="D\ln x=\frac{1}{x}{,}\ kun\ 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;"--></div> <span>b)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cln%5Cleft%7Cx%5Cright%7C%3D%5Cfrac%7B1%7D%7Bx%7D%7B%2C%7D%5C%20kun%5C%20x%5Cne0" alt="D\ln\left|x\right|=\frac{1}{x}{,}\ kun\ x\ne0"/></div> <div> </div> <span>Esim. Derivoi, kun x&gt;0</span> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D2%5Cln%20x%5E3%3DD2%5Ccdot3%5Cln%20x%3D6D%5Cln%20x%3D6%5Ccdot%5Cfrac%7B1%7D%7Bx%7D%3D%5Cfrac%7B6%7D%7Bx%7D" alt="D2\ln x^3=D2\cdot3\ln x=6D\ln x=6\cdot\frac{1}{x}=\frac{6}{x}"/></div> <div>b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5C%20%5Cfrac%7B7x%7D%7B3%7D" alt="\ln\ \frac{7x}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cln%5C%20%5Cfrac%7B7x%7D%7B3%7D%3DD%5Cln%5C%20%5Cfrac%7B7%7D%7B3%7Dx%3D%5Cfrac%7B1%7D%7B%5Cfrac%7B7x%7D%7B3%7D%7D%5Ccdot%5Cfrac%7B7%7D%7B3%7D%3D%5Cleft(%5Cfrac%7B7x%7D%7B3%7D%5Cright)%5E%7B-1%7D%5Ccdot%5Cfrac%7B7%7D%7B3%7D%3D%5Cfrac%7B3%7D%7B7x%7D%5Ccdot%5Cfrac%7B7%7D%7B3%7D%3D%5Cfrac%7B1%7D%7Bx%7D" alt="D\ln\ \frac{7x}{3}=D\ln\ \frac{7}{3}x=\frac{1}{\frac{7x}{3}}\cdot\frac{7}{3}=\left(\frac{7x}{3}\right)^{-1}\cdot\frac{7}{3}=\frac{3}{7x}\cdot\frac{7}{3}=\frac{1}{x}"/><br/> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5Cleft(3%5Csqrt%5B%5D%7Bx%7D%5Cright)%7B%2C%7D%5C%20u%5Cleft(x%5Cright)%3D%5Cln%20x%7B%2C%7D%5C%20u%27%5Cleft(x%5Cright)%3D%5Cfrac%7B1%7D%7Bx%7D%7B%2C%7D%5C%20s%5Cleft(x%5Cright)%3D3%5Csqrt%5B%5D%7Bx%7D%3Dx%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7B%2C%7D%5C%20s%27%5Cleft(x%5Cright)%3D%5Cfrac%7B1%7D%7B3%7Dx%5E%7B-%5Cfrac%7B2%7D%7B3%7D%7D" alt="\ln\left(3\sqrt[]{x}\right){,}\ u\left(x\right)=\ln x{,}\ u'\left(x\right)=\frac{1}{x}{,}\ s\left(x\right)=3\sqrt[]{x}=x^{\frac{1}{3}}{,}\ s'\left(x\right)=\frac{1}{3}x^{-\frac{2}{3}}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cln%5Cleft(%5Csqrt%5B3%5D%7Bx%7D%5Cright)%3D%5Cfrac%7B1%7D%7Bx%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D%5Ccdot%5Cfrac%7B1%7D%7B3%7Dx%5E%7B-%5Cfrac%7B2%7D%7B3%7D%7D%3Dx%5E%7B-%5Cfrac%7B1%7D%7B3%7D%7D%5Ccdot%5Cfrac%7B1%7D%7B3%7Dx%5E%7B-%5Cfrac%7B2%7D%7B3%7D%7D%3D%5Cfrac%7B1%7D%7B3%7D%5Ccdot%5Cleft(x%5E%7B%5E%7B-%5Cfrac%7B1%7D%7B3%7D%2B%5Cleft(-%5Cfrac%7B2%7D%7B3%7D%5Cright)%7D%7D%5Cright)%3D%5Cfrac%7B1%7D%7B3%7D%5Ccdot%20x%5E%7B-1%7D%3D%5Cfrac%7B1%7D%7B3x%7D" alt="D\ln\left(\sqrt[3]{x}\right)=\frac{1}{x^{\frac{1}{3}}}\cdot\frac{1}{3}x^{-\frac{2}{3}}=x^{-\frac{1}{3}}\cdot\frac{1}{3}x^{-\frac{2}{3}}=\frac{1}{3}\cdot\left(x^{^{-\frac{1}{3}+\left(-\frac{2}{3}\right)}}\right)=\frac{1}{3}\cdot x^{-1}=\frac{1}{3x}"/></div> <div><br/> <div>482.</div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cln%5Cleft(x%5E3-x%5Cright)" alt="f\left(x\right)=\ln\left(x^3-x\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3-x%3E0%7B%2C%7D%5C%20kun%5C%20-1%3Cx%3C0%5C%20tai%5C%20x%3E1%5C%20%5Cleft(laskin%5Cright)" alt="x^3-x&gt;0{,}\ kun\ -1&lt;x&lt;0\ tai\ x&gt;1\ \left(laskin\right)"/><!--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><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Cfrac%7B1%7D%7Bx%5E3-x%7D%5Ccdot%5Cleft(3x%5E2-1%5Cright)%3D%5Cfrac%7B3x%5E2-1%7D%7Bx%5E3-x%7D" alt="f'\left(x\right)=\frac{1}{x^3-x}\cdot\left(3x^2-1\right)=\frac{3x^2-1}{x^3-x}"/></div> <span>Ratkaistaan derivaattafunktion nollakohdat</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-%5Csqrt%5B%5D%7B3%7D%7D%7B3%7D%5C%20tai%5C%20x%3D%5Cfrac%7B%5Csqrt%5B%5D%7Bx%7D%7D%7B3%7D" alt="x=\frac{-\sqrt[]{3}}{3}\ tai\ x=\frac{\sqrt[]{x}}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-0%7B%2C%7D57735%5C%20tai%5C%20x%3D0%7B%2C%7D57735" alt="x=-0{,}57735\ tai\ x=0{,}57735"/></div> <div>Ratkaisuista toteuttaa määrittelyehdon vain</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B%5Csqrt%5B%5D%7B3%7D%7D%7B3%7D%5Capprox-0%7B%2C%7D577" alt="x=-\frac{\sqrt[]{3}}{3}\approx-0{,}577"/><br/> Laaditaan funktion f(x) kulkukaavio, Selvitetään kulkukaaviota varten derivaatan merkit testikohtien avulal. Nimetään f'(x)=g(x)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(2%5Cright)%3D%5Cfrac%7B11%7D%7B6%7D" alt="g\left(2\right)=\frac{11}{6}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(-0%7B%2C%7D1%5Cright)%3D-9%7B%2C%7D79798" alt="g\left(-0{,}1\right)=-9{,}79798"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(-0%7B%2C%7D7%5Cright)%3D1%7B%2C%7D31653" alt="g\left(-0{,}7\right)=1{,}31653"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26-1%26%26-%5Cfrac%7B%5Csqrt%5B%5D%7B3%7D%7D%7B3%7D%26%260%26%261%26%5C%5C%0A%5Chline%0Af%27%5Cleft(x%5Cright)%26%26%2B%26%26-%26%26%5Csetminus%26%26%2B%5C%5C%0Af%5Cleft(x%5Cright)%26%26%5Cnearrow%26%26%5Csearrow%26%26%5Csetminus%26%26%5Cnearrow%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;-1&amp;&amp;-\frac{\sqrt[]{3}}{3}&amp;&amp;0&amp;&amp;1&amp;\\&#10;\hline&#10;f'\left(x\right)&amp;&amp;+&amp;&amp;-&amp;&amp;\setminus&amp;&amp;+\\&#10;f\left(x\right)&amp;&amp;\nearrow&amp;&amp;\searrow&amp;&amp;\setminus&amp;&amp;\nearrow&#10;\end{array}"/></div> <div>Funktiolla on paikallinen maksimiarvo kohdassa </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B%5Csqrt%5B%5D%7B3%7D%7D%7B3%7D" alt="x=-\frac{\sqrt[]{3}}{3}"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-%5Cfrac%7B%5Csqrt%5B%5D%7B3%7D%7D%7B3%7D%5Cright)%3D%5Cfrac%7B%5Cln%5C%20%5Cfrac%7B4%7D%7B27%7D%7D%7B2%7D" alt="f\left(-\frac{\sqrt[]{3}}{3}\right)=\frac{\ln\ \frac{4}{27}}{2}"/><br/> <div>Sievennetään laskimen antaman tulos</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cln%5C%20%5Cfrac%7B4%7D%7B27%7D%7D%7B2%7D%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cln%5C%20%5Cleft(%5Cfrac%7B4%7D%7B27%7D%5Cright)%3D%5Cln%5C%20%5Cleft(%5Cfrac%7B4%7D%7B27%7D%5Cright)%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%3D%5Csqrt%5B%5D%7B%5Cfrac%7B4%7D%7B27%7D%7D%3D%5Cln%5C%20%5Cfrac%7B%5Csqrt%5B%5D%7B4%7D%7D%7B%5Csqrt%5B%5D%7B27%7D%7D%3D%5Cln%5C%20%5Cfrac%7B2%7D%7B%5Csqrt%5B%5D%7B3%5Ccdot9%7D%7D%3D%5Cln%5C%20%5Cfrac%7B2%7D%7B3%5Csqrt%5B%5D%7B3%7D%7D" alt="\frac{\ln\ \frac{4}{27}}{2}=\frac{1}{2}\cdot\ln\ \left(\frac{4}{27}\right)=\ln\ \left(\frac{4}{27}\right)^{\frac{1}{2}}=\sqrt[]{\frac{4}{27}}=\ln\ \frac{\sqrt[]{4}}{\sqrt[]{27}}=\ln\ \frac{2}{\sqrt[]{3\cdot9}}=\ln\ \frac{2}{3\sqrt[]{3}}"/></div> <br/> <span>Lause</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Clog_ax%3D%5Cfrac%7B1%7D%7B%5Cln%20a%7D%5Ccdot%5Cfrac%7B1%7D%7Bx%7D%7B%2C%7D%5C%20kun%5C%20x%3E0" alt="D\log_ax=\frac{1}{\ln a}\cdot\frac{1}{x}{,}\ kun\ 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;"--></div> 2019-02-11T09:19:54+02:00 https://peda.net/id/738756c429e 2019-02-06T09:54:37+02:00 <div>Lause</div> <div>Olkoon a&gt;0, a≠1.</div> <div>Logaritmifunktio <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_ax" alt="\log_ax"/> on määriteltty, kun x&gt;0</div> <div> </div> <div>- Sen arvojoukko on ℝ.</div> <div>- Logaritmifunktio on jatkuva ja </div> <div>* Kasvava, kun a&gt;1</div> <div>*Vähenevä, kun 0&lt;a&lt;1</div> <div> </div> <div>Huom:</div> <div>-Briggsin logaritmi:<img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_%7B10%7Dx%3D%5Clg%5C%20x" alt="\log_{10}x=\lg\ x"/></div> <div>-Binäärilogaritmi: <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_2x%3Dlb%5C%20x" alt="\log_2x=lb\ x"/></div> <div> </div> <div>Esim. </div> <div>455. Määritä funktion määrittelyjoukko ja nollakohdat</div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cln%5Cleft(x%5E2-3%5Cright)" alt="f\left(x\right)=\ln\left(x^2-3\right)"/></div> Määrittelyjoukko <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-3%3E0" alt="x^2-3&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;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-3%3D0" alt="x^2-3=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpm%5Csqrt%5B%5D%7B3%7D" alt="x=\pm\sqrt[]{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-3%3E0%7B%2C%7D%5C%20kun%5C%20x%3C-%5Csqrt%5B%5D%7B3%7D%5C%20tai%5C%20x%3E%5Csqrt%5B%5D%7B3%7D" alt="x^2-3&gt;0{,}\ kun\ x&lt;-\sqrt[]{3}\ tai\ x&gt;\sqrt[]{3}"/><!--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>Nollakohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D0" alt="f\left(x\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5Cleft(x%5E2-3%5Cright)%3D0" alt="\ln\left(x^2-3\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5Cleft(x%5E2-3%5Cright)%3D0%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Clog_ax%3Dy%5C%20%5Cleftrightarrow%5C%20a%5Ey%3Dx" alt="\ln\left(x^2-3\right)=0\ \ \ \ \ \left|\right|\log_ax=y\ \leftrightarrow\ a^y=x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-3%3De%5E0" alt="x^2-3=e^0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-3%3D1" alt="x^2-3=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D4" alt="x^2=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpm2" alt="x=\pm2"/><br/> <div> </div> </div> <div>Esim. Ratkaise yhtälö</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_%7B0%7B%2C%7D5%7D2x%3D%5Clog_%7B0%7B%2C%7D5%7D%5Cleft(3x-1%5Cright)" alt="\log_{0{,}5}2x=\log_{0{,}5}\left(3x-1\right)"/></div> Määrittelyehto <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%3E0%5C%20eli%5C%20x%3E0" alt="2x&gt;0\ eli\ 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;"--></div> ja  <div><img src="https://math-demo.abitti.fi/math.svg?latex=3x-1%3E0%5C%20eli%5C%20x%3E%5Cfrac%7B1%7D%7B3%7D" alt="3x-1&gt;0\ eli\ x&gt;\frac{1}{3}"/><!--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>On siis oltava</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3E%5Cfrac%7B1%7D%7B3%7D" alt="x&gt;\frac{1}{3}"/><!--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=%5Clog_%7B0%7B%2C%7D5%7D2x%3D%5Clog_%7B0%7B%2C%7D5%7D%5Cleft(3x-1%5Cright)" alt="\log_{0{,}5}2x=\log_{0{,}5}\left(3x-1\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%3D3x-1" alt="2x=3x-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-x%3D-1" alt="-x=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1" alt="x=1"/> <div> </div> <div>b) </div> <img src="https://math-demo.abitti.fi/math.svg?latex=3%5Cln%20x%3D%5Cln%5Cleft(x%5E3-x%5E2%2B1%5Cright)" alt="3\ln x=\ln\left(x^3-x^2+1\right)"/> <div>Määrittelyehto</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3E0" alt="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;"--> ja <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3-x%5E2%2B1%3E0" alt="x^3-x^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;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3E-0%7B%2C%7D755%5C%20%5Cleft(laskin%5Cright)" alt="x&gt;-0{,}755\ \left(laskin\right)"/><!--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>Siis</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3E0" alt="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;"--></div> <div> <div><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3%5Cln%20x%3D%5Cln%5Cleft(x%5E3-x%5E2%2B1%5Cright)" alt="3\ln x=\ln\left(x^3-x^2+1\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%20x%5E3%3D%5Cln%5Cleft(x%5E3-x%5E2%2B1%5Cright)" alt="\ln x^3=\ln\left(x^3-x^2+1\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3%3Dx%5E3-x%5E2%2B1" alt="x^3=x^3-x^2+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-x%5E2%2B1%3D0" alt="-x^2+1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D1" alt="x^2=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpm1" alt="x=\pm1"/></div> <div>Hylätään negatiivinen tulos</div> <div><br/> <div><span> </span><br/> <br/> </div> </div> 2019-02-06T09:54:37+02:00 https://peda.net/id/73639f7a284 2019-02-04T09:14:03+02:00 <div>Määritelmä</div> <div>e-kantaista logaritmia luvusta x&gt;0 kutstaan luonnolliseksi logaritmiksi. Merkitään</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_ex%3D%5Cln%5C%20x" alt="\log_ex=\ln\ x"/></div> <br/> <div>Esim. Ratkaise yhtälö</div> <div>a)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=6e%5Ex-24%3D0" alt="6e^x-24=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex%3D4%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Clog_ax%3Dy%5C%20%5C%20%5Cleftrightarrow%5C%20%5C%20a%5Ey%3Dx" alt="e^x=4\ \ \ \ \ \left|\right|\log_ax=y\ \ \leftrightarrow\ \ a^y=x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cln%5C%204%3D%5Cln2%5E2%3D2%5Cln2" alt="x=\ln\ 4=\ln2^2=2\ln2"/> <div><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5C%20x%3D-1" alt="\ln\ x=-1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3De%5E%7B-1%7D%3D%5Cfrac%7B1%7D%7Be%7D" alt="x=e^{-1}=\frac{1}{e}"/></div> <div><br/> c) <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B5x%7D-1000e%5E%7B2x%7D%3D0" alt="e^{5x}-1000e^{2x}=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B3x%2B2x%7D-1000e%5E%7B2x%7D%3D0" alt="e^{3x+2x}-1000e^{2x}=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B3x%7D%5Ccdot%20e%5E%7B2x%7D-1000e%5E%7B2x%7D%3D0" alt="e^{3x}\cdot e^{2x}-1000e^{2x}=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B2x%7D%5Cleft(e%5E%7B3x%7D-1000%5Cright)%3D0" alt="e^{2x}\left(e^{3x}-1000\right)=0"/> <div>Tulon nollasäännön nojalla</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B2x%7D%3D0%5Cleft(ei%5C%20ratkaisua%5Cright)" alt="e^{2x}=0\left(ei\ ratkaisua\right)"/><br/> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B3x%7D-1000%3D0" alt="e^{3x}-1000=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B3x%7D%3D1000" alt="e^{3x}=1000"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D%5Cln%5C%201000%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3A3" alt="3x=\ln\ 1000\ \ \ \ \ \left|\right|:3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cln%5C%201000%7D%7B3%7D%3D%5Cfrac%7B%5Cln10%5E3%7D%7B3%7D%3D%5Cfrac%7B3%5Cln10%7D%7B3%7D%3D%5Cln10" alt="x=\frac{\ln\ 1000}{3}=\frac{\ln10^3}{3}=\frac{3\ln10}{3}=\ln10"/><br/> <div> </div> <div>Lause</div> <div>Kun a, b&gt;0 ja kantaluku k&gt;0, k≠1, niin</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_ka%3D%5Clog_kb%5C%20%5C%20%5Cleftrightarrow%5C%20%5C%20a%3Db" alt="\log_ka=\log_kb\ \ \leftrightarrow\ \ a=b"/></div> <div> </div> 436. Lääkeaineen pitoisuus alussa 50 mg/l <div>pitoisuus alenee 30% tunnissa eli 0,7-kertaiseksi.</div> <div>x tunnin kuluttua pitoisuus on </div> <img src="https://math-demo.abitti.fi/math.svg?latex=50%5Ccdot0%7B%2C%7D7%5Ex" alt="50\cdot0{,}7^x"/><br/> <div>Ratkaise milloi pitoisuus on 5% alkuperäisestä eli</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D05%5Ccdot50%3D2%7B%2C%7D5" alt="0{,}05\cdot50=2{,}5"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=50%5Ccdot0%7B%2C%7D7%5Ex%3D0%7B%2C%7D05%5Ccdot50" alt="50\cdot0{,}7^x=0{,}05\cdot50"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D7%5Ex%3D0%7B%2C%7D05" alt="0{,}7^x=0{,}05"/></div> Tapa 1 <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Clog_%7B0%7B%2C%7D7%7D0%7B%2C%7D05%5Capprox8%7B%2C%7D399" alt="x=\log_{0{,}7}0{,}05\approx8{,}399"/></div> <div>Tapa 2</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D7%5Ex%3D0%7B%2C%7D05" alt="0{,}7^x=0{,}05"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln0%7B%2C%7D7%5Ex%3D%5Cln0%7B%2C%7D05" alt="\ln0{,}7^x=\ln0{,}05"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cln0%7B%2C%7D7%3D%5Cln0%7B%2C%7D05" alt="x\ln0{,}7=\ln0{,}05"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cln0%7B%2C%7D05%7D%7B%5Cln0%7B%2C%7D7%7D%5Capprox8%7B%2C%7D399" alt="x=\frac{\ln0{,}05}{\ln0{,}7}\approx8{,}399"/><br/> <div>Alkuperäisestä lääkeainepitoisuudesta on jälellä alle 5% 9 tunnin kuluttua.</div> <div> </div> <div>Huom: Logaritmin kantaluku voidaan vaihtaa</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_ab%3D%5Cfrac%7B%5Clog_cb%7D%7B%5Clog_ca%7D" alt="\log_ab=\frac{\log_cb}{\log_ca}"/></div> <div> </div> Esim. Sievnnä <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_35%2B%5Clog_925%3D%5Clog_35%2B%5Cfrac%7B%5Clog_325%7D%7B%5Clog_39%7D%3D%5Clog_3%2B%5Cfrac%7B%5Clog_35%5E2%7D%7B2%7D%3D%5Clog_35%2B%5Cfrac%7B2%5Clog_35%7D%7B2%7D%3D%5Clog_35%2B%5Clog_35%3D2%5Clog_35" alt="\log_35+\log_925=\log_35+\frac{\log_325}{\log_39}=\log_3+\frac{\log_35^2}{2}=\log_35+\frac{2\log_35}{2}=\log_35+\log_35=2\log_35"/></div> <div> </div> <div>Lause</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=Da%5Ex%3D%5Cleft(%5Cln%20a%5Cright)a%5Ex%7B%2C%7D%5C%20a%3E0" alt="Da^x=\left(\ln a\right)a^x{,}\ a&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;"--><br/> <div> <div>Todistus</div> </div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B%5Cln%20a%7D%3Da" alt="e^{\ln a}=a"/>, joten </div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Ex%3D%5Cleft(e%5E%7B%5Cln%20a%7D%5Cright)%5Ex%3De%5E%7B%5Cleft(%5Cln%20a%5Cright)%5Ccdot%20x%7D" alt="a^x=\left(e^{\ln a}\right)^x=e^{\left(\ln a\right)\cdot x}"/><br/> <div>Nimetään</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3De%5E%7B%5Cleft(%5Cln%20a%5Cright)%5Ccdot%20x%7D" alt="f\left(x\right)=e^{\left(\ln a\right)\cdot x}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Du%5Cleft(s%5Cleft(x%5Cright)%5Cright)%7B%2C%7D%5C%20kun%5C%20s%5Cleft(x%5Cright)%3D%5Cleft(%5Cln%20a%5Cright)%5Ccdot%20x%7B%2C%7D%5C%20u%5Cleft(x%5Cright)%3De%5Ex" alt="f\left(x\right)=u\left(s\left(x\right)\right){,}\ kun\ s\left(x\right)=\left(\ln a\right)\cdot x{,}\ u\left(x\right)=e^x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3Du%27%5Cleft(s%5Cleft(x%5Cright)%5Cright)%5Ccdot%20s%27%5Cleft(x%5Cright)%3De%5E%7B%5Cleft(%5Cln%20a%5Cright)x%7D%5Ccdot%5Cleft(%5Cln%20a%5Cright)%3Da%5Ex%5Ccdot%5Cln%5Cleft(a%5Cright)" alt="f'\left(x\right)=u'\left(s\left(x\right)\right)\cdot s'\left(x\right)=e^{\left(\ln a\right)x}\cdot\left(\ln a\right)=a^x\cdot\ln\left(a\right)"/><br/> <div>siis</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=Da%5Ex%3D%5Cleft(%5Cln%20a%5Cright)a%5Ex" alt="Da^x=\left(\ln a\right)a^x"/></div> </div> 2019-02-04T09:14:02+02:00 https://peda.net/id/9c0b3c64246 2019-02-01T08:57:17+02:00 <div>Määritelmä:</div> <div> </div> <div>Olkoon </div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3E0%7B%2C%7D%5C%20a%5Cne1%5C%20ja%5C%20b%3E0" alt="a&gt;0{,}\ a\ne1\ ja\ b&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; 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>a-kantainen logaritmi <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_ab" alt="\log_ab"/>tarkoittaa lukua x, joka toteuttaa eksponenttiyhtälön <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Ex%3Db" alt="a^x=b"/></div> <div>Toisin sanoen</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%5Ex%3Db%5C%20%5Cleftrightarrow%5C%20%5Clog_ab%3Dx" alt="a^x=b\ \leftrightarrow\ \log_ab=x"/></div> <div> </div> <div>Vastaa kysymykseen: ''Mihin potenssiin kantalukua a täytyy korottaa, että tulokseksi saadaan b?''</div> <div> </div> <div>Logaritmin ominaisuuksia</div> <div>1) </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_a1%3D0%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cleft(a%5E%7B_0%7D%3D1%5Cright)%5Cright%7C" alt="\log_a1=0\ \ \ \ \ \left|\left(a^{_0}=1\right)\right|"/> <div>2)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_aa%3D1%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cleft(a%5E1%3Da%5Cright)%5Cright%7C" alt="\log_aa=1\ \ \ \ \left|\left(a^1=a\right)\right|"/><br/> <div> <div>3)</div> </div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_aa%5Ex%3Dx%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7Ca%5Ex%3Da%5Ex%5Cright%7C" alt="\log_aa^x=x\ \ \ \ \ \left|a^x=a^x\right|"/></div> <span>4)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%5E%7B%5Clog_ax%7D%3Dx" alt="a^{\log_ax}=x"/>  |(logaritmin määritelmän nojalla)|</div> <div> </div> <div>Huomautus</div> <div>Jos logaritmin kantaluku on 10, merkitään</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_%7B10%7D%5E%7B%20%7Dx%3D%5Clg%5C%20x%5Cleft(%3D%5Clog%5C%20x%5Cright)" alt="\log_{10}^{ }x=\lg\ x\left(=\log\ x\right)"/></div> <span>Esim. Määritä</span> <div>a)<img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_28" alt="\log_28"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ex%3D8" alt="2^x=8"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ex%3D2%5E3" alt="2^x=2^3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D3" alt="x=3"/> <div>On se luku, johon 2 pitää korottaa, jotta saadaa 8.</div> <div>Koska <img src="https://math-demo.abitti.fi/math.svg?latex=2%5E3%3D8" alt="2^3=8"/>, niin <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_28%3D3" alt="\log_28=3"/></div> <div><br/> <div>b)<img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_5%5Cfrac%7B1%7D%7B5%7D" alt="\log_5\frac{1}{5}"/></div> <div>Merkitään </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_5%5Cfrac%7B1%7D%7B5%7D%3Dy" alt="\log_5\frac{1}{5}=y"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=5%5Ey%3D%5Cfrac%7B1%7D%7B5%7D" alt="5^y=\frac{1}{5}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=5%5Ey%3D5%5E%7B-1%7D" alt="5^y=5^{-1}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D-1" alt="y=-1"/><br/> <div> </div> <div>Esim. Ratkaise yhtälö</div> <div>a) </div> <img src="https://math-demo.abitti.fi/math.svg?latex=3%5Ex%3D8" alt="3^x=8"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Clog_38%5Capprox1%7B%2C%7D89" alt="x=\log_38\approx1{,}89"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot5%5E%7B3x%7D%3D30%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3A2" alt="2\cdot5^{3x}=30\ \ \ \ \ \left|\right|:2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=5%5E%7B3x%7D%3D15" alt="5^{3x}=15"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D%5Clog_515%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3A3" alt="3x=\log_515\ \ \ \ \ \left|\right|:3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Clog_515%7D%7B3%7D%5Capprox0%7B%2C%7D56" alt="x=\frac{\log_515}{3}\approx0{,}56"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_3x%3D4" alt="\log_3x=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D3%5E4%3D81" alt="x=3^4=81"/><br/> <span>Lause</span> <div>Olkoon a&gt;0, a≠1. Kaikilla x&gt;0 ja y&gt;0 pätee</div> <div>a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_a%5Cleft(xy%5Cright)%3D%5Clog_ax%2B%5Clog_ay" alt="\log_a\left(xy\right)=\log_ax+\log_ay"/></div> <div>b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_a%5Cfrac%7Bx%7D%7By%7D%3D%5Clog_ax-%5Clog_ay" alt="\log_a\frac{x}{y}=\log_ax-\log_ay"/></div> <span>c)<br/> </span><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_ax%5Er%3Dr%5Clog_ax" alt="\log_ax^r=r\log_ax"/> <div> </div> <div>Todistus: a) ks. Kirja s.99</div> <div>Merkitään</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_ax%3Du%5C%20%5Crightarrow%5C%20a%5Eu%3Dx" alt="\log_ax=u\ \rightarrow\ a^u=x"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_ay%3Dv%5C%20%5Crightarrow%5C%20a%5Ev%3Dy" alt="\log_ay=v\ \rightarrow\ a^v=y"/> <div>Tällöin<br/> <div>b)<img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_a%5Cfrac%7Bx%7D%7By%7D%3D%5Clog_a%5Cfrac%7Ba%5Eu%7D%7Ba%5Ev%7D%3D%5Clog_aa%5E%7Bu-v%7D%3Du-v%3D%5Clog_ax-%5Clog_ay" alt="\log_a\frac{x}{y}=\log_a\frac{a^u}{a^v}=\log_aa^{u-v}=u-v=\log_ax-\log_ay"/></div> <div><br/> <div>c)<img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_ax%5Er%3D%5Clog_a%5Cleft(a%5Eu%5Cright)%5Er%3D%5Clog_aa%5E%7Bur%7D%3Dur%3Dru%3Dr%5Clog_ax" alt="\log_ax^r=\log_a\left(a^u\right)^r=\log_aa^{ur}=ur=ru=r\log_ax"/></div> <div> </div> </div> </div> <div>Esim. Sievennä</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_42%2B%5Clog_432%3D%5Clog_4%5Cleft(2%5Ccdot32%5Cright)%3D%5Clog_464%3D3" alt="\log_42+\log_432=\log_4\left(2\cdot32\right)=\log_464=3"/></div> <span>koska</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4%5E3%3D64" alt="4^3=64"/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_5%5Csqrt%5B7%5D%7B25%7D%3D%5Clog_525%5E%7B%5Cfrac%7B1%7D%7B7%7D%7D%3D%5Cfrac%7B1%7D%7B7%7D%5Clog_525%3D%5Cfrac%7B1%7D%7B7%7D%5Ccdot2%3D%5Cfrac%7B2%7D%7B7%7D" alt="\log_5\sqrt[7]{25}=\log_525^{\frac{1}{7}}=\frac{1}{7}\log_525=\frac{1}{7}\cdot2=\frac{2}{7}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_e3e%5E2%3D%5Clog_e3%2B%5Clog_ee%5E2%3D%5Clog_e3%2B2%5Clog_ee%3D%5Clog_e3%2B2%5Ccdot1%3D%5Clog_e3%2B2" alt="\log_e3e^2=\log_e3+\log_ee^2=\log_e3+2\log_ee=\log_e3+2\cdot1=\log_e3+2"/></div> <br/> <br/> <br/> <br/> <br/> <div> </div> </div> </div> 2019-01-30T09:42:35+02:00 https://peda.net/id/706f664a22c 2019-01-28T09:07:57+02:00 <div>Tehtävä:</div> <div>Tutki geogebralla onko olemassa kantalukua a, jolle eksponenttifunktio</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Da%5Ex" alt="f\left(x\right)=a^x"/></div> <div>ja sen dervaattafunktio f' ovat täsmälleen samat.</div> <div> </div> <div>V: On olemassa. Tämä luku on Neperin luku e=2,71828...</div> <div> </div> <div>Lause</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20e%5Ex%3De%5Ex" alt="D\ e^x=e^x"/></div> <br/> <span>Esim. Derivoi</span> <div>a)<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3e%5Ex%7D%7B4%7D" alt="\frac{3e^x}{4}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7B3e%5Ex%7D%7B4%7D%3D%5Cfrac%7B3%7D%7B4%7DD%5C%20e%5Ex%3D%5Cfrac%7B3%7D%7B4%7De%5Ex" alt="D\ \frac{3e^x}{4}=\frac{3}{4}D\ e^x=\frac{3}{4}e^x"/></div> <div> </div> <div>b)<img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(4e%5E%7B6x%7D%2B2e%5Cright)" alt="D\left(4e^{6x}+2e\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(4e%5E%7B6x%7D%2B2e%5Cright)%3D4D%5C%20e%5E%7B6x%7D%2B2D%5C%20e%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7Cs%5Cleft(x%5Cright)%3D6x%7B%2C%7D%5C%20u%5Cleft(x%5Cright)%3De%5Ex%7B%2C%7D%5C%20s%27%5Cleft(x%5Cright)%3D6" alt="D\left(4e^{6x}+2e\right)=4D\ e^{6x}+2D\ e\ \ \ \ \ \left|\right|s\left(x\right)=6x{,}\ u\left(x\right)=e^x{,}\ s'\left(x\right)=6"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D4%5Ccdot%20e%5E%7B6x%7D%5Ccdot6%2B0%3D24e%5E%7B6x%7D" alt="=4\cdot e^{6x}\cdot6+0=24e^{6x}"/> <div><br/> <div>c)<img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(-%5Cfrac%7B5%7D%7Be%5Ex%7D%5Cright)%3D-5%5Ccdot%20D%5C%20%5Cfrac%7B1%7D%7Be%5Ex%7D%3D-5e%5E%7B-x%7D" alt="D\left(-\frac{5}{e^x}\right)=-5\cdot D\ \frac{1}{e^x}=-5e^{-x}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D-5e%5E%7B-x%7D%3D-5%5Ccdot%20e%5E%7B-x%7D%5Ccdot-1%3D5e%5E%7B-x%7D%3D%5Cfrac%7B5%7D%7Be%5Ex%7D" alt="D-5e^{-x}=-5\cdot e^{-x}\cdot-1=5e^{-x}=\frac{5}{e^x}"/></div> <div> </div> <div>d)<img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2x%5E3e%5Ex" alt="f\left(x\right)=2x^3e^x"/>ja etsi paikalliset ääriarvokohdat.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D6x%5E2%5Ccdot%20e%5Ex%2B2x%5E3%5Ccdot%20e%5Ex%3De%5Ex%5Cleft(6x%5E2%2B3x%5E3%5Cright)" alt="f'\left(x\right)=6x^2\cdot e^x+2x^3\cdot e^x=e^x\left(6x^2+3x^3\right)"/></div> <div>Derivaatan nollakohdat</div> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex%3D0" alt="e^x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex%5Cne0" alt="e^x\ne0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(e%5Ex%3E0%5C%20kaikilla%5C%20x%5C%20arvoilla%5Cright)" alt="\left(e^x&gt;0\ kaikilla\ x\ arvoilla\right)"/><!--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>tai<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=6x%5E2%2B2x%5E3%3D0" alt="6x^2+2x^3=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%5Cleft(3%2Bx%5Cright)%3D0" alt="2x^2\left(3+x\right)=0"/> <div><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%3D0" alt="2x^2=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0" alt="x=0"/> <div>tai<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3%2Bx%3D0" alt="3+x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-3" alt="x=-3"/></div> <div> </div> <div>Selvitetään derivaatan f' merkit testipisteiden avulla.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0Ax%26f%27%5Cleft(x%5Cright)%26merkki%5C%5C%0A%5Chline%0A-4%26-32e%5E%7B-4%7D%26-%5C%5C%0A-1%264e%5E%7B-1%7D%26%2B%5C%5C%0A1%268e%26%2B%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;x&amp;f'\left(x\right)&amp;merkki\\&#10;\hline&#10;-4&amp;-32e^{-4}&amp;-\\&#10;-1&amp;4e^{-1}&amp;+\\&#10;1&amp;8e&amp;+&#10;\end{array}"/></div> <div>Kulkukaavio</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26%26-3%26%260%26%5C%5C%0A%5Chline%0Af%27%5Cleft(x%5Cright)%26-%26%26%2B%26%26%2B%5C%5C%0Af%5Cleft(x%5Cright)%26%5Csearrow%26%26%5Cnearrow%26%26%5Cnearrow%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;&amp;-3&amp;&amp;0&amp;\\&#10;\hline&#10;f'\left(x\right)&amp;-&amp;&amp;+&amp;&amp;+\\&#10;f\left(x\right)&amp;\searrow&amp;&amp;\nearrow&amp;&amp;\nearrow&#10;\end{array}"/></div> <div>Paikallinen minimikohta x=-3</div> <div>(x=0 on terassikohta)</div> <div> </div> <div> <div> </div> <div> <div> </div> <div> </div> <div> </div> </div> </div> </div> </div> </div> 2019-01-28T09:07:57+02:00 https://peda.net/id/efebda00207 2019-01-25T09:15:04+02:00 <div>Määritelmä</div> <div>Funktio </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Da%5Ex%7B%2C%7D%5C%20a%3E0%7B%2C%7D%5C%20a%5Cne1" alt="f\left(x\right)=a^x{,}\ a&gt;0{,}\ a\ne1"/><!--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;"--><span>on eksponenttifunktio</span> <div>- f(x) on jatkuva ja määritelty kaikilla x∈ℝ</div> <div>- Arvojoukko ]0,∞[</div> <div>- f(x) on kasvava, kun kantaluku a&gt;1 ja vähenevä 0&lt;a&lt;1</div> <div> </div> <div>314.</div> <div>Turkin väkiluku kasvaa vuosittain 700 000:lla eli 0,7 miljoonalla henkilöllä. Tansanian väkiluku kasvaa vuosittain 3% eli tulee 1,03 kertaiseksi. </div> <div>Taulukoidaan maiden väkilukuja</div> <div> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0AVuosi%26Turkki%26Tansania%5C%5C%0A%5Chline%0A2013%2675%5C%20%5Cleft(milj.%5Cright)%2649%5Cleft(milj.%5Cright)%5C%5C%0A2014%2675%2B0%7B%2C%7D7%2649%5Ccdot1%7B%2C%7D03%5C%5C%0A2015%2675%2B0%7B%2C%7D7%2B0%7B%2C%7D7%3D75%2B2%5Ccdot0%7B%2C%7D7%2649%5Ccdot1%7B%2C%7D03%5Ccdot1%7B%2C%7D03%3D49%5Ccdot1%7B%2C%7D03%5E2%5C%5C%0A2016%2675%2B3%5Ccdot0%7B%2C%7D7%2649%5Ccdot1.03%5E3%5C%5C%0A...%26%26%5C%5C%0A2013%2Bx%2675%2Bx%5Ccdot0%7B%2C%7D7%2649%5Ccdot1%7B%2C%7D03%5Ex%5C%5C%0A%26%26%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;Vuosi&amp;Turkki&amp;Tansania\\&#10;\hline&#10;2013&amp;75\ \left(milj.\right)&amp;49\left(milj.\right)\\&#10;2014&amp;75+0{,}7&amp;49\cdot1{,}03\\&#10;2015&amp;75+0{,}7+0{,}7=75+2\cdot0{,}7&amp;49\cdot1{,}03\cdot1{,}03=49\cdot1{,}03^2\\&#10;2016&amp;75+3\cdot0{,}7&amp;49\cdot1.03^3\\&#10;...&amp;&amp;\\&#10;2013+x&amp;75+x\cdot0{,}7&amp;49\cdot1{,}03^x\\&#10;&amp;&amp;&#10;\end{array}"/> <div>Turkin väkilukua kuvaa lineaarinen malli</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D0%7B%2C%7D7x%2B75" alt="f\left(x\right)=0{,}7x+75"/></div> <div>Tansanian väkilukua kuvaa eksponentiaalinen malli</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D49%5Ccdot1%7B%2C%7D03%5Ex" alt="g\left(x\right)=49\cdot1{,}03^x"/></div> <div>x on aika vuosina vuodesta 2013</div> <div>Väliluvut ovat yhtä suuret, kun on kulunut 20 vuotta vuodesta 2013 eli vuonna 2033</div> <div> </div> <div>312</div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3%5Ex%3D%5Cfrac%7B1%7D%7B9%7D" alt="3^x=\frac{1}{9}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=3%5Ex%3D%5Cfrac%7B1%7D%7B3%5E2%7D" alt="3^x=\frac{1}{3^2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3%5Ex%3D3%5E%7B-2%7D" alt="3^x=3^{-2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-2" alt="x=-2"/> 2019-01-25T09:15:04+02:00 https://peda.net/id/d91e969e207 2019-01-25T09:28:45+02:00 <span>270. </span> <div>Paraabelin pisteen P= (x,y) etäisyys origosta on </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=d%3D%5Csqrt%5B%5D%7B%5Cleft(x-0%5Cright)%5E2%2B%5Cleft(y-0%5Cright)%5E2%3D%5Csqrt%5B%5D%7Bx%5E2%2By%5E2%7D%7D" alt="d=\sqrt[]{\left(x-0\right)^2+\left(y-0\right)^2=\sqrt[]{x^2+y^2}}"/></div> <div>Piste P on paraabelilla, joten </div> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D3x-5x%5E2" alt="y=3x-5x^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=d%3D%5Csqrt%5B%5D%7Bx%5E2%2B%5Cleft(3x-5x%5E2%5Cright)%5E2%7D%7B%2C%7D%5C%20d%5Cge0%5C%20kaikilla%5C%20x%5Cin%5Cmathbb%7BR%7D" alt="d=\sqrt[]{x^2+\left(3x-5x^2\right)^2}{,}\ d\ge0\ kaikilla\ x\in\mathbb{R}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=d%3D%5Csqrt%5B%5D%7B25x%5E4-30x%5E3%2B10x%5E2%7D" alt="d=\sqrt[]{25x^4-30x^3+10x^2}"/><br/> <div> </div> <div>Etäisyys on suurin, kun juurrettava on suurin. Tutkitaan funktion </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D25x%5E4-30x%5E3%2B10x%5E2" alt="f\left(x\right)=25x^4-30x^3+10x^2"/></div> <span>Kulkua derivaattafunktion avulla</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D100x%5E3-90x%5E2%2B2x" alt="f'\left(x\right)=100x^3-90x^2+2x"/>(laskin)</div> <div>Funktio f on jatkuva välillä [<img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D%5C%20%5Cfrac%7B1%7D%7B2%7D" alt="0{,}\ \frac{1}{2}"/>] ja dervoituva välillä ]<img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D%5C%20%5Cfrac%7B1%7D%7B2%7D" alt="0{,}\ \frac{1}{2}"/>[. Funktio f saa suurimman arvonsa välin päätepisteissä tai välillä olevissa derivaatan nollakohdissa.</div> <div>Ratkaistaan derivaatan nollakohdat laskimella</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D0" alt="f'\left(x\right)=0"/>, kun x=0 tai<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B2%7D%7B5%7D" alt="x=\frac{2}{5}"/>tai<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1%7D%7B2%7D" alt="x=\frac{1}{2}"/></div> <div>Laketaan funktion f arvot derivaatan nollakohdissa ja välin [<img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D%5C%20%5Cfrac%7B1%7D%7B2%7D" alt="0{,}\ \frac{1}{2}"/>] päätepisteissä</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(0%5Cright)%3D0" alt="f\left(0\right)=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(%5Cfrac%7B2%7D%7B5%7D%5Cright)%3D%5Cfrac%7B8%7D%7B25%7D%5Capprox0%7B%2C%7D32" alt="f\left(\frac{2}{5}\right)=\frac{8}{25}\approx0{,}32"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%3D%5Cfrac%7B5%7D%7B16%7D%5Capprox0%7B%2C%7D3125" alt="f\left(\frac{1}{2}\right)=\frac{5}{16}\approx0{,}3125"/> <div>Pisteen P y-koordinaatti, kun x=<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%7D%7B5%7D" alt="\frac{2}{5}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=y%3D3%5Ccdot%5Cfrac%7B2%7D%7B5%7D-5%5Ccdot%5Cleft(%5Cfrac%7B2%7D%7B5%7D%5Cright)%5E2%3D%5Cfrac%7B2%7D%7B5%7D" alt="y=3\cdot\frac{2}{5}-5\cdot\left(\frac{2}{5}\right)^2=\frac{2}{5}"/></div> <div>V: Piste (<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%7D%7B5%7D%7B%2C%7D%5C%20%5Cfrac%7B2%7D%7B5%7D" alt="\frac{2}{5}{,}\ \frac{2}{5}"/>) on kauimpana origosta<br/> <div> </div> <div> </div> <div> </div> <div> </div> </div> 2019-01-25T09:28:45+02:00 https://peda.net/id/3a3d24ac17c 2019-01-14T09:30:11+02:00 <div>Määritelmä</div> <div>Kun a&gt;0 ja n=2,3,4, ... ,niin<img src="https://math-demo.abitti.fi/math.svg?latex=a%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D%3D%5Csqrt%5Bn%5D%7Ba%7D" alt="a^{\frac{1}{n}}=\sqrt[n]{a}"/></div> <div>Määritelmä</div> <div>Olkoon a&gt;0, m kokonaisluku ja n= 2,3,4, ... Tällöin</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%5E%7B%5Cfrac%7Bm%7D%7Bn%7D%7D%3D%5Csqrt%5Bn%5D%7Ba%5Em%7D%3D%5Cleft(%5Csqrt%5Bn%5D%7Ba%7D%5Cright)%5Em" alt="a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m"/></div> <div> </div> <div>Esim. Esitä murtopotenssia käyttäen.</div> <div>a) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7B7%5E2%7D%3D7%5E%7B%5Cfrac%7B2%7D%7B2%7D%7D%3D7" alt="\sqrt[]{7^2}=7^{\frac{2}{2}}=7"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Csqrt%5B5%5D%7B2%5E3%7D%3D2%5Ccdot2%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D%3D2%5E%7B1%2B%5Cfrac%7B5%7D%7B3%7D%7D%3D2%5E%7B%5Cfrac%7B8%7D%7B5%7D%7D" alt="2\sqrt[5]{2^3}=2\cdot2^{\frac{5}{3}}=2^{1+\frac{5}{3}}=2^{\frac{8}{5}}"/></div> <div>c) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3%7D%7B%5Csqrt%5B8%5D%7B9%7D%7D%3D%5Cfrac%7B3%7D%7B9%5E%7B%5Cfrac%7B1%7D%7B8%7D%7D%7D%3D3%5Ccdot%5Cfrac%7B1%7D%7B9%5E%7B%5Cfrac%7B1%7D%7B8%7D%7D%7D%3D3%5Ccdot9%5E%7B-%5Cfrac%7B1%7D%7B8%7D%7D%3D3%5Ccdot%5Cleft(3%5E2%5Cright)%5E%7B-%5Cfrac%7B1%7D%7B8%7D%7D%3D3%5E1%5Ccdot3%5E%7B%5E%7B-%5Cfrac%7B1%7D%7B4%7D%7D%7D%3D3%5E%7B%5E%7B%5Cfrac%7B3%7D%7B4%7D%7D%7D" alt="\frac{3}{\sqrt[8]{9}}=\frac{3}{9^{\frac{1}{8}}}=3\cdot\frac{1}{9^{\frac{1}{8}}}=3\cdot9^{-\frac{1}{8}}=3\cdot\left(3^2\right)^{-\frac{1}{8}}=3^1\cdot3^{^{-\frac{1}{4}}}=3^{^{\frac{3}{4}}}"/></div> <div> <div>d)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%5Csqrt%5B%5D%7Bx%7D%7D%7B3x%5E2%7D%3D%5Cfrac%7B2x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%7B3x%5E2%7D%3D%5Cfrac%7B2%7D%7B3%7D%5Ccdot%5Cfrac%7Bx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%7Bx%5E2%7D%3D%5Cfrac%7B2%7D%7B3%7Dx%5E%7B%5Cfrac%7B1%7D%7B2%7D-2%7D%3D%5Cfrac%7B2%7D%7B3%7Dx%5E%7B-%5Cfrac%7B3%7D%7B2%7D%7D" alt="\frac{2\sqrt[]{x}}{3x^2}=\frac{2x^{\frac{1}{2}}}{3x^2}=\frac{2}{3}\cdot\frac{x^{\frac{1}{2}}}{x^2}=\frac{2}{3}x^{\frac{1}{2}-2}=\frac{2}{3}x^{-\frac{3}{2}}"/><br/> <div> </div> <div>Esim. Esitä juurimerkijtää käyttäen, kun x&gt;0</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D%5Csqrt%5B4%5D%7Bx%7D" alt="x^{\frac{1}{4}}=\sqrt[4]{x}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E%7B-%5Cfrac%7B5%7D%7B2%7D%7D%3D%5Cfrac%7B1%7D%7Bx%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D%7D%3D%5Cfrac%7B1%7D%7B%5Csqrt%5B5%5D%7Bx%7D%7D%3D%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7Bx%5E2%5Ccdot%20x%5E2%5Ccdot%20x%7D%7D%3D%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7Bx%5E2%7D%5Ccdot%5Csqrt%5B%5D%7Bx%5E2%7D%5Ccdot%5Csqrt%5B%5D%7Bx%7D%7D%3D%5Cfrac%7B1%7D%7Bx%5E2%5Csqrt%5B%5D%7Bx%7D%7D" alt="x^{-\frac{5}{2}}=\frac{1}{x^{\frac{5}{2}}}=\frac{1}{\sqrt[5]{x}}=\frac{1}{\sqrt[]{x^2\cdot x^2\cdot x}}=\frac{1}{\sqrt[]{x^2}\cdot\sqrt[]{x^2}\cdot\sqrt[]{x}}=\frac{1}{x^2\sqrt[]{x}}"/></div> <div> Esim. Sievennä</div> <div>a)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B5%5D%7B-%5Cfrac%7B1%7D%7B32%7D%7D%3D%5Cfrac%7B%5Csqrt%5B5%5D%7B-1%7D%7D%7B%5Csqrt%5B5%5D%7B32%7D%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7COsam%C3%A4%C3%A4r%C3%A4n%5C%20juuri%3A%5C%20%5Csqrt%5Bn%5D%7B%5Cfrac%7Ba%7D%7Bb%7D%7D%3D%5Cfrac%7B%5Csqrt%5Bn%5D%7Ba%7D%7D%7B%5Csqrt%5Bn%5D%7Bb%7D%7D" alt="\sqrt[5]{-\frac{1}{32}}=\frac{\sqrt[5]{-1}}{\sqrt[5]{32}}\ \ \ \ \ \left|\right|Osamäärän\ juuri:\ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B-1%7D%7B%5Csqrt%5B5%5D%7B2%5E5%7D%7D%3D-%5Cfrac%7B1%7D%7B2%7D" alt="=\frac{-1}{\sqrt[5]{2^5}}=-\frac{1}{2}"/><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B4%5D%7B4%7D%3D%5Csqrt%5B4%5D%7B2%5Ccdot2%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7CTulon%5C%20juuri%3A%5C%20%5Csqrt%5Bn%5D%7Bab%7D%3D%5Csqrt%5Bn%5D%7Ba%7D%5Ccdot%5Csqrt%5Bn%5D%7Bb%7D" alt="\sqrt[4]{4}=\sqrt[4]{2\cdot2}\ \ \ \ \ \left|\right|Tulon\ juuri:\ \sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Csqrt%5B4%5D%7B2%7D%5Ccdot%5Csqrt%5B4%5D%7B2%7D%3D2%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%5Ccdot2%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D2%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%3D%5Csqrt%5B%5D%7B2%7D" alt="=\sqrt[4]{2}\cdot\sqrt[4]{2}=2^{\frac{1}{4}}\cdot2^{\frac{1}{4}}=2^{\frac{1}{2}}=\sqrt[]{2}"/></div> 2019-01-14T09:30:11+02:00