https://peda.net/id/5bc95cdcc3d 2019-08-21T08:27:25+03:00 https://peda.net/:static/535/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/a53ab44adef 2019-09-24T20:42:14+03:00 <span>202 </span> <div>a) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D3%5Ccdot%5Cleft(3x-1%5Cright)%5E5dx%3D%5Cfrac%7B1%7D%7B6%7D%5Cleft(3x-1%5Cright)%5E6%2BC" alt="\int_{ }^{ }3\cdot\left(3x-1\right)^5dx=\frac{1}{6}\left(3x-1\right)^6+C"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D6x%5Cleft(3x%5E2-5%5Cright)%5E4dx%3D%5Cfrac%7B1%7D%7B5%7D%5Cleft(3x%5E2-5%5Cright)%5E5%2BC" alt="\int_{ }^{ }6x\left(3x^2-5\right)^4dx=\frac{1}{5}\left(3x^2-5\right)^5+C"/></div> <div> </div> <div>203</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D15%5Cleft(3x%2B2%5Cright)%5E3dx%3D%5Cfrac%7B5%7D%7B4%7D%5Cleft(3x%2B2%5Cright)%5E4%2BC" alt="\int_{ }^{ }15\left(3x+2\right)^3dx=\frac{5}{4}\left(3x+2\right)^4+C"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(4x-6%5Cright)%5E4dx%3D%5Cfrac%7B1%7D%7B20%7D%5Cleft(4x-6%5Cright)%5E5%2BC" alt="\int_{ }^{ }\left(4x-6\right)^4dx=\frac{1}{20}\left(4x-6\right)^5+C"/></div> <div> </div> <div>204</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D6x%5E2%5Cleft(x%5E3%2B3%5Cright)%5E6dx%3D2%5Cint_%7B%20%7D%5E%7B%20%7D3x%5E2%5Cleft(x%5E3%2B3%5Cright)%5E6%3D%5Cfrac%7B2%7D%7B7%7D%5Cleft(x%5E3%2B3%5Cright)%5E7%2BC" alt="\int_{ }^{ }6x^2\left(x^3+3\right)^6dx=2\int_{ }^{ }3x^2\left(x^3+3\right)^6=\frac{2}{7}\left(x^3+3\right)^7+C"/></div> <span>b)</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(%5Cfrac%7Bx%7D%7B3%7D%2B1%5Cright)%5E5dx%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft(%5Cfrac%7Bx%7D%7B3%7D%2B1%5Cright)%5E6%2BC" alt="\int_{ }^{ }\left(\frac{x}{3}+1\right)^5dx=\frac{1}{2}\left(\frac{x}{3}+1\right)^6+C"/><br/> 2019-09-24T20:42:14+03:00 https://peda.net/id/ecb078f0d5a 2019-09-24T20:18:24+03:00 <div>534</div> <div>Poikkileikkauksen pinta-ala: <img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(x%5Cright)%3Dy%5E2%3D%5Cleft(2%5Ccdot4%7B%2C%7D25e%5E%7B2%7B%2C%7D67-0%7B%2C%7D889x%7D%5Cright)%5E2" alt="A\left(x\right)=y^2=\left(2\cdot4{,}25e^{2{,}67-0{,}889x}\right)^2"/></div> <div>Tilavuus välillä </div> <img src="https://math-demo.abitti.fi/math.svg?latex=0%5Cle%20x%5Cle300" alt="0\le x\le300"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cint_0%5E%7B300%7D%5Cleft(2%5Ccdot4%7B%2C%7D25e%5E%7B%5E%7B2%7B%2C%7D67-0%7B%2C%7D0089x%7D%7D%5Cright)%5E2dx%3D842%5C%20292m%5E3" alt="V=\int_0^{300}\left(2\cdot4{,}25e^{^{2{,}67-0{,}0089x}}\right)^2dx=842\ 292m^3"/><br/> <div>Tilavuus olisi n.840 000m³.</div> <div> </div> <div>535</div> <div> <div>a) Poikkileikkausneliön sivun pituus kohdassa x on x²</div> <div>b) Poikkileikkausneliön pinta-ala on<img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(x%5Cright)%3Dx%5E2%5Ccdot%20x%5E2%3Dx%5E4" alt="A\left(x\right)=x^2\cdot x^2=x^4"/></div> <div>c) Kappaleen tilavuus on </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5E1x%5E4dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E1%5Cfrac%7B1%7D%7B5%7Dx%5E5%3D%5Cfrac%7B1%7D%7B5%7D" alt="\int_0^1x^4dx=\bigg/_{\!\!\!\!\!0}^1\frac{1}{5}x^5=\frac{1}{5}"/><br/> <div> </div> <div>537</div> <div>a) Korkeus on molemmissa kappaleissa sama, 2. </div> <div>Kappale A: Poikkileikkaukset ovat ympyröitä, joiden halkaisija kohdassa x on <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%7D%7Bx%7D" alt="\frac{2}{x}"/></div> <div>Kappale B: Poikkileikkaukset ovat ympyröitä, joiden halkaisija x on <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot%5Cfrac%7B1%7D%7Bx%7D%3D%5Cfrac%7B2%7D%7Bx%7D" alt="2\cdot\frac{1}{x}=\frac{2}{x}"/>.</div> <div>Koska molempien poikkileikkausympyröiden halkaisijat ovat samat, ovat myös pinta-alat samat ja siten myös kappaleiden tilavuudet ovat samat.</div> <div> </div> <div>b) Tilavuus on</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%5Cleft(x%5Cright)%3D%5Cpi%5Cint_1%5E3%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)%5E2dx%3D%5Cfrac%7B2%5Cpi%7D%7B3%7D" alt="V\left(x\right)=\pi\int_1^3\left(\frac{1}{x}\right)^2dx=\frac{2\pi}{3}"/>(Laskin)</div> <div> </div> </div> <div> </div> </div> 2019-09-13T00:11:44+03:00 https://peda.net/id/ac6af4f8d25 2019-09-24T19:46:45+03:00 <div>504</div> <div>a)</div> <div>Tilavuus<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cpi%5Cint_1%5E5%5Cleft(%5Csqrt%5B%5D%7Bx-1%7D%5Cright)%5E2dx%3D8%5Cpi" alt="V=\pi\int_1^5\left(\sqrt[]{x-1}\right)^2dx=8\pi"/><span>(Laskin)</span></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cpi%5Cint_1%5E4%5Cleft(x-2%5Cright)%5E2dx%3D%5Cpi%5Cint_1%5E4%5Cleft(x%5E2-4x%2B4%5Cright)dx%3D%5Cpi%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!1%7D%5E4%5Cleft(%5Cfrac%7B1%7D%7B3%7Dx%5E3-2x%5E2%2B4x%5Cright)%3D3%5Cpi" alt="V=\pi\int_1^4\left(x-2\right)^2dx=\pi\int_1^4\left(x^2-4x+4\right)dx=\pi\bigg/_{\!\!\!\!\!1}^4\left(\frac{1}{3}x^3-2x^2+4x\right)=3\pi"/></div> <br/> <div> </div> <div>502</div> <div>a) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3D2%5Ccdot%5Cpi%5Cint_0%5E2f%5Cleft(x%5Cright)%5E2dx%3D2%5Cpi%5C%20%5Cint_0%5E2x%5E2dx%3D2%5Cpi%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E2%5Cfrac%7B1%7D%7B3%7Dx%5E3%3D2%5Cpi%5Cleft(%5Cfrac%7B1%7D%7B3%7D2%5E3-%5Cfrac%7B1%7D%7B3%7D0%5E3%5Cright)%3D%5Cfrac%7B16%5Cpi%7D%7B3%7D" alt="V=2\cdot\pi\int_0^2f\left(x\right)^2dx=2\pi\ \int_0^2x^2dx=2\pi\bigg/_{\!\!\!\!\!0}^2\frac{1}{3}x^3=2\pi\left(\frac{1}{3}2^3-\frac{1}{3}0^3\right)=\frac{16\pi}{3}"/></div> <span>b)</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cpi%5Cint_%7B-1%7D%5E2f%5Cleft(x%5Cright)%5E2dx%3D%5Cpi%5Cint_%7B-1%7D%5E2%5Cleft(%5Cfrac%7B1%7D%7B2%7Dx%5E3%5Cright)%5E2dx%3D%5Cpi%5Cint_%7B-1%7D%5E2%5Cfrac%7B1%7D%7B4%7Dx%5E6dx%3D%5Cpi%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5E2%5Cfrac%7B1%7D%7B28%7Dx%5E7%3D%5Cpi%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7B28%7D%5Ccdot2%5E7-%5Cfrac%7B1%7D%7B28%7D%5Ccdot%5Cleft(-1%5Cright)%5E7%5Cright)%3D%5Cfrac%7B129%7D%7B28%7D%5Cpi" alt="V=\pi\int_{-1}^2f\left(x\right)^2dx=\pi\int_{-1}^2\left(\frac{1}{2}x^3\right)^2dx=\pi\int_{-1}^2\frac{1}{4}x^6dx=\pi\bigg/_{\!\!\!\!\!{-1}}^2\frac{1}{28}x^7=\pi\cdot\left(\frac{1}{28}\cdot2^7-\frac{1}{28}\cdot\left(-1\right)^7\right)=\frac{129}{28}\pi"/><br/> <div> </div> <div>503</div> <div>a) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cpi%5Cint_1%5E3f%5Cleft(x%5Cright)%5E2dx%3D%5Cpi%5Cint_1%5E3%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)%5E2dx%3D%5Cpi%5Cint_1%5E3x%5E%7B-2%7Ddx%3D%5Cpi%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!1%7D%5E3-%5Cfrac%7B1%7D%7Bx%7D%3D%5Cpi%5Cleft(-%5Cfrac%7B1%7D%7B3%7D-%5Cleft(-1%5Cright)%5Cright)%3D%5Cfrac%7B2%5Cpi%7D%7B3%7D" alt="V=\pi\int_1^3f\left(x\right)^2dx=\pi\int_1^3\left(\frac{1}{x}\right)^2dx=\pi\int_1^3x^{-2}dx=\pi\bigg/_{\!\!\!\!\!1}^3-\frac{1}{x}=\pi\left(-\frac{1}{3}-\left(-1\right)\right)=\frac{2\pi}{3}"/></div> <span>b)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cpi%5C%20%5Cint_1%5E2f%5Cleft(x%5Cright)%5E2dx%3D%5Cpi%5Cint_1%5E2%5Cleft(%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7Bx%7D%7D%5Cright)%5E2dx%3D%5Cpi%5Cint_1%5E2%5Cleft(x%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D%5Cright)%5E2%3D%5Cpi%5Cint_1%5E2x%5E%7B-1%7Ddx%3D%5Cpi%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!1%7D%5E2%5Cln%20x%3D%5Cpi%5Cleft(%5Cln2-%5Cln1%5Cright)%3D%5Cpi%5Cln2" alt="V=\pi\ \int_1^2f\left(x\right)^2dx=\pi\int_1^2\left(\frac{1}{\sqrt[]{x}}\right)^2dx=\pi\int_1^2\left(x^{-\frac{1}{2}}\right)^2=\pi\int_1^2x^{-1}dx=\pi\bigg/_{\!\!\!\!\!1}^2\ln x=\pi\left(\ln2-\ln1\right)=\pi\ln2"/><br/> <br/> <div>505</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cint_0%5E1%5Cpi%5Cleft(e%5Ex-2%5Cright)%5E2dx%3D%5Cpi%5Cint_0%5E1%5Cleft(e%5E%7B2x%7D-4e%5Ex%2B4%5Cright)dx%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cleft(e%5E2-8e%2B15%5Cright)" alt="V=\int_0^1\pi\left(e^x-2\right)^2dx=\pi\int_0^1\left(e^{2x}-4e^x+4\right)dx=\frac{\pi}{2}\left(e^2-8e+15\right)"/></div> 508<br/> <div>Lasketaan funktioiden leikkauspisteet</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%2B1%3D%5Cleft(x-1%5Cright)%5E2" alt="x+1=\left(x-1\right)^2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2B1%3Dx%5E2-2x%2B1" alt="x+1=x^2-2x+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3Dx%5E2-2x" alt="x=x^2-2x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-3x%3D0" alt="x^2-3x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0%5C%20tai%5C%20x%3D3" alt="x=0\ tai\ x=3"/><span>(Laskin)</span> <div>Lasketaan rajattu alueen pinta-ala välillä [0,3]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cpi%5Cint_0%5E3%5Cleft(x%5E2-3x%5Cright)%5E2dx%3D%5Cfrac%7B72%5Cpi%7D%7B5%7D" alt="\pi\int_0^3\left(x^2-3x\right)^2dx=\frac{72\pi}{5}"/>(Laskin)<br/> <br/> </div> <br/> 509<br/> <div>Lasketaan funktioiden leikkauspisteet</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B4%7Dx%5E2%2B2%3D%5Cfrac%7B1%7D%7B2%7Dx%5E2%2B1" alt="\frac{1}{4}x^2+2=\frac{1}{2}x^2+1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-2%5C%20tai%5C%20x%3D2" alt="x=-2\ tai\ x=2"/><br/> <div>Lasketaan pyörähdyskappaleen tilavuus välillä [-2,2]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cpi%5Cint_%7B-2%7D%5E2%5Cleft(%5Cfrac%7B1%7D%7B4%7Dx%5E2%2B2%5Cright)%5E2dx-%5Cpi%5Cint_%7B-2%7D%5E2%5Cleft(%5Cfrac%7B1%7D%7B2%7Dx%5E2%2B1%5Cright)%5E2dx%3D%5Cfrac%7B48%5Cpi%7D%7B5%7D" alt="V=\pi\int_{-2}^2\left(\frac{1}{4}x^2+2\right)^2dx-\pi\int_{-2}^2\left(\frac{1}{2}x^2+1\right)^2dx=\frac{48\pi}{5}"/><br/> <br/> </div> 510<br/> <div>a)</div> <span>Käyrien </span><img src="https://math-demo.abitti.fi/math.svg?latex=y_1%3De%5E%7B%5Cfrac%7Bx%7D%7B2%7D%7D" alt="y_1=e^{\frac{x}{2}}"/><span>ja </span><img src="https://math-demo.abitti.fi/math.svg?latex=y_2%3De%5E%7B-%5Cfrac%7Bx%7D%7B2%7D%7D" alt="y_2=e^{-\frac{x}{2}}"/><span>leikkauskohdat</span><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B%5Cfrac%7Bx%7D%7B2%7D%7D%3De%5E%7B-%5Cfrac%7Bx%7D%7B2%7D%7D" alt="e^{\frac{x}{2}}=e^{-\frac{x}{2}}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%7D%7B2%7D%3D-%5Cfrac%7Bx%7D%7B2%7D" alt="\frac{x}{2}=-\frac{x}{2}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-x" alt="x=-x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0" alt="x=0"/><br/> <div> </div> <div>Lasketaan kappaleen tilaavus välillä [0,1]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cpi%5Cint_0%5E1%5Cleft(y_1%5Cright)%5E2dx-%5Cpi%5Cint_0%5E1%5Cleft(y_2%5Cright)%5E2dx%3D%5Cpi%5Cleft(e-1%5Cright)-%5Cleft(-%5Cpi%5Cleft(%5Cfrac%7B1%7D%7Be%7D-1%5Cright)%5Cright)%3D%5Cpi%5Cleft(e%2B%5Cfrac%7B1%7D%7Be%7D-2%5Cright)" alt="\pi\int_0^1\left(y_1\right)^2dx-\pi\int_0^1\left(y_2\right)^2dx=\pi\left(e-1\right)-\left(-\pi\left(\frac{1}{e}-1\right)\right)=\pi\left(e+\frac{1}{e}-2\right)"/></div> <div>b)</div> <div>Käyrät eivät leikkaa, koska yhtälöllä<img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7Bx%7D%3D%5Csqrt%5B%5D%7Bx%2B1%7D" alt="\sqrt[]{x}=\sqrt[]{x+1}"/>ei ole ratkaisuja</div> <div>Käyrän <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7Bx%7D" alt="y=\sqrt[]{x}"/>nollakohta on x=0 ja käyrän <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7Bx%2B1%7D" alt="y=\sqrt[]{x+1}"/>nollakohta on x=-1</div> <div>Koska välillä [−1, 1] olevassa testipisteessä x=0, </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7B0%2B1%7D%3D1%3E%5Csqrt%5B%5D%7B0%7D%3D0" alt="\sqrt[]{0+1}=1&gt;\sqrt[]{0}=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;"--><span>, on käyrä </span><img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7Bx%2B1%7D" alt="y=\sqrt[]{x+1}"/><span>ylempänä kuin käyrä</span><img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7Bx%7D" alt="y=\sqrt[]{x}"/><span>.</span><br/> <div>Käyrä <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7Bx%7D" alt="y=\sqrt[]{x}"/> pyörähtää välillä [0,1] ja käyrä <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7Bx%2B1%7D" alt="y=\sqrt[]{x+1}"/>välillä [-1,1]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_1%3D%5Cpi%5Cint_%7B-1%7D%5E1%5Cleft(%5Csqrt%5B%5D%7Bx%2B1%7D%5Cright)%5E2dx%3D2%5Cpi" alt="V_1=\pi\int_{-1}^1\left(\sqrt[]{x+1}\right)^2dx=2\pi"/>(Laskin)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_2%3D%5Cpi%5Cint_0%5E1%5Cleft(%5Csqrt%5B%5D%7Bx%7D%5Cright)%5E2dx%3D%5Cfrac%7B1%7D%7B2%7D%5Cpi" alt="V_2=\pi\int_0^1\left(\sqrt[]{x}\right)^2dx=\frac{1}{2}\pi"/>(Laskin)</div> <div>Kysytyn pyörähdyskappaleen tilavuus on</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3DV_1-V_2%3D%5Cfrac%7B3%5Cpi%7D%7B2%7D" alt="V=V_1-V_2=\frac{3\pi}{2}"/></div> <br/> <div>512</div> <div>a)</div> <div>Muutetaan ellipsin yhtälö <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2B1%7B%2C%7D7y%5E2%3D100" alt="x^2+1{,}7y^2=100"/>muotoon <img src="https://math-demo.abitti.fi/math.svg?latex=y%3Df%5Cleft(x%5Cright)" alt="y=f\left(x\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2B1%7B%2C%7D7y%5E2%3D100" alt="x^2+1{,}7y^2=100"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%7B%2C%7D7y%5E2%3D100-x%5E2" alt="1{,}7y^2=100-x^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3Df%5Cleft(x%5Cright)%3D%5Csqrt%5B%5D%7B%5Cfrac%7B100-x%5E2%7D%7B1%7B%2C%7D7%7D%7D" alt="y=f\left(x\right)=\sqrt[]{\frac{100-x^2}{1{,}7}}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cpi%5Cint_%7B-10%7D%5E%7B10%7D%5Cleft(%5Csqrt%5B%5D%7B%5Cfrac%7B100-x%5E2%7D%7B1%7B%2C%7D7%7D%7D%5Cright)%5E2dx%3D%5Cpi%5Cint_%7B-10%7D%5E%7B10%7D%5Cfrac%7B100-x%5E2%7D%7B1%7B%2C%7D7%7Ddx%3D2463.9942...cm%5E3%5Capprox2500cm%5E3%3D2%7B%2C%7D5l" alt="V=\pi\int_{-10}^{10}\left(\sqrt[]{\frac{100-x^2}{1{,}7}}\right)^2dx=\pi\int_{-10}^{10}\frac{100-x^2}{1{,}7}dx=2463.9942...cm^3\approx2500cm^3=2{,}5l"/></div> <div>b)</div> <div>Tulos on järkevä, sillä se on hieman vähemmän kuin 2640cm³</div> <div>c)</div> <div>Emmä jaksa<br/> <br/> 513<br/> a)<br/> <span>Pyörähdyskappaleen tilavuus ratkaistaan integroimalla muuttujan y suhteen.  </span><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cpi%5Cint_1%5E2%5Cleft(2y%5Cright)%5E2dy%3D%5Cfrac%7B28%7D%7B3%7D%5Cpi" alt="V=\pi\int_1^2\left(2y\right)^2dy=\frac{28}{3}\pi"/><br/> b)</div> Pyörähdyskappaleen tilavuus ratkaistaan<br/> integroimalla muuttujan y suhteen.<br/> Ratkaistaan käyrän yhtälö muuttujan x<br/> suhteen. <br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7Bx%7D" alt="y=\sqrt[]{x}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3Dy%5E2" alt="x=y^2"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cpi%5Cint_0%5E2%5Cleft(y%5E2%5Cright)%5E2dy%3D%5Cfrac%7B32%5Cpi%7D%7B5%7D" alt="V=\pi\int_0^2\left(y^2\right)^2dy=\frac{32\pi}{5}"/>(Laskin)</div> <br/> <span>514</span> <div>a)Lasketaan kahvimukin vetoisuus eli sisätilavus.</div> <div>Sisätilavuus saadaan, kun sisempi suora y=6x-18 pyörähtää y-akselin ympäri</div> <div> </div> <div>Ratkaistaan suoran yhtälöstä x</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=y%3D6x-18" alt="y=6x-18"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=6x%3Dy%2B18" alt="6x=y+18"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1%7D%7B6%7Dy%2B3" alt="x=\frac{1}{6}y+3"/> <div>Mukin pohjan paksuus on 1,0cm, joten integroimisväli on [1,11]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_%7Bsis%C3%A4%7D%3D%5Cpi%5Cint_1%5E%7B11%7D%5Cleft(%5Cfrac%7B1%7D%7B6%7Dy%2B3%5Cright)%5E%7B%5E2%7Ddy%3D%5Cfrac%7B8765%5Cpi%7D%7B54%7D%5Capprox509%7B%2C%7D927%5C%20%5Cleft(Laskin%5Cright)" alt="V_{sisä}=\pi\int_1^{11}\left(\frac{1}{6}y+3\right)^{^2}dy=\frac{8765\pi}{54}\approx509{,}927\ \left(Laskin\right)"/></div> <div>Kahvin vetoisuus on 509,927cm³=0,509927dm³=</div> <div>b)</div> <div>Lasketaan ulkotilavuus</div> <div>Ulkotilavuus saadaan, kun ulompi suora y=6x-20 pyörähtää y-akselin ympäri</div> <div>Ratkaistaan suoran yhtälöstä x</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=y%3D6x-20" alt="y=6x-20"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1%7D%7B6%7Dy%2B%5Cfrac%7B10%7D%7B3%7D" alt="x=\frac{1}{6}y+\frac{10}{3}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_%7Bulko%7D%3D%5Cpi%5Cint_0%5E%7B11%7D%5Cleft(%5Cfrac%7B1%7D%7B6%7Dy%2B%5Cfrac%7B10%7D%7B3%7D%5Cright)%5E2dy%3D%5Cfrac%7B21791%7D%7B108%7D%5Cpi%5Capprox633%7B%2C%7D874%5Cleft(Laskin%5Cright)" alt="V_{ulko}=\pi\int_0^{11}\left(\frac{1}{6}y+\frac{10}{3}\right)^2dy=\frac{21791}{108}\pi\approx633{,}874\left(Laskin\right)"/></div> <div> <div>Posliinin määrä saadaan tilavuuksien erotuksena.</div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_%7Bulko%7D-V_%7Bsis%C3%A4%7D%3D%5Cfrac%7B21791%5Cpi%7D%7B108%7D-%5Cfrac%7B8765%5Cpi%7D%7B54%7D%3D%5Cfrac%7B4261%5Cpi%7D%7B108%7D%5Capprox123%7B%2C%7D947" alt="V_{ulko}-V_{sisä}=\frac{21791\pi}{108}-\frac{8765\pi}{54}=\frac{4261\pi}{108}\approx123{,}947"/></div> <div>Posliinin määrä on n.120cm³</div> </div> <div><br/> 516<br/> <div>Kappaleen ulompi osa on ”kiekko”, joka syntyy, kun suora x = 5 pyörähtää y-akselin ympäri ja sisempi osa on ”malja”, joka syntyy, kun käyrä<img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7B3x-6%7D" alt="y=\sqrt[]{3x-6}"/> pyörähtää y-akselin ympäri</div> <div>Sisempi osa:</div> <div>Ratkaistaan yhtälöstä<img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7B3x-6%7D" alt="y=\sqrt[]{3x-6}"/>muuttuja x:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7B3x-6%7D" alt="y=\sqrt[]{3x-6}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=y%5E2%3D3x-6" alt="y^2=3x-6"/><span>, josta </span><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7By%5E2%2B6%7D%7B3%7D" alt="x=\frac{y^2+6}{3}"/><br/> <div>Integroinin yläraja, kun x=5</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7B3%5Ccdot5-6%7D%3D%5Csqrt%5B%5D%7B9%7D%3D3" alt="y=\sqrt[]{3\cdot5-6}=\sqrt[]{9}=3"/></div> <div>Sisemmän pyörähdyskappaleen tilavuus on </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_%7Bsisempi%7D%3D%5Cpi%5Cint_0%5E3%5Cleft(%5Cfrac%7By%5E2%2B6%7D%7B3%7D%5Cright)%5E2dy%3D%5Cfrac%7B147%5Cpi%7D%7B5%7D" alt="V_{sisempi}=\pi\int_0^3\left(\frac{y^2+6}{3}\right)^2dy=\frac{147\pi}{5}"/>(Laskin)</div> <div>Ulompi pyörähdyskappale on suora lieriö, jonka pohjan säde on 5 ja korkeus on 3. </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_%7Bulompi%7D%3D%5Cpi%5Ccdot5%5E2%5Ccdot3%3D75%5Cpi" alt="V_{ulompi}=\pi\cdot5^2\cdot3=75\pi"/></div> <div>Koko kappaleen tilavuus</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_%7Bulompi%7D-V_%7Bsisempi%7D-75%5Cpi-%5Cfrac%7B147%5Cpi%7D%7B5%7D-%5Cfrac%7B228%5Cpi%7D%7B5%7D" alt="V_{ulompi}-V_{sisempi}-75\pi-\frac{147\pi}{5}-\frac{228\pi}{5}"/></div> </div> <div>517</div> <div> <div>Lasketaan funktio <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7B2x-1%7D" alt="y=\sqrt[]{2x-1}"/>nollakohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7B2x-1%7D%3D0" alt="\sqrt[]{2x-1}=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=2x-1%3D0" alt="2x-1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%3D1" alt="2x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1%7D%7B2%7D" alt="x=\frac{1}{2}"/> <div>Eli funktion ylä-raja alkaa pisteestä x=1/2 ja ala-raja on jokin piste a</div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cpi%5Cint_%7B%5Cfrac%7B1%7D%7B2%7D%7D%5Ea%5Cleft(%5Csqrt%5B%5D%7B2x-1%7D%5Cright)%5E2dx%3D%5Cfrac%7B%5Cleft(4a%5E2-4a%2B1%5Cright)%5Ccdot%5Cpi%7D%7B4%7D" alt="V=\pi\int_{\frac{1}{2}}^a\left(\sqrt[]{2x-1}\right)^2dx=\frac{\left(4a^2-4a+1\right)\cdot\pi}{4}"/>(Laskin)</div> <div>Ratkaistaan yhtälö</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3D4%5Cpi" alt="V=4\pi"/><br/> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft(4a%5E2-4a%2B1%5Cright)%5Ccdot%5Cpi%7D%7B4%7D%3D4%5Cpi%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3A%5Cpi" alt="\frac{\left(4a^2-4a+1\right)\cdot\pi}{4}=4\pi\ \ \ \ \ \left|\right|:\pi"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D-%5Cfrac%7B3%7D%7B2%5C%20%7D%5C%20tai%5C%20a%3D%5Cfrac%7B5%7D%7B2%7D" alt="a=-\frac{3}{2\ }\ tai\ a=\frac{5}{2}"/></div> </div> </div> <br/> 518<br/> <span>Lasketaan ensin ulomman kappaleen tilavuus, joka syntyy kun suora y=4 pyörähtää x-akselin ympäri ja sitten sisemmän kappaleen tilavuus, joka syntyy kun paraabeli </span><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1%2By%5E2" alt="x=1+y^2"/><span>pyörähtää x-akselin ympäri.  </span> <div><br/> <div>Ratkaistaan yhtälö muuttujan y suhteen</div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1%2By%5E2" alt="x=1+y^2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=y%5E2%3Dx-1" alt="y^2=x-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Cpm%5Csqrt%5B%5D%7Bx-1%7D" alt="y=\pm\sqrt[]{x-1}"/><br/> <div> </div> <div>Voidaan valita pyörähtäväks käyräksi </div> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7Bx-1%7D" alt="y=\sqrt[]{x-1}"/><br/> <div>Käyrän <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D4" alt="y=4"/>ja<img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7Bx-1%7D" alt="y=\sqrt[]{x-1}"/> leikkauskohta on </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4%3D%5Csqrt%5B%5D%7Bx-1%7D" alt="4=\sqrt[]{x-1}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x-1%3D16" alt="x-1=16"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D17" alt="x=17"/> <div> </div> <div>Käyrän <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Csqrt%5B%5D%7Bx-1%7D" alt="y=\sqrt[]{x-1}"/>ja x-akselin leikkauskohta on x=1</div> <div>Tilavuus saadaan pyörähdyskappaleiden tilavuutena käyristä y=4 välillä [1,17]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_%7Bulompi%7D%3D%5Cpi%5Cint_0%5E%7B17%7D4%5E2dx%3D272%5Cpi" alt="V_{ulompi}=\pi\int_0^{17}4^2dx=272\pi"/>(Laskin)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_%7Bsisempi%7D%3D%5Cpi%5Cint_1%5E%7B17%7D%5Csqrt%5B%5D%7Bx-1%7Ddx%3D128%5Cpi" alt="V_{sisempi}=\pi\int_1^{17}\sqrt[]{x-1}dx=128\pi"/>(Laskin)<br/> <div> </div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_%7Bulompi%7D-V_%7Bsisempi%7D%3D289%5Cpi-128%5Cpi%3D144%5Cpi%5Capprox452%7B%2C%7D389" alt="V_{ulompi}-V_{sisempi}=289\pi-128\pi=144\pi\approx452{,}389"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=452%7B%2C%7D389cm%5E3%3D0%7B%2C%7D452389dm%5E3%3D0%7B%2C%7D000452389m%5E3" alt="452{,}389cm^3=0{,}452389dm^3=0{,}000452389m^3"/><br/> <div>Lasketaan maljakon massa m tiheyden ρ avulla</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Crho%3D%5Cfrac%7Bm%7D%7BV%7D%7B%2C%7D%5C%20josta%5C%20m%3D%5Crho%20V" alt="\rho=\frac{m}{V}{,}\ josta\ m=\rho V"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=m%3D3600%5C%20%5Cfrac%7Bkg%7D%7Bm%5E3%7D%5Ccdot0%7B%2C%7D000452389m%5E3%3D1%7B%2C%7D6286kg%5Capprox1%7B%2C%7D6kg" alt="m=3600\ \frac{kg}{m^3}\cdot0{,}000452389m^3=1{,}6286kg\approx1{,}6kg"/><br/> <div>Maljakon massa on 1,6 kg</div> <br/> 520<br/> <span>Tangentin kulmakerroin:</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cfrac%7B1%7D%7Bx%7D%3DDx%5E%7B-1%7D%3D-x%5E%7B-2%7D%3D-%5Cfrac%7B1%7D%7Bx%5E2%7D" alt="D\frac{1}{x}=Dx^{-1}=-x^{-2}=-\frac{1}{x^2}"/></div> <div>Kulmakertoimen arvo, kun x=1.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=k%3D-%5Cfrac%7B1%7D%7B1%5E2%7D%3D-1" alt="k=-\frac{1}{1^2}=-1"/> Tangentti kulkee pisteen (1,1) kautta</div> <div>Tangentin yhtälö on y-1=-(x-1), josta y=-x+1</div> <div>Pyörähtävä alue jää käyrän y=1/x ja y=-x+2 väliin välillä [1,2]</div> <div>Kappaleen tilavuus saadaan kun ulomman käyrän y=1/x pyörähtäessä syntyvästä kappaleesta vähennetään tangentin y=−x + 2 pyörähtäessä syntyvän kappaleen tilavuus. Pyörähdyskappaleen tilavuus:  </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_%7Bulompi%7D%3D%5Cpi%5Cint_1%5E2%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)%5E2dx%3D%5Cfrac%7B1%7D%7B2%7D%5Cpi" alt="V_{ulompi}=\pi\int_1^2\left(\frac{1}{x}\right)^2dx=\frac{1}{2}\pi"/>(Laskin)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_%7Bsisempi%7D%3D%5Cpi%5Cint_1%5E2%5Cleft(-x%2B2%5Cright)%5E2dx%3D%5Cfrac%7B1%7D%7B3%7D%5Cpi" alt="V_{sisempi}=\pi\int_1^2\left(-x+2\right)^2dx=\frac{1}{3}\pi"/>(Laskin)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V_%7Bulompi%7D-V_%7Bsisempi%7D%3D%5Cfrac%7B1%7D%7B2%7D%5Cpi-%5Cfrac%7B1%7D%7B3%7D%5Cpi%3D%5Cfrac%7B1%7D%7B6%7D%5Cpi%3D%5Cfrac%7B%5Cpi%7D%7B6%7D" alt="V_{ulompi}-V_{sisempi}=\frac{1}{2}\pi-\frac{1}{3}\pi=\frac{1}{6}\pi=\frac{\pi}{6}"/></div> <div>Kappaleen tilavuus on π/6</div> </div> <div>524<br/> <span>Käyrän </span><img src="https://math-demo.abitti.fi/math.svg?latex=y%3De%5Ex" alt="y=e^x"/><span> pyörähtäminen suoran y=c ympäri on sama tilanne, kuin jos käyrä </span><img src="https://math-demo.abitti.fi/math.svg?latex=y%3De%5Ex-c" alt="y=e^x-c"/><span>pyörähtäisi x-akselin ympäri</span><br/> <div>Määritetään pyörähdyskappaleen tilavuus välillä [-1,1]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cpi%5Cint_%7B-1%7D%5E1%5Cleft(e%5Ex-c%5Cright)%5E2%3D%5Cpi%5Cint_%7B-1%7D%5E1%5Cleft(e%5E%7B2x%7D-2ce%5Ex%2Bc%5E2%5Cright)dx%3DV%5Cleft(c%5Cright)" alt="\pi\int_{-1}^1\left(e^x-c\right)^2=\pi\int_{-1}^1\left(e^{2x}-2ce^x+c^2\right)dx=V\left(c\right)"/></div> <span>Tilavuusfunktio on muuttujan c toisen asteen funktio. Sen kuvaaja on ylöspäin aukeava paraabeli, jonka pienin arvo saavutetaan huipussa. Huippu on derivaatan nollakohdassa</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Cpi%5Cleft(2c%5E2-%5Cleft(2e-%5Cfrac%7B2%7D%7Be%7D%5Cright)c-%5Cfrac%7B1%7D%7B2e%5E2%7D%2B%5Cfrac%7B1%7D%7B2%7De%5E2%5Cright)%5Cright)%3D%5Cpi%5Cleft(4c-2e%2B%5Cfrac%7B2%7D%7Be%7D%5Cright)" alt="D\left(\pi\left(2c^2-\left(2e-\frac{2}{e}\right)c-\frac{1}{2e^2}+\frac{1}{2}e^2\right)\right)=\pi\left(4c-2e+\frac{2}{e}\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cpi%5Cleft(4c-2e%2B%5Cfrac%7B2%7D%7Be%7D%5Cright)%3D0" alt="\pi\left(4c-2e+\frac{2}{e}\right)=0"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=c%3D%5Cfrac%7Be%7D%7B2%7D-%5Cfrac%7B1%7D%7B2e%7D%3D%5Cfrac%7Be%5E2-1%7D%7B2e%7D" alt="c=\frac{e}{2}-\frac{1}{2e}=\frac{e^2-1}{2e}"/></div> <div>Vakion c tulee olla <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Be%5E2-1%7D%7B2e%7D" alt="\frac{e^2-1}{2e}"/>, jotta tilavuus olisi pienin </div> <div> </div> </div> 2019-09-08T20:11:18+03:00 https://peda.net/id/53af5452ced 2019-09-08T20:10:20+03:00 <span>431</span> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2B2x-5%3D7-x%5E2" alt="x^2+2x-5=7-x^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-3%5C%20tai%5C%20x%3D2" alt="x=-3\ tai\ x=2"/></div> <div>Otetaan testipisteeksi välin keskiarvo<img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B1%7D%7B2%7D" alt="-\frac{1}{2}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E2%2B2x-5-7" alt="f\left(x\right)=x^2+2x-5-7"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D7-x%5E2" alt="g\left(x\right)=7-x^2"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-%5Cfrac%7B1%7D%7B2%7D%5Cright)%3D-%5Cfrac%7B23%7D%7B4%7D" alt="f\left(-\frac{1}{2}\right)=-\frac{23}{4}"/>(Laskin)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(-%5Cfrac%7B1%7D%7B2%7D%5Cright)%3D%5Cfrac%7B27%7D%7B4%7D" alt="g\left(-\frac{1}{2}\right)=\frac{27}{4}"/><span>(Laskin)</span> <div>Laskujen perusteella funktio g(x) on ylempänä, koska sen arvo pisteessä <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B1%7D%7B2%7D" alt="x=-\frac{1}{2}"/>on suurempi</div> <div>b) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cint_%7B-3%7D%5E2%5Cleft(f%5Cleft(x%5Cright)-g%5Cleft(x%5Cright)%5Cright)%3D%5Cfrac%7B125%7D%7B3%7D%3D41%5Cfrac%7B2%7D%7B3%7D" alt="A=\int_{-3}^2\left(f\left(x\right)-g\left(x\right)\right)=\frac{125}{3}=41\frac{2}{3}"/><br/> <br/> <div> <div>432</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3-x%5E2-x%3Dx" alt="x^3-x^2-x=x"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%E2%88%921%5C%20tai%5C%20x%3D0%5C%20tai%5C%20x%3D2" alt="x=−1\ tai\ x=0\ tai\ x=2"/>(laskin)</div> <div>b)</div> <div>f(x) on ylempänä välillä ]-1,0[, ja g(x) on ylempänä välillä ]0,2[</div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A_1%3D%5Cint_%7B-1%7D%5E0%5Cleft(f%5Cleft(x%5Cright)-g%5Cleft(x%5Cright)%5Cright)%3D%5Cfrac%7B5%7D%7B12%7D" alt="A_1=\int_{-1}^0\left(f\left(x\right)-g\left(x\right)\right)=\frac{5}{12}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A_2%3D%5Cint_0%5E2%5Cleft(g%5Cleft(x%5Cright)-f%5Cleft(x%5Cright)%5Cright)%3D%5Cfrac%7B8%7D%7B3%7D" alt="A_2=\int_0^2\left(g\left(x\right)-f\left(x\right)\right)=\frac{8}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A_%7Bkok%7D%3D%5Cfrac%7B5%7D%7B12%7D%2B%5Cfrac%7B8%7D%7B3%7D%3D%5Cfrac%7B37%7D%7B12%7D%3D3%5Cfrac%7B1%7D%7B12%7D" alt="A_{kok}=\frac{5}{12}+\frac{8}{3}=\frac{37}{12}=3\frac{1}{12}"/><br/> <br/> <div> <div> <div>433</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A_1%3D%5Cint_%7B-3%7D%5E1f%5Cleft(x%5Cright)-g%5Cleft(x%5Cright)dx%3D10-2%3D8" alt="A_1=\int_{-3}^1f\left(x\right)-g\left(x\right)dx=10-2=8"/></div> b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cint_1%5E7f%5Cleft(x%5Cright)-g%5Cleft(x%5Cright)%3D23-%5Cleft(-5%5Cright)%3D28" alt="A=\int_1^7f\left(x\right)-g\left(x\right)=23-\left(-5\right)=28"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D28%2B8%3D36" alt="A=28+8=36"/></div> <div><br/> <div>434</div> <div>a)</div> <div>Lasketaan funktioiden leikkauspisteet</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%2B6%3Dx%5E2" alt="x+6=x^2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=-x%5E2%2Bx%2B6%3D0" alt="-x^2+x+6=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-1%5Cpm%5Csqrt%5B%5D%7B1%5E2-4%5Ccdot%5Cleft(-1%5Cright)%5Ccdot6%7D%7D%7B2%5Ccdot%5Cleft(-1%5Cright)%7D%3D%5Cfrac%7B-1%5Cpm5%7D%7B-2%7D" alt="x=\frac{-1\pm\sqrt[]{1^2-4\cdot\left(-1\right)\cdot6}}{2\cdot\left(-1\right)}=\frac{-1\pm5}{-2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-1%2B5%7D%7B-2%7D%3D-2%5C%20tai%5C%20x%3D%5Cfrac%7B-1-5%7D%7B-2%7D%3D3" alt="x=\frac{-1+5}{-2}=-2\ tai\ x=\frac{-1-5}{-2}=3"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-2%7D%5E3%5Cleft(-x%5E2%2Bx%2B6%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-2%7D%7D%5E3-%5Cfrac%7B1%7D%7B3%7Dx%5E3%2B%5Cfrac%7B1%7D%7B2%7Dx%5E2%2B6x%3D%5Cleft(-%5Cfrac%7B1%7D%7B3%7D%5Ccdot3%5E3%2B%5Cfrac%7B1%7D%7B2%7D%5Ccdot3%5E2%2B6%5Ccdot3%5Cright)-%5Cleft(-%5Cfrac%7B1%7D%7B3%7D%5Ccdot%5Cleft(-2%5Cright)%5E3%2B%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cleft(-2%5Cright)%5E2%2B6%5Cleft(-2%5Cright)%5Cright)%3D%5Cfrac%7B125%7D%7B6%7D%3D20%5Cfrac%7B5%7D%7B6%7D" alt="\int_{-2}^3\left(-x^2+x+6\right)dx=\bigg/_{\!\!\!\!\!{-2}}^3-\frac{1}{3}x^3+\frac{1}{2}x^2+6x=\left(-\frac{1}{3}\cdot3^3+\frac{1}{2}\cdot3^2+6\cdot3\right)-\left(-\frac{1}{3}\cdot\left(-2\right)^3+\frac{1}{2}\cdot\left(-2\right)^2+6\left(-2\right)\right)=\frac{125}{6}=20\frac{5}{6}"/></div> </div> <div>b)</div> <div>Suoran nollakohta on</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%2B6%3D0" alt="x+6=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-6" alt="x=-6"/><br/> <div>Lasketaan suora, y-akseli ja x-akseli rajaaman alueen pinta-ala</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A_a%3D%5Cfrac%7B6%5Ccdot6%7D%7B2%7D%3D18" alt="A_a=\frac{6\cdot6}{2}=18"/></div> <div>Lasketaan paraapelin ja suoran rajaaman alueen pinta-ala välillä [-2,0]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A_b%3D%5Cint_%7B-2%7D%5E0%5Cleft(-x%5E2%2Bx%2B6%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-2%7D%7D%5E0-%5Cfrac%7B1%7D%7B3%7Dx%5E3%2B%5Cfrac%7B1%7D%7B2%7Dx%5E2%2B6x%3D%5Cleft(-%5Cfrac%7B1%7D%7B3%7D%5Ccdot0%5E3%2B%5Cfrac%7B1%7D%7B2%7D%5Ccdot0%5E2%2B6%5Ccdot0%5Cright)-%5Cleft(-%5Cfrac%7B1%7D%7B3%7D%5Ccdot%5Cleft(-2%5Cright)%5E3%2B%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cleft(-2%5Cright)%5E2%2B6%5Ccdot%5Cleft(-2%5Cright)%5Cright)%3D7%5Cfrac%7B1%7D%7B3%7D" alt="A_b=\int_{-2}^0\left(-x^2+x+6\right)dx=\bigg/_{\!\!\!\!\!{-2}}^0-\frac{1}{3}x^3+\frac{1}{2}x^2+6x=\left(-\frac{1}{3}\cdot0^3+\frac{1}{2}\cdot0^2+6\cdot0\right)-\left(-\frac{1}{3}\cdot\left(-2\right)^3+\frac{1}{2}\cdot\left(-2\right)^2+6\cdot\left(-2\right)\right)=7\frac{1}{3}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A_2%3DA_a-A_b%3D10%5Cfrac%7B2%7D%7B3%7D" alt="A_2=A_a-A_b=10\frac{2}{3}"/></div> <div> </div> <div>436</div> <div>Lasketaan funktioiden leikkauspisteet</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex%3De%5E%7B2x-1%7D" alt="e^x=e^{2x-1}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex-e%5E%7B2x-1%7D%3D0" alt="e^x-e^{2x-1}=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1" alt="x=1"/></div> <div>Lasketan funktioiden rajaama alueen pinta-ala välillä [0,1]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5E1%5Cleft(e%5Ex-e%5E%7B2x-1%7D%5Cright)dx%3D%5Cfrac%7B%5Cleft(e%5E2-2e%2B1%5Cright)%5Ccdot%20e%5E%7B-1%7D%7D%7B2%7D" alt="\int_0^1\left(e^x-e^{2x-1}\right)dx=\frac{\left(e^2-2e+1\right)\cdot e^{-1}}{2}"/><br/> <br/> <a href="https://peda.net/p/kirin_porsti/ma/ma9p/teht%C3%A4v%C3%A4t/4-2/436-png#top" title="436.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma9p/teht%C3%A4v%C3%A4t/4-2/436-png:file/photo/ca02e720fb15952e66050f4353534904d6bc01bb/436.PNG" alt="" title="436.PNG" class="inline" loading="lazy"/></a><br/> <br/> 438 <div> <div>a)</div> <div>Leikkauskohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D%5Ccos%20x" alt="\sin x=\cos x"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Csin%20x%7D%7B%5Ccos%20x%7D%3D1" alt="\frac{\sin x}{\cos x}=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D1" alt="\tan x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D%2Bn%5Ccdot%5Cpi" alt="x=\frac{\pi}{4}+n\cdot\pi"/></div> <div>Leikkauskohdista välillä 0≤x≤π/2 on x=π/4</div> <div>Alue koostuu kahdesta osasta. Välillä [0,π/4] käyrän sinx ja x-akselin rajaamasta osasta ja välillä [π/4,π/2] käyrän cosx ja x-akselin rajaamasta osasta</div> <div><br/> <div>Käyrien rajoittaman alueen pinta-ala on</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5E%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5Cleft(%5Csin%20x%5Cright)dx%2B%5Cint_%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5E%7B%5Cfrac%7B%5Cpi%7D%7B2%7D%7D%5Cleft(%5Ccos%20x%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5Cleft(-%5Ccos%20x%5Cright)%2B%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%7D%5E%7B%5Cfrac%7B%5Cpi%7D%7B2%7D%7D%5Cleft(%5Csin%20x%5Cright)" alt="\int_0^{\frac{\pi}{4}}\left(\sin x\right)dx+\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\left(\cos x\right)dx=\bigg/_{\!\!\!\!\!0}^{\frac{\pi}{4}}\left(-\cos x\right)+\bigg/_{\!\!\!\!\!{\frac{\pi}{4}}}^{\frac{\pi}{2}}\left(\sin x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(-%5Ccos%5Cfrac%7B%5Cpi%7D%7B4%7D%2B%5Ccos0%5Cright)%2B%5Cleft(%5Csin%5Cfrac%7B%5Cpi%7D%7B2%7D-%5Csin%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cright)" alt="=\left(-\cos\frac{\pi}{4}+\cos0\right)+\left(\sin\frac{\pi}{2}-\sin\frac{\pi}{4}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(-%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7B2%7D%7D%2B1%5Cright)%2B%5Cleft(1-%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7B2%7D%7D%5Cright)" alt="=\left(-\frac{1}{\sqrt[]{2}}+1\right)+\left(1-\frac{1}{\sqrt[]{2}}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D2-%5Csqrt%5B%5D%7B2%7D" alt="=2-\sqrt[]{2}"/></div> <div>b)</div> <div>Käyrien ja y-akselin väliin jäävä alue on käyrien y=sinx ja y=cosx väliin välillä [0,π/4] jäävä alue</div> <div>Käyrien väliin jäävä pinta-ala on</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cint_0%5E%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5Cleft(%5Ccos%20x-%5Csin%20x%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5Cleft(%5Csin%20x%2B%5Ccos%20x%5Cright)" alt="A=\int_0^{\frac{\pi}{4}}\left(\cos x-\sin x\right)dx=\bigg/_{\!\!\!\!\!0}^{\frac{\pi}{4}}\left(\sin x+\cos x\right)"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Csin%5Cfrac%7B%5Cpi%7D%7B4%7D%2B%5Ccos%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cright)-%5Cleft(%5Csin0%2B%5Ccos0%5Cright)" alt="=\left(\sin\frac{\pi}{4}+\cos\frac{\pi}{4}\right)-\left(\sin0+\cos0\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7B2%7D%7D%2B%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7B2%7D%7D%5Cright)-%5Cleft(0%2B1%5Cright)%3D%5Cfrac%7B%5E%7B%5Cleft(%5Csqrt%5B%5D%7B2%7D%5Cright)%7D2%7D%7B%5Csqrt%5B%5D%7B2%7D%7D-1%3D%5Cfrac%7B2%5Csqrt%5B%5D%7B2%7D%7D%7B2%7D-1%3D%5Csqrt%5B%5D%7B2%7D-1" alt="=\left(\frac{1}{\sqrt[]{2}}+\frac{1}{\sqrt[]{2}}\right)-\left(0+1\right)=\frac{^{\left(\sqrt[]{2}\right)}2}{\sqrt[]{2}}-1=\frac{2\sqrt[]{2}}{2}-1=\sqrt[]{2}-1"/><br/> <br/> 443 <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-1%7D%5E1%5Cleft(%5Cleft(-x%5E2%2B1%5Cright)-%5Cleft(x%2Bb%5Cright)%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5E1%5Cleft(%5Cleft(-%5Cfrac%7B1%7D%7B3%7Dx%5E3%2Bx%5Cright)-%5Cleft(%5Cfrac%7B1%7D%7B2%7Dx%5E2%2Bbx%5Cright)%5Cright)%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5E1%5Cleft(-%5Cfrac%7B1%7D%7B3%7Dx%5E3-%5Cfrac%7B1%7D%7B2%7Dx%5E2%2Bx-bx%5Cright)" alt="\int_{-1}^1\left(\left(-x^2+1\right)-\left(x+b\right)\right)dx=\bigg/_{\!\!\!\!\!{-1}}^1\left(\left(-\frac{1}{3}x^3+x\right)-\left(\frac{1}{2}x^2+bx\right)\right)=\bigg/_{\!\!\!\!\!{-1}}^1\left(-\frac{1}{3}x^3-\frac{1}{2}x^2+x-bx\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cleft(-%5Cfrac%7B1%7D%7B3%7D-%5Cfrac%7B1%7D%7B2%7D%2B1-b%5Cright)-%5Cleft(%5Cfrac%7B1%7D%7B3%7D-%5Cfrac%7B1%7D%7B2%7D-1%2Bb%5Cright)%3D%5Cfrac%7B4%7D%7B3%7D-2b" alt="A=\left(-\frac{1}{3}-\frac{1}{2}+1-b\right)-\left(\frac{1}{3}-\frac{1}{2}-1+b\right)=\frac{4}{3}-2b"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B4%7D%7B3%7D-2b%3D8" alt="\frac{4}{3}-2b=8"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=-2b%3D%5Cfrac%7B20%7D%7B3%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3A%5Cleft(-2%5Cright)" alt="-2b=\frac{20}{3}\ \ \ \ \ \left|\right|:\left(-2\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=b%3D-%5Cfrac%7B10%7D%7B3%7D%3D-3%5Cfrac%7B1%7D%7B3%7D" alt="b=-\frac{10}{3}=-3\frac{1}{3}"/></div> </div> </div> 2019-09-04T09:29:19+03:00 https://peda.net/id/450c1e70ce1 2019-09-08T16:35:03+03:00 401 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E2-1" alt="f\left(x\right)=x^2-1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D-%5Cint_%7B-1%7D%5E1f%5Cleft(x%5Cright)dx%3D-%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5E1%5Cleft(%5Cfrac%7B1%7D%7B3%7Dx%5E3-x%5Cright)" alt="A=-\int_{-1}^1f\left(x\right)dx=-\bigg/_{\!\!\!\!\!{-1}}^1\left(\frac{1}{3}x^3-x\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D-%5Cleft(f%5Cleft(1%5Cright)-f%5Cleft(-1%5Cright)%5Cright)%3D-%5Cleft(%5Cfrac%7B1%7D%7B3%7D-1-%5Cleft(-%5Cfrac%7B1%7D%7B3%7D%5Cright)-1%5Cright)%3D-%5Cleft(-%5Cfrac%7B4%7D%7B3%7D%5Cright)%3D%5Cfrac%7B4%7D%7B3%7D" alt="A=-\left(f\left(1\right)-f\left(-1\right)\right)=-\left(\frac{1}{3}-1-\left(-\frac{1}{3}\right)-1\right)=-\left(-\frac{4}{3}\right)=\frac{4}{3}"/><br/> b) <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E3-2x%5E2" alt="f\left(x\right)=x^3-2x^2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D-%5Cint_0%5E2%5Cleft(x%5E3-2x%5E2%5Cright)%3D-%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E2%5Cleft(%5Cfrac%7B1%7D%7B4%7Dx%5E4-%5Cfrac%7B2%7D%7B3%7Dx%5E3%5Cright)" alt="A=-\int_0^2\left(x^3-2x^2\right)=-\bigg/_{\!\!\!\!\!0}^2\left(\frac{1}{4}x^4-\frac{2}{3}x^3\right)"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D-%5Cleft(%5Cfrac%7B1%7D%7B4%7D%5Ccdot2%5E4-%5Cfrac%7B2%7D%7B3%7D%5Ccdot2%5E3-0%5Cright)%3D-%5Cleft(-%5Cfrac%7B4%7D%7B3%7D%5Cright)%3D%5Cfrac%7B4%7D%7B3%7D" alt="A=-\left(\frac{1}{4}\cdot2^4-\frac{2}{3}\cdot2^3-0\right)=-\left(-\frac{4}{3}\right)=\frac{4}{3}"/><br/> <br/> <div> <div>403</div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E2-4x-5" alt="f\left(x\right)=x^2-4x-5"/></div> <span>a)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D5" alt="x=5"/>tai<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1" alt="x=-1"/>(Laskin)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(2%5Cright)%3D2%5E2-8-5%3D-9" alt="f\left(2\right)=2^2-8-5=-9"/></div> <span>b)</span> <div>Pinta-alaa rajaa y-akseli, joten väli on x=0</div> <div>Pinta-alaa rajaa suora x=1, joten kysytty pinta-ala on välillä [0,1]</div> <div>Edellisessä tehtävässä lasketun nollakohdan puolivälin arvo on negatiivinen, joiten rajattu alue on myös negatiivinen(ylöspäin aukeava)</div> <div>Tällöin</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D-%5Cint_0%5E1f%5Cleft(x%5Cright)dx%3D-%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E1%5Cleft(%5Cfrac%7B1%7D%7B3%7Dx%5E3-2x%5E2-5x%5Cright)" alt="A=-\int_0^1f\left(x\right)dx=-\bigg/_{\!\!\!\!\!0}^1\left(\frac{1}{3}x^3-2x^2-5x\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D-%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Ccdot1%5E3-2%5Ccdot1%5E2-5%5Ccdot1-0%5Cright)%3D-%5Cleft(-%5Cfrac%7B20%7D%7B3%7D%5Cright)%3D%5Cfrac%7B20%7D%7B3%7D%3D6%5Cfrac%7B2%7D%7B3%7D" alt="A=-\left(\frac{1}{3}\cdot1^3-2\cdot1^2-5\cdot1-0\right)=-\left(-\frac{20}{3}\right)=\frac{20}{3}=6\frac{2}{3}"/><br/> <br/> <div> <div>404</div> </div> <div>a)<br/> <a href="https://peda.net/p/kirin_porsti/ma/ma9p/teht%C3%A4v%C3%A4t/4-1/404-a-png#top" title="404 a.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma9p/teht%C3%A4v%C3%A4t/4-1/404-a-png:file/photo/21c4a10be85ec88a8f9c5c792314ef611690d963/404%20a.PNG" alt="" title="404 a.PNG" class="inline" loading="lazy"/></a></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5E%7B2%5Cpi%7Df%5Cleft(x%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E%7B2%5Cpi%7D%5Cleft(-%5Ccos%20x%5Cright)" alt="\int_0^{2\pi}f\left(x\right)dx=\bigg/_{\!\!\!\!\!0}^{2\pi}\left(-\cos x\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-1-%5Cleft(-1%5Cright)%3D0" alt="=-1-\left(-1\right)=0"/><br/> <div>c)</div> <div>Tehtävässä halutaan laskea pinta-alaa välillä [0,2π]</div> <div>Merkkikaavion perusteella voidaan oleta, että funktio välillä [0,π] on positiivinen, ja [π,2π] negatiivinen</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cint_0%5E%7B%5C%202%5Cpi%7D%5Cleft%7C%5Csin%20x%5Cright%7Cdx%3D4" alt="A=\int_0^{\ 2\pi}\left|\sin x\right|dx=4"/>(laskin)<br/> <br/> <div>407<br/> <a href="https://peda.net/p/kirin_porsti/ma/ma9p/teht%C3%A4v%C3%A4t/4-1/407-png#top" title="407.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma9p/teht%C3%A4v%C3%A4t/4-1/407-png:file/photo/2d9010e682610557b5b50d97e2c94dd522519a2b/407.PNG" alt="" title="407.PNG" class="inline" loading="lazy"/></a></div> <div>Suora 2x+y-8=0 on ratkaistussa muodossa y=-2x+8. Paraabeli y=x²</div> <div> </div> <div>Lasketaan ensi funktioiden leikkauspisteet</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-2x%2B8%3Dx%5E2%5C%20%5C%20%5C%20%5C%20%5C%20" alt="-2x+8=x^2\ \ \ \ \ "/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=-x%5E2-2x%2B8%3D0" alt="-x^2-2x+8=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%E2%88%924" alt="x=−4"/>tai<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D2" alt="x=2"/> <div> <div>Ja seuraavaksi funktioiden nollakohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-2x%2B8%3D0" alt="-2x+8=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-2x%3D-8%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3A%5Cleft(-2%5Cright)" alt="-2x=-8\ \ \ \ \ \left|\right|:\left(-2\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D4" alt="x=4"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D0" alt="x^2=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0" alt="x=0"/></div> <div>Koska väli on positiivisella puolella, leikkauspiste pisteessä x=-4 ei huomioitaan.<br/> <div>Eli halutaan saada suora -2x+8 ja paraabelin x² pinta-ala välillä [0,4] </div> </div> <div>Tätä väliä voidaan jakaa kahteen eri osaa: [0,2] ja [2,4]</div> </div> <div>Lasketaan paraapelin avulla välin [0,2] pinta-ala ja suoran avulla pinta-ala välillä [2,4]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A_1%3D%5Cint_0%5E2f%5Cleft(x%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E2%5Cleft(%5Cfrac%7B1%7D%7B3%7Dx%5E3%5Cright)" alt="A_1=\int_0^2f\left(x\right)dx=\bigg/_{\!\!\!\!\!0}^2\left(\frac{1}{3}x^3\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=A_1%3D%5Cleft(%5Cfrac%7B1%7D%7B3%7D2%5E3-0%5Cright)%3D%5Cfrac%7B8%7D%7B3%7D" alt="A_1=\left(\frac{1}{3}2^3-0\right)=\frac{8}{3}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A_2%3D%5Cint_2%5E4%5Cleft(-2x%2B8%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!2%7D%5E4%5Cleft(-x%5E2%2B8x%5Cright)" alt="A_2=\int_2^4\left(-2x+8\right)dx=\bigg/_{\!\!\!\!\!2}^4\left(-x^2+8x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A_2%3D%5Cleft(%5Cleft(-4%5E2%2B8%5Ccdot4%5Cright)-%5Cleft(-2%5E2%2B8%5Ccdot2%5Cright)%5Cright)%3D%5Cleft(16-12%5Cright)%3D4" alt="A_2=\left(\left(-4^2+8\cdot4\right)-\left(-2^2+8\cdot2\right)\right)=\left(16-12\right)=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A_%7Bkok%7D%3D%5Cfrac%7B8%7D%7B3%7D%2B4%3D%5Cfrac%7B20%7D%7B3%7D%3D6%5Cfrac%7B2%7D%7B3%7D" alt="A_{kok}=\frac{8}{3}+4=\frac{20}{3}=6\frac{2}{3}"/><br/> <br/> <div>408</div> <div>a) <img src="https://math-demo.abitti.fi/math.svg?latex=-3" alt="-3"/><br/> b) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B5%7D%7B2%7D" alt="\frac{5}{2}"/><br/> c)<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B5%7D%7B2%7D%2B%5Cleft(-3%5Cright)%3D-%5Cfrac%7B1%7D%7B2%7D" alt="\frac{5}{2}+\left(-3\right)=-\frac{1}{2}"/></div> <div>d)<img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B2%7D%7B3%7D" alt="-\frac{2}{3}"/><br/> e)<img src="https://math-demo.abitti.fi/math.svg?latex=0" alt=""/></div> <div> </div> <div>409</div> <div>a)</div> <div>Käyrä muodostaa akselien kanssa alueen, joten ensimmäinen aluetta rajaava suora on kohdassa x=0</div> <div>Toisen suoran saadaan laskemalla käyrän nollakohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex-2%3D0" alt="e^x-2=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex%3D2" alt="e^x=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_e2%3Dx" alt="\log_e2=x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cln2" alt="x=\ln2"/> <div>Koska aro kohdassa x=ln2 on 0, tarvitaan kohdan x=0 arvo</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5E0-2%3D1-2%3D-1" alt="e^0-2=1-2=-1"/></div> <div>Koska arvo on negatiivinen, janojen rajaama alue on negatiivinen<br/> <div>Halutaan laskea se alue joka on välillä [0,ln2]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D-%5Cint_0%5E%7B%5Cln2%7D%5Cleft(e%5Ex-2%5Cright)dx%3D-%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E%7B%5Cln2%7D%5Cleft(e%5Ex-2x%5Cright)" alt="A=-\int_0^{\ln2}\left(e^x-2\right)dx=-\bigg/_{\!\!\!\!\!0}^{\ln2}\left(e^x-2x\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D-%5Cleft(e%5E%7B%5Cln2%7D-2%5Ccdot%5Cleft(%5Cln2%5Cright)-e%5E0-2%5Ccdot0%5Cright)%3D%5Cleft(2-2%5Cln2-1%5Cright)%3D2%5Cln2-1" alt="A=-\left(e^{\ln2}-2\cdot\left(\ln2\right)-e^0-2\cdot0\right)=\left(2-2\ln2-1\right)=2\ln2-1"/></div> </div> <div>b)</div> <div>Aluetta rajaa suorat x=3, x-akseli ja funktio f(x)</div> <div>Aluetta rajaava suora on funktio f(x) ja x-akseli leikkauspisteellä</div> <div>Lasketaan funktion nollakohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7Bx%7D-1%3D0" alt="\frac{1}{x}-1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7Bx%7D%3D1" alt="\frac{1}{x}=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1" alt="x=1"/><br/> <br/> <div>Koska aro kohdassa x=1 on 0, tarvitaan kohdan x=3 arvo<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B3%7D-1%3D-%5Cfrac%7B2%7D%7B3%7D" alt="\frac{1}{3}-1=-\frac{2}{3}"/><br/> Arvo on negatiivinen, janojen rajaama alue on negatiivinen<br/> Halutaan laskea se alue joka on välillä [1,3]<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D-%5Cint_1%5E3%5Cleft(%5Cfrac%7B1%7D%7Bx%7D-1%5Cright)dx%3D-%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!1%7D%5E3%5Cleft(%5Cln%5Cleft%7Cx%5Cright%7C-x%5Cright)" alt="A=-\int_1^3\left(\frac{1}{x}-1\right)dx=-\bigg/_{\!\!\!\!\!1}^3\left(\ln\left|x\right|-x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D-%5Cleft(%5Cln3-3-%5Cleft(-1%5Cright)%5Cright)%3D-%5Cleft(%5Cln3-2%5Cright)%3D2-%5Cln3" alt="A=-\left(\ln3-3-\left(-1\right)\right)=-\left(\ln3-2\right)=2-\ln3"/><br/> <br/> </div> <div>410 <div> <div> <div>Lasketaan funktion f(x) nollakohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7B3-x%7D%3D0" alt="\sqrt[]{3-x}=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=3-x%3D0" alt="3-x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-x%3D-3" alt="-x=-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D3" alt="x=3"/></div> <div>Lasketaan suoran y nollakohta</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%2B3%3D0" alt="x+3=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-3" alt="x=-3"/></div> <div>Lasketaan funktioiden leikkaupisteet</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7B3-x%7D%3Dx%2B3" alt="\sqrt[]{3-x}=x+3"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1" alt="x=-1"/>(laskin)</div> <div> <div>Koska suora on nouseva, ja funktiolla f(x) on vain ratkaisuja kun x≤0, joten alue on positiivinen</div> <div>Alueet ovat välillä [-3,-1] ja [-1,3]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A_1%3D%5Cint_%7B-3%7D%5E%7B-1%7D%5Cleft(x%2B3%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-3%7D%7D%5E%7B-1%7D%5Cleft(%5Cfrac%7B1%7D%7B2%7Dx%5E2%2B3x%5Cright)" alt="A_1=\int_{-3}^{-1}\left(x+3\right)dx=\bigg/_{\!\!\!\!\!{-3}}^{-1}\left(\frac{1}{2}x^2+3x\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=A_1%3Df%5Cleft(-1%5Cright)-f%5Cleft(-3%5Cright)%3D2" alt="A_1=f\left(-1\right)-f\left(-3\right)=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A_2%3D%5Cint_%7B-1%7D%5E3%5Cleft(%5Csqrt%5B%5D%7B3-x%7D%5Cright)dx%3D%5Cint_%7B-1%7D%5E3%5Cleft(%5Cleft(3-x%5C%20%5Cright)%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5E3%5Cleft(%5Cfrac%7B-2%5Cleft(3-x%5Cright)%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%7B3%7D%5Cright)" alt="A_2=\int_{-1}^3\left(\sqrt[]{3-x}\right)dx=\int_{-1}^3\left(\left(3-x\ \right)^{\frac{1}{2}}\right)dx=\bigg/_{\!\!\!\!\!{-1}}^3\left(\frac{-2\left(3-x\right)^{\frac{3}{2}}}{3}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A_2%3Df%5Cleft(3%5Cright)-f%5Cleft(-1%5Cright)%3D%5Cfrac%7B16%7D%7B3%7D" alt="A_2=f\left(3\right)-f\left(-1\right)=\frac{16}{3}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A_%7Bkok%7D%3D2%2B%5Cfrac%7B16%7D%7B3%7D%3D%5Cfrac%7B22%7D%7B3%7D%3D7%5Cfrac%7B1%7D%7B3%7D" alt="A_{kok}=2+\frac{16}{3}=\frac{22}{3}=7\frac{1}{3}"/></div> </div> <div>413<br/> <div> <div><span>Lasketaan suorakulmion pinta-ala</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A_s%3D2%5Ccdot1%3D2" alt="A_s=2\cdot1=2"/><br/> <span>Käyrän nollakohdat välillä [-1,1] on</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos2x%3D0%7B%2C%7D%5C%20%5Cleft%5B-1%7B%2C%7D1%5Cright%5D" alt="\cos2x=0{,}\ \left[-1{,}1\right]"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D%5C%20ja%5C%20x%3D-%5Cfrac%7B%5Cpi%7D%7B4%7D" alt="x=\frac{\pi}{4}\ ja\ x=-\frac{\pi}{4}"/><span>(laskin)</span> <div>Käyrän pinta-ala välillä [<img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20%5Cfrac%7B%5Cpi%7D%7B4%7D" alt="-\frac{\pi}{4}{,}\ \frac{\pi}{4}"/>]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cint_%7B-%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5E%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5Cleft(%5Ccos2x%5Cright)dx%3D1" alt="A=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\left(\cos2x\right)dx=1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%7D%7B2%7D%3D1" alt="\frac{2}{2}=1"/></div> <div>Joten käyrän yläpuolella on suorakulmiosta yhtä paljo kuin alapuolella.</div> <a href="https://peda.net/p/kirin_porsti/ma/ma9p/teht%C3%A4v%C3%A4t/4-1/413-png#top" title="413.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma9p/teht%C3%A4v%C3%A4t/4-1/413-png:file/photo/7cd1eb3a7bd4eddc1b85be5db53259c54cefcb2f/413.PNG" alt="" title="413.PNG" class="inline" loading="lazy"/></a></div> </div> </div> <div>416<br/> <span>a)</span> <div>1,5 tai -1,5</div> <div>b)</div> <div>Koska käyrän pinta-ala voi olla 2 kun se on joko ylöspäin tai alaspäin aukeava</div> <div>joten sen pinta-ala funktio voi olla joko</div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-1%7D%5E1%5Cleft(ax%5E2-a%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5E1%5Cfrac%7Ba%7D%7B2%2B1%7Dx%5E%7B2%2B1%7D-ax%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5E1%5Cfrac%7Ba%7D%7B3%7Dx%5E3-ax" alt="\int_{-1}^1\left(ax^2-a\right)dx=\bigg/_{\!\!\!\!\!{-1}}^1\frac{a}{2+1}x^{2+1}-ax=\bigg/_{\!\!\!\!\!{-1}}^1\frac{a}{3}x^3-ax"/></div> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cint_%7B-1%7D%5E1%5Cleft(ax%5E2-a%5Cright)dx%3D-%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5E1%5Cfrac%7Ba%7D%7B2%2B1%7Dx%5E%7B2%2B1%7D-ax%3D-%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5E1%5Cfrac%7Ba%7D%7B3%7Dx%5E3-ax" alt="-\int_{-1}^1\left(ax^2-a\right)dx=-\bigg/_{\!\!\!\!\!{-1}}^1\frac{a}{2+1}x^{2+1}-ax=-\bigg/_{\!\!\!\!\!{-1}}^1\frac{a}{3}x^3-ax"/> </div> <div>Tällöin a voi olla</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3Df%5Cleft(1%5Cright)-f%5Cleft(-1%5Cright)%3D2" alt="A=f\left(1\right)-f\left(-1\right)=2"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%3D-1%7B%2C%7D5" alt="a=-1{,}5"/></div> <span>tai</span><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D-%5Cleft(f%5Cleft(1%5Cright)-f%5Cleft(-1%5Cright)%5Cright)%3D2" alt="A=-\left(f\left(1\right)-f\left(-1\right)\right)=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D1%7B%2C%7D5" alt="a=1{,}5"/></div> </div> <div>420<br/> <div>Käänetään oikealle avautuva paraapeli x-akselin suuntaan</div> <div>Tällöin sen funktio on </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=y%3Df%5Cleft(x%5Cright)%3Dx%5E2%2Bx-2" alt="y=f\left(x\right)=x^2+x-2"/></div> <span>Lasketaan paraabelin nollakohdat x-akselilla</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2Bx-2%3D0" alt="x^2+x-2=0"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-2%5C%20tai%5C%20x%3D1" alt="x=-2\ tai\ x=1"/>(Laskin)</div> <div>Koska paraapeli on ylöspäin aukeava, ja sillä on nollakohdat</div> <div>Tällöin rajattu alue on x-akselin alapuolella eli on lisättävä integraalifuntion eteen ''-'' merkki</div> Lasketaan paraapelin ja x-akselin rajoittaman alueen pinta-ala välillä [-2,1]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cint_%7B-2%7D%5E1%5Cleft(x%5E2%2Bx-2%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-2%7D%7D%5E1%5Cleft(%5Cfrac%7B1%7D%7B3%7Dx%5E3%2Bx%5E2-2x%5Cright)" alt="-\int_{-2}^1\left(x^2+x-2\right)dx=\bigg/_{\!\!\!\!\!{-2}}^1\left(\frac{1}{3}x^3+x^2-2x\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D-%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-2%7D%7D%5E1%5Cleft(%5Cfrac%7B1%7D%7B3%7Dx%5E3%2Bx%5E2-2x%5Cright)%3D%5Cfrac%7B9%7D%7B2%7D%3D4%5Cfrac%7B1%7D%7B2%7D" alt="A=-\bigg/_{\!\!\!\!\!{-2}}^1\left(\frac{1}{3}x^3+x^2-2x\right)=\frac{9}{2}=4\frac{1}{2}"/></div> </div> </div> </div> 2019-09-03T09:08:37+03:00 https://peda.net/id/9533ac12c95 2019-09-02T08:10:11+03:00 <div> <div> <div> <div>319 <div>a) C</div> <div>b) A</div> <div>c) B</div> <div> </div> <div>320</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_1%5E3%5Cleft(6x-x%5Cright)%5E2dx%3D%5Cint_1%5E3%5Cleft(5x%5Cright)%5E2dx%3D%5Cint_1%5E325x%5E2dx%3D25%5Ccdot%5Cint_1%5E3x%5E2%3D25%5Ccdot%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!1%7D%5E3%5Cfrac%7B1%7D%7B3%7Dx%5E3%3D%5Cfrac%7B25x%5E3%7D%7B3%7D" alt="\int_1^3\left(6x-x\right)^2dx=\int_1^3\left(5x\right)^2dx=\int_1^325x^2dx=25\cdot\int_1^3x^2=25\cdot\bigg/_{\!\!\!\!\!1}^3\frac{1}{3}x^3=\frac{25x^3}{3}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(3%5Cright)-f%5Cleft(1%5Cright)%3D%5Cfrac%7B25%5Ccdot3%5E3%7D%7B3%7D-%5Cfrac%7B25%5Ccdot1%5E3%7D%7B3%7D%3D%5Cfrac%7B650%7D%7B3%7D" alt="f\left(3\right)-f\left(1\right)=\frac{25\cdot3^3}{3}-\frac{25\cdot1^3}{3}=\frac{650}{3}"/> <div> </div> <div>321</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)-f%5Cleft(0%5Cright)%3De-1" alt="f\left(1\right)-f\left(0\right)=e-1"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(e%5Cright)-f%5Cleft(%5Cfrac%7B1%7D%7Be%7D%5Cright)%3D2%7B%2C%7D35" alt="f\left(e\right)-f\left(\frac{1}{e}\right)=2{,}35"/></div> <div> </div> <div>322</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5E13e%5Exdx%3D3%5Ccdot%5Cint_0%5E1e%5Exdx%3D3%5Ccdot%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E1e%5Ex%3D3e%5Ex" alt="\int_0^13e^xdx=3\cdot\int_0^1e^xdx=3\cdot\bigg/_{\!\!\!\!\!0}^1e^x=3e^x"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)-f%5Cleft(0%5Cright)%3D3e-3" alt="f\left(1\right)-f\left(0\right)=3e-3"/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5E%7B%5Cpi%7D%5Csin%20xdx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E%7B%5Cpi%7D-%5Ccos%20x" alt="\int_0^{\pi}\sin xdx=\bigg/_{\!\!\!\!\!0}^{\pi}-\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(%5Cpi%5Cright)-f%5Cleft(0%5Cright)%3D" alt="f\left(\pi\right)-f\left(0\right)="/><br/> <div> </div> </div> <br/> 324</div> </div> </div> </div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4%5Cfrac%7B1%7D%7B3%7D" alt="4\frac{1}{3}"/></div> <div>b) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-4%5Cfrac%7B1%7D%7B3%7D" alt="-4\frac{1}{3}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4%5Cfrac%7B1%7D%7B3%7D%2B%5Cfrac%7B4%7D%7B3%7D%3D%5Cfrac%7B13%7D%7B3%7D%2B%5Cfrac%7B4%7D%7B3%7D%3D%5Cfrac%7B17%7D%7B3%7D%3D5%5Cfrac%7B2%7D%7B3%7D" alt="4\frac{1}{3}+\frac{4}{3}=\frac{13}{3}+\frac{4}{3}=\frac{17}{3}=5\frac{2}{3}"/></div> <div>d) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4" alt="4"/></div> <span>e)</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=0" alt=""/><br/> <span>f)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4%5Cfrac%7B1%7D%7B3%7D" alt="4\frac{1}{3}"/><br/> <div> </div> <div>332</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_1%5Ee%5Cfrac%7B1%7D%7Bx%7Ddx%2B%5Cint_1%5Ee%5Cleft(1-%5Cfrac%7B1%7D%7Bx%7D%5Cright)dx%3D%5Cint_1%5Eex%5E%7B-1%7Ddx%2B%5Cint_1%5Ee1-x%5E%7B-1%7Ddx%3D%5Cint_1%5Eex%5E%7B-1%7D-x%5E%7B-1%7D%2B1%3D%5Cint_1%5Ee1%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!1%7D%5Eex" alt="\int_1^e\frac{1}{x}dx+\int_1^e\left(1-\frac{1}{x}\right)dx=\int_1^ex^{-1}dx+\int_1^e1-x^{-1}dx=\int_1^ex^{-1}-x^{-1}+1=\int_1^e1=\bigg/_{\!\!\!\!\!1}^ex"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(e%5Cright)-f%5Cleft(1%5Cright)%3De-1" alt="f\left(e\right)-f\left(1\right)=e-1"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5E1%5Cleft(3x%5E2-2x%2B1%5Cright)dx%2B%5Cint_1%5E2%5Cleft(3x%5E2-2x%2B1%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E1%5Cleft(x%5E3-x%5E2%2Bx%5Cright)%2B%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!1%7D%5E2%5Cleft(x%5E3-x%5E2%2Bx%5Cright)" alt="\int_0^1\left(3x^2-2x+1\right)dx+\int_1^2\left(3x^2-2x+1\right)dx=\bigg/_{\!\!\!\!\!0}^1\left(x^3-x^2+x\right)+\bigg/_{\!\!\!\!\!1}^2\left(x^3-x^2+x\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E2x%5E3-x%5E2%2Bx" alt="=\bigg/_{\!\!\!\!\!0}^2x^3-x^2+x"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(2%5Cright)-f%5Cleft(0%5Cright)%3D6" alt="f\left(2\right)-f\left(0\right)=6"/></div> <br/> <div>333</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_5%5E52x%5Csqrt%5B%5D%7Bx%5E2%2B1%7Ddx%3D0" alt="\int_5^52x\sqrt[]{x^2+1}dx=0"/></div> b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-1%7D%5E1x%5Csin%5E2xdx%2B%5Cint_%7B-1%7D%5E1x%5Ccos%5E2xdx%3D%5Cint_%7B-1%7D%5E1%5Cleft(x%5Csin%5E2x%5Cright)%2B%5Cleft(x%5Ccos%5E2x%5Cright)dx%3D%5Cint_%7B-1%7D%5E1%5Cleft(x%5Cleft(%5Csin%5E2x%2B%5Ccos%5E2x%5Cright)%5Cright)dx%3D%5Cint_%7B-1%7D%5E1xdx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5E1%5Cfrac%7B1%7D%7B2%7Dx%5E2" alt="\int_{-1}^1x\sin^2xdx+\int_{-1}^1x\cos^2xdx=\int_{-1}^1\left(x\sin^2x\right)+\left(x\cos^2x\right)dx=\int_{-1}^1\left(x\left(\sin^2x+\cos^2x\right)\right)dx=\int_{-1}^1xdx=\bigg/_{\!\!\!\!\!{-1}}^1\frac{1}{2}x^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)-f%5Cleft(-1%5Cright)%3D0" alt="f\left(1\right)-f\left(-1\right)=0"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5E1%5Cleft(x-1%5Cright)dx%2B%5Cint_0%5E1%5Cleft(x%5E2-x%5Cright)dx%2B%5Cint_0%5E1%5Cleft(x%5E3-x%5E2%5Cright)dx%2B...%2B%5Cint_0%5E1%5Cleft(x%5E9-x%5E8%5Cright)dx%3D%5Cint_0%5E1%5Cleft(x-1%2Bx%5E2-x%2Bx%5E3-x%5E2%2B...%2Bx%5E9-x%5E8%5Cright)dx%3D%5Cint_0%5E1%5Cleft(-1%2Bx%5E9%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E1-x%2B%5Cfrac%7B1%7D%7B10%7Dx%5E%7B10%7D" alt="\int_0^1\left(x-1\right)dx+\int_0^1\left(x^2-x\right)dx+\int_0^1\left(x^3-x^2\right)dx+...+\int_0^1\left(x^9-x^8\right)dx=\int_0^1\left(x-1+x^2-x+x^3-x^2+...+x^9-x^8\right)dx=\int_0^1\left(-1+x^9\right)dx=\bigg/_{\!\!\!\!\!0}^1-x+\frac{1}{10}x^{10}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)-f%5Cleft(0%5Cright)%3D-%5Cfrac%7B9%7D%7B10%7D" alt="f\left(1\right)-f\left(0\right)=-\frac{9}{10}"/></div> <div><br/> <div>334</div> <div>a)<br/> <a href="https://peda.net/id/82fb834ec95">https://peda.net/id/82fb834ec95</a></div> </div> <div>Kun x&lt;2, funktio saa negatiivisia arvoja, tällöin</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C2x-4%5Cright%7C%3D-%5Cleft(2x-4%5Cright)%3D-2x%2B4" alt="\left|2x-4\right|=-\left(2x-4\right)=-2x+4"/></div> <div>Kun x≥2, funktio saa positiivisia arvoja, tällöin</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C2x-4%5Cright%7C%3D2x-4" alt="\left|2x-4\right|=2x-4"/></div> <div>Eli tässä tapauksessa pitää laskea pinta-alat välillä [0,2] [2,3]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5E3%5Cleft%7C2x-4%5Cright%7Cdx%3D%5Cint_0%5E2%5Cleft(-2x%2B4%5Cright)dx%2B%5Cint_2%5E3%5Cleft(2x-4%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E3%5Cleft(-x%5E2%2B4x%5Cright)%2B%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E3%5Cleft(x%5E2-4x%5Cright)%3D%5Cleft(-2%5E2%2B4%5Ccdot2%5Cright)-%5Cleft(-0%5E2%2B4%5Ccdot0%5Cright)%2B%5Cleft(3%5E2-4%5Ccdot3%5Cright)-%5Cleft(2%5E2-4%5Ccdot2%5Cright)%3D4%2B8%2B9-12-4%2B8%3D5" alt="\int_0^3\left|2x-4\right|dx=\int_0^2\left(-2x+4\right)dx+\int_2^3\left(2x-4\right)dx=\bigg/_{\!\!\!\!\!0}^3\left(-x^2+4x\right)+\bigg/_{\!\!\!\!\!0}^3\left(x^2-4x\right)=\left(-2^2+4\cdot2\right)-\left(-0^2+4\cdot0\right)+\left(3^2-4\cdot3\right)-\left(2^2-4\cdot2\right)=4+8+9-12-4+8=5"/></div> <span>b)<br/> <a href="https://peda.net/id/8a70e5c4c95">https://peda.net/id/8a70e5c4c95</a><br/> </span> <div>Kun x≤-2 tai x≥2, funktio saa positiivisia arvoja, tällöin<span> </span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7Cx%5E2-4%5Cright%7C%3Dx%5E2-4" alt="\left|x^2-4\right|=x^2-4"/><br/> Kun -2&lt;x&lt;2, funktio saa negatiivisia arvoja, tällöin<span> </span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7Cx%5E2-4%5Cright%7C%3D-%5Cleft(x%5E2-4%5Cright)%3D-x%5E2%2B4" alt="\left|x^2-4\right|=-\left(x^2-4\right)=-x^2+4"/><br/> Eli tässä tapauksessa pitää laskea pinta-alat välillä [-3,-2] [-2,2] ja [2,3] <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-3%7D%5E3%5Cleft%7Cx%5E2-4%5Cright%7C%3D%5Cint_%7B-3%7D%5E%7B-2%7D%5Cleft(x%5E2-4%5Cright)dx%2B%5Cint_%7B-2%7D%5E2%5Cleft(-x%5E2%2B4%5Cright)dx%2B%5Cint_2%5E3%5Cleft(x%5E2-4%5Cright)dx" alt="\int_{-3}^3\left|x^2-4\right|=\int_{-3}^{-2}\left(x^2-4\right)dx+\int_{-2}^2\left(-x^2+4\right)dx+\int_2^3\left(x^2-4\right)dx"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-3%7D%7D%5E%7B-2%7D%5Cleft(%5Cfrac%7B1%7D%7B3%7Dx%5E3-4x%5Cright)%2B%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-2%7D%7D%5E2%5Cleft(-%5Cfrac%7B1%7D%7B3%7Dx%5E3%2B4x%5Cright)%2B%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!2%7D%5E3%5Cleft(%5Cfrac%7B1%7D%7B3%7Dx%5E3-4x%5Cright)" alt="=\bigg/_{\!\!\!\!\!{-3}}^{-2}\left(\frac{1}{3}x^3-4x\right)+\bigg/_{\!\!\!\!\!{-2}}^2\left(-\frac{1}{3}x^3+4x\right)+\bigg/_{\!\!\!\!\!2}^3\left(\frac{1}{3}x^3-4x\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Ccdot%5Cleft(-2%5Cright)%5E3-4%5Ccdot%5Cleft(-2%5Cright)%5Cright)-%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Ccdot%5Cleft(-3%5Cright)%5E3-4%5Ccdot%5Cleft(-3%5Cright)%5Cright)%5Cright)%2B%5Cleft(%5Cleft(-%5Cfrac%7B1%7D%7B3%7D2%5E3%2B4%5Ccdot2%5Cright)-%5Cleft(-%5Cfrac%7B1%7D%7B3%7D%5Cleft(-2%5Cright)%5E3%2B4%5Ccdot%5Cleft(-2%5Cright)%5Cright)%5Cright)%2B%5Cleft(%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Ccdot3%5E3-4%5Ccdot3%5Cright)-%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Ccdot2%5E3-4%5Ccdot2%5Cright)%5Cright)" alt="=\left(\left(\frac{1}{3}\cdot\left(-2\right)^3-4\cdot\left(-2\right)\right)-\left(\frac{1}{3}\cdot\left(-3\right)^3-4\cdot\left(-3\right)\right)\right)+\left(\left(-\frac{1}{3}2^3+4\cdot2\right)-\left(-\frac{1}{3}\left(-2\right)^3+4\cdot\left(-2\right)\right)\right)+\left(\left(\frac{1}{3}\cdot3^3-4\cdot3\right)-\left(\frac{1}{3}\cdot2^3-4\cdot2\right)\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Cfrac%7B16%7D%7B3%7D-3%5Cright)%2B%5Cleft(%5Cfrac%7B16%7D%7B3%7D-%5Cleft(-%5Cfrac%7B16%7D%7B3%7D%5Cright)%5Cright)%2B%5Cleft(-3-%5Cleft(-%5Cfrac%7B16%7D%7B3%7D%5Cright)%5Cright)%3D%5Cfrac%7B46%7D%7B3%7D%3D15%5Cfrac%7B1%7D%7B3%7D" alt="=\left(\frac{16}{3}-3\right)+\left(\frac{16}{3}-\left(-\frac{16}{3}\right)\right)+\left(-3-\left(-\frac{16}{3}\right)\right)=\frac{46}{3}=15\frac{1}{3}"/> 2019-08-28T09:46:37+03:00 https://peda.net/id/d17e0166c89 2019-09-12T22:47:32+03:00 <div> <div> <div> <div>301</div> </div> </div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=16%7B%2C%7D7-6%7B%2C%7D6%3D10%7B%2C%7D1" alt="16{,}7-6{,}6=10{,}1"/></div> <br/> <div>302</div> <div>a) Appletin avulla saattiin pinta-ala välillä [0,3], joka on siis <img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(3%5Cright)%3D21" alt="A\left(3\right)=21"/> , pinta-ala välillä [0,1] on appletin mukaan </div> <img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(1%5Cright)%3D1" alt="A\left(1\right)=1"/> <div>Joten ylläolevien tietojen mukaan pinta-ala välillä [1,3] on</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=21-1%3D20" alt="21-1=20"/><br/> <div> <div>b) </div> <div>A-kohdan perusteella voidaan oleta, että pinta-ala välillä [1,3] on funktio A(3) ja A(1) erotus</div> <div>Määritetään funktiot A(3) ja A(1)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(3%5Cright)%3D3%5E3-3%5E2%2B3%3D21" alt="A\left(3\right)=3^3-3^2+3=21"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(1%5Cright)%3D1%5E3-1%5E2%2B1%3D1" alt="A\left(1\right)=1^3-1^2+1=1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(3%5Cright)-A%5Cleft(1%5Cright)%3D20" alt="A\left(3\right)-A\left(1\right)=20"/></div> <div>c)</div> <div>Koska lauseen mukaan pinta-alafunktio A on funktion f eräs integraalifunktio, eli </div> <img src="https://math-demo.abitti.fi/math.svg?latex=A%27%5Cleft(x%5Cright)%3Df%5Cleft(x%5Cright)" alt="A'\left(x\right)=f\left(x\right)"/><br/> <div>Derivoitaan funktio A(x)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%27%5Cleft(x%5Cright)%3Df%5Cleft(x%5Cright)%3D3x%5E2-2x%2B1" alt="A'\left(x\right)=f\left(x\right)=3x^2-2x+1"/><br/>  <br/> <div>304</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(x%5Cright)%3D%5Cfrac%7B%5Cleft(1%2Bx%2B1%5Cright)%5Ccdot%20x%7D%7B2%7D%3D%5Cfrac%7Bx%5E2%2B2x%7D%7B2%7D%3D%5Cfrac%7Bx%5E2%7D%7B2%7D%2Bx" alt="A\left(x\right)=\frac{\left(1+x+1\right)\cdot x}{2}=\frac{x^2+2x}{2}=\frac{x^2}{2}+x"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%27%5Cleft(x%5Cright)%3Df%5Cleft(x%5Cright)" alt="A'\left(x\right)=f\left(x\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=A%27%5Cleft(x%5Cright)%3D" alt="A'\left(x\right)="/><br/> <br/> </div> <div> <div>b)</div> <div> Koska <img src="https://math-demo.abitti.fi/math.svg?latex=A%27%5Cleft(x%5Cright)%3Df%5Cleft(x%5Cright)" alt="A'\left(x\right)=f\left(x\right)"/>, pinta-alafunktio A on funktion f eräs integraalifunktio, joten voidaan saada integroimalla f(x) kautta funktio A(x)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(x%5Cright)%3D%5Cint_%7B%20%7D%5E%7B%20%7Df%5Cleft(x%5Cright)dx%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(x%2B1%5Cright)dx%3D%5Cfrac%7B1%7D%7B2%7Dx%5E2%2Bx%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="A\left(x\right)=\int_{ }^{ }f\left(x\right)dx=\int_{ }^{ }\left(x+1\right)dx=\frac{1}{2}x^2+x+C{,}\ C\in\mathbb{R}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(0%5Cright)%3D%5Cfrac%7B1%7D%7B2%7D0%5E2%2B0%2BC%3D0" alt="A\left(0\right)=\frac{1}{2}0^2+0+C=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=C%3D0" alt="C=0"/></div> <div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(2%5Cright)-A%5Cleft(1%5Cright)%3D%5Cfrac%7B5%7D%7B2%7D%3D2%5Cfrac%7B1%7D%7B2%7D%3D2%7B%2C%7D5" alt="A\left(2\right)-A\left(1\right)=\frac{5}{2}=2\frac{1}{2}=2{,}5"/></div> <br/> <div>306</div> <div>a)</div> <div>Koska <img src="https://math-demo.abitti.fi/math.svg?latex=A%27%5Cleft(x%5Cright)%3Df%5Cleft(x%5Cright)" alt="A'\left(x\right)=f\left(x\right)"/>, pinta-alafunktio A on funktion f eräs integraalifunktio, joten voidaan saada integroimalla f(x) kautta funktio A(x) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(x%5Cright)%3D%5Cint_%7B%20%7D%5E%7B%20%7Df%5Cleft(x%5Cright)dx%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(x-1%5Cright)dx%3D%5Cfrac%7B1%7D%7B2%7Dx%5E2-x%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="A\left(x\right)=\int_{ }^{ }f\left(x\right)dx=\int_{ }^{ }\left(x-1\right)dx=\frac{1}{2}x^2-x+C{,}\ C\in\mathbb{R}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(2%5Cright)%3D%5Cfrac%7B1%7D%7B2%7D2%5E2-2%2BC%3D2" alt="A\left(2\right)=\frac{1}{2}2^2-2+C=2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=C%3D0" alt="C=0"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(5%5Cright)-A%5Cleft(3%5Cright)%3D7%7B%2C%7D5-1%7B%2C%7D5%3D6" alt="A\left(5\right)-A\left(3\right)=7{,}5-1{,}5=6"/></div> <div><br/> <div>308</div> <div>Koska <img src="https://math-demo.abitti.fi/math.svg?latex=A%27%5Cleft(x%5Cright)%3Df%5Cleft(x%5Cright)" alt="A'\left(x\right)=f\left(x\right)"/>, pinta-alafunktio A on funktion f eräs integraalifunktio, joten voidaan saada integroimalla f(x) kautta funktio A(x) <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft(x%5Cright)%3D%5Cint_%7B%20%7D%5E%7B%20%7Df%5Cleft(x%5Cright)dx%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(x-1%5Cright)dx%3D%5Cfrac%7B1%7D%7B2%7Dx%5E2-x%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="A\left(x\right)=\int_{ }^{ }f\left(x\right)dx=\int_{ }^{ }\left(x-1\right)dx=\frac{1}{2}x^2-x+C{,}\ C\in\mathbb{R}"/></div> <div> </div> <div>309<br/> <div>a)</div> <div>Tässä halutaan laskea funktion y nollakohdat</div> <div>eli</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-x%5E2%2Bx%2B2%3D0" alt="-x^2+x+2=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%5C%20tai%5C%20x%3D2" alt="x=-1\ tai\ x=2"/>(Laskin) <div>eli pisteessö (-1,0) ja (2,0)</div> <div> <div>b) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-1%7D%5E2%5Cleft(-x%5E2%2Bx%2B2%3D0%5Cright)dx%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5E2%5Cleft(-1%5Ccdot%5Cfrac%7B1%7D%7B2%2B1%7Dx%5E%7B2%2B1%7D%2B1%5Ccdot%5Cfrac%7B1%7D%7B1%2B1%7Dx%5E%7B1%2B1%7D%2B2x%5Cright)%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!%7B-1%7D%7D%5E2%5Cleft(-%5Cfrac%7B1%7D%7B3%7Dx%5E3%2B%5Cfrac%7B1%7D%7B2%7Dx%5E2%2B2x%5Cright)" alt="\int_{-1}^2\left(-x^2+x+2=0\right)dx=\bigg/_{\!\!\!\!\!{-1}}^2\left(-1\cdot\frac{1}{2+1}x^{2+1}+1\cdot\frac{1}{1+1}x^{1+1}+2x\right)=\bigg/_{\!\!\!\!\!{-1}}^2\left(-\frac{1}{3}x^3+\frac{1}{2}x^2+2x\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(-%5Cfrac%7B1%7D%7B3%7D%5Ccdot2%5E3%2B%5Cfrac%7B1%7D%7B2%7D%5Ccdot2%5E2%2B2%5Ccdot2%5Cright)-%5Cleft(-%5Cfrac%7B1%7D%7B3%7D%5Ccdot%5Cleft(-1%5Cright)%5E3%2B%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cleft(-1%5Cright)%5E2%2B2%5Ccdot%5Cleft(-1%5Cright)%5Cright)%3D%5Cfrac%7B10%7D%7B3%7D-%5Cleft(-%5Cfrac%7B7%7D%7B6%7D%5Cright)%3D%5Cfrac%7B9%7D%7B2%7D%3D4%5Cfrac%7B1%7D%7B2%7D" alt="=\left(-\frac{1}{3}\cdot2^3+\frac{1}{2}\cdot2^2+2\cdot2\right)-\left(-\frac{1}{3}\cdot\left(-1\right)^3+\frac{1}{2}\cdot\left(-1\right)^2+2\cdot\left(-1\right)\right)=\frac{10}{3}-\left(-\frac{7}{6}\right)=\frac{9}{2}=4\frac{1}{2}"/></div> </div> </div> </div> <div>311<br/> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5E3%5Cleft(3t%2B4%5Cright)dt%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E3%5Cleft(3%5Ccdot%5Cfrac%7B1%7D%7B1%2B1%7Dt%5E%7B1%2B1%7D%2B4t%5Cright)%3D%5Cbigg%2F_%7B%5C!%5C!%5C!%5C!%5C!0%7D%5E3%5Cleft(%5Cfrac%7B3%7D%7B2%7Dt%5E2%2B4t%5Cright)" alt="\int_0^3\left(3t+4\right)dt=\bigg/_{\!\!\!\!\!0}^3\left(3\cdot\frac{1}{1+1}t^{1+1}+4t\right)=\bigg/_{\!\!\!\!\!0}^3\left(\frac{3}{2}t^2+4t\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Cfrac%7B3%7D%7B2%7D%5Ccdot3%5E2%2B4%5Ccdot3%5Cright)-%5Cleft(%5Cfrac%7B3%7D%7B2%7D%5Ccdot0%5E2%2B4%5Ccdot0%5Cright)%3D25%7B%2C%7D5mm" alt="=\left(\frac{3}{2}\cdot3^2+4\cdot3\right)-\left(\frac{3}{2}\cdot0^2+4\cdot0\right)=25{,}5mm"/></div> <div>b)</div> <div>jos x-akseli on smalla ajan kuvaava t-akseli, veden määrä kolmen ensimmäisen tunnin aikana on siis sama asia kuin f kuvaajan ja t-akselin rajoittaman alueen pinta-ala välillä [0,3]</div> <div>ja se on 25.5mm</div> <div>c)C</div> </div> <div>312<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5Ea%5Cleft(3x%5E2-4x%2B2%5Cright)dx%3D4" alt="\int_0^a\left(3x^2-4x+2\right)dx=4"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%3D2" alt="a=2"/>(Laskin)<br/> <br/> </div> </div> <div>314<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(4x-1%5Cright)dx%3D2x%5E2-x%2BC" alt="\int_{ }^{ }\left(4x-1\right)dx=2x^2-x+C"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)%3D3" alt="f\left(1\right)=3"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot1%5E2-1%2BC%3D3" alt="2\cdot1^2-1+C=3"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=2-1%2BC%3D3" alt="2-1+C=3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=C%3D2" alt="C=2"/><br/> <div>eli</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2x%5E2-x%2B2" alt="f\left(x\right)=2x^2-x+2"/></div> <span> Halutaan laskea kuvaajan f ja x-akselin rajoittaman alueen pinta-ala välillä [1,4] </span><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_1%5E4%5Cleft(2x%5E2-x%2B2%5Cright)dx%3D%5Cfrac%7B81%7D%7B2%7D%3D40%5Cfrac%7B1%7D%7B2%7D" alt="\int_1^4\left(2x^2-x+2\right)dx=\frac{81}{2}=40\frac{1}{2}"/>(Laskin)</div> </div> </div> </div> </div> 2019-08-27T09:35:10+03:00 https://peda.net/id/182a4fcac5a 2019-09-12T22:07:44+03:00 <span>238</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7De%5E%7B2x%7D%3D%5Cfrac%7B1%7D%7B2%7De%5E%7B2x%7D%2BC" alt="\int_{ }^{ }e^{2x}=\frac{1}{2}e^{2x}+C"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=H%5Cleft(0%5Cright)%3D%5Cfrac%7B1%7D%7B2%7De%5E%7B2%5Ccdot0%7D%2BC" alt="H\left(0\right)=\frac{1}{2}e^{2\cdot0}+C"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B2%7De%5E%7B2%5Ccdot0%7D%2BC%3D-1" alt="\frac{1}{2}e^{2\cdot0}+C=-1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=C%3D-%5Cfrac%7B3%7D%7B2%7D" alt="C=-\frac{3}{2}"/> <div><br/> <div>243</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D-6%5Csin3x%5C%20dx%3D%5Cint_%7B%20%7D%5E%7B%20%7D-2%5Ccdot3%5Ccdot%5Csin3x%5C%20dx%3D-2%5Cint_%7B%20%7D%5E%7B%20%7D3%5Ccdot%5Csin3x%5C%20dx%3D-2%5Ccdot%5Cleft(-%5Ccos3x%5Cright)%2BC%3D2%5Ccos3x%2BC" alt="\int_{ }^{ }-6\sin3x\ dx=\int_{ }^{ }-2\cdot3\cdot\sin3x\ dx=-2\int_{ }^{ }3\cdot\sin3x\ dx=-2\cdot\left(-\cos3x\right)+C=2\cos3x+C"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=G%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%7B%2C%7D1%5Cright)%3D2%5Ccos%5Cleft(3%5Ccdot%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%2BC" alt="G\left(\frac{\pi}{2}{,}1\right)=2\cos\left(3\cdot\frac{\pi}{2}\right)+C"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos%5Cleft(3%5Ccdot%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%2BC%3D1" alt="2\cos\left(3\cdot\frac{\pi}{2}\right)+C=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot0%2BC%3D1" alt="2\cdot0+C=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=C%3D1" alt="C=1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=G%5Cleft(x%5Cright)%3D2%5Ccos3x%2B1" alt="G\left(x\right)=2\cos3x+1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos3x%2B1%3D2" alt="2\cos3x+1=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos3x%3D1" alt="2\cos3x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos3x%3D%5Cfrac%7B1%7D%7B2%7D" alt="\cos3x=\frac{1}{2}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D%5Cpm%5Cfrac%7B%5Cpi%7D%7B3%7D%2Bn%5Ccdot2%5Cpi%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3A3" alt="3x=\pm\frac{\pi}{3}+n\cdot2\pi\ \ \ \ \ \left|\right|:3"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpm%5Cfrac%7B%5Cpi%7D%7B9%7D%2Bn%5Ccdot%5Cfrac%7B2%5Cpi%7D%7B3%7D%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\pm\frac{\pi}{9}+n\cdot\frac{2\pi}{3}{,}\ n\in\mathbb{Z}"/><br/> <div><br/> <div> <div> <div>250</div> <div>a)</div> <div>Koska 2,5 tunti on 150 min, on jäljellä n. 6 eliötä/min</div> <div>b) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g%27%5Cleft(x%5Cright)%3D54%7B%2C%7D06%5Ccdot%20e%5E%7B-0%7B%2C%7D01x%7D" alt="g'\left(x\right)=54{,}06\cdot e^{-0{,}01x}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D%5Cint_%7B%20%7D%5E%7B%20%7D54%7B%2C%7D06%5Ccdot%20e%5E%7B-0%7B%2C%7D01x%7Ddx%3D-3624%7B%2C%7D53e%5E%7B-0%7B%2C%7D01x%7D%2BC" alt="g\left(x\right)=\int_{ }^{ }54{,}06\cdot e^{-0{,}01x}dx=-3624{,}53e^{-0{,}01x}+C"/></div> <div>Koska tunti on 60 min</div> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(0%5Cright)%3D-3624%7B%2C%7D53e%5E%7B-0%7B%2C%7D01%5Ccdot60%7D%2BC%3D500" alt="g\left(0\right)=-3624{,}53e^{-0{,}01\cdot60}+C=500"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=C%3D4124.53" alt="C=4124.53"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(60%5Cright)%3D2135.3457606281...%5Capprox2135" alt="g\left(60\right)=2135.3457606281...\approx2135"/>  <br/> <br/> </div> </div> <div>257 <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Ctan%20xdx%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(%5Cfrac%7B%5Csin%20x%7D%7B%5Ccos%20x%7D%5Cright)dx%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(%5Csin%20x%5Ccdot%5Cleft(%5Ccos%20x%5Cright)%5E%7B-1%7D%5Cright)dx%3D-%5Cint_%7B%20%7D%5E%7B%20%7D-%5Csin%20x%5Cleft(%5Ccos%20x%5Cright)%5E%7B-1%7Ddx" alt="\int_{ }^{ }\tan xdx=\int_{ }^{ }\left(\frac{\sin x}{\cos x}\right)dx=\int_{ }^{ }\left(\sin x\cdot\left(\cos x\right)^{-1}\right)dx=-\int_{ }^{ }-\sin x\left(\cos x\right)^{-1}dx"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D%5Ccos%20x%7B%2C%7D%5C%20s%27%5Cleft(x%5Cright)%3D-%5Csin%20x%7B%2C%7D%5C%20U%3D%5Cln%5Cleft%7Cx%5Cright%7C" alt="s\left(x\right)=\cos x{,}\ s'\left(x\right)=-\sin x{,}\ U=\ln\left|x\right|"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Cln%5Cleft%7C%5Ccos%20x%5Cright%7C%2BC" alt="=-\ln\left|\cos x\right|+C"/></div> <div>Koska <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3E0%7B%2C%7D%5C%20-%5Cfrac%7B%5Cpi%7D%7B2%7D%3Cx%3C%5Cfrac%7B%5Cpi%7D%7B2%7D" alt="\cos x&gt;0{,}\ -\frac{\pi}{2}&lt;x&lt;\frac{\pi}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Cln%5Cleft(%5Ccos%20x%5Cright)%2BC" alt="=-\ln\left(\cos x\right)+C"/></div> </div> </div> 2019-08-23T15:32:39+03:00 https://peda.net/id/76f2b580c3d 2019-09-12T22:13:00+03:00 <div>145</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7B1%7D%7Bx%5E3%7Ddx%3D%5Cint_%7B%20%7D%5E%7B%20%7Dx%5E%7B-3%7Ddx%3D1%5Ccdot%5Cfrac%7B1%7D%7B-3%2B1%7Dx%5E%7B-3%2B1%7D%2BC" alt="\int_{ }^{ }\frac{1}{x^3}dx=\int_{ }^{ }x^{-3}dx=1\cdot\frac{1}{-3+1}x^{-3+1}+C"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Cfrac%7B1%7D%7B2x%5E%7B-2%7D%7D%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="=-\frac{1}{2x^{-2}}+C{,}\ C\in\mathbb{R}"/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7B9%7D%7Bx%5E2%7Ddx%3D%5Cint_%7B%20%7D%5E%7B%20%7D9x%5E%7B-2%7Ddx%3D9%5Ccdot%5Cfrac%7B1%7D%7B-2%2B1%7Dx%5E%7B-2%2B1%7D%2BC" alt="\int_{ }^{ }\frac{9}{x^2}dx=\int_{ }^{ }9x^{-2}dx=9\cdot\frac{1}{-2+1}x^{-2+1}+C"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D-9x%5E%7B-1%7D%2BC%3D-%5Cfrac%7B9%7D%7Bx%7D%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="=-9x^{-1}+C=-\frac{9}{x}+C{,}\ C\in\mathbb{R}"/></div> <div>c) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7B4%7D%7Bx%7Ddx%3D%5Cint_%7B%20%7D%5E%7B%20%7D4%5Ccdot%5Cfrac%7B1%7D%7Bx%7Ddx%3D4%5Ccdot%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7B1%7D%7Bx%7Ddx%3D4%5Cln%20x%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="\int_{ }^{ }\frac{4}{x}dx=\int_{ }^{ }4\cdot\frac{1}{x}dx=4\cdot\int_{ }^{ }\frac{1}{x}dx=4\ln x+C{,}\ C\in\mathbb{R}"/></div> <div><br/> <div> <div>147</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7Bx%2B2%7D%7Bx%7Ddx%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7Bx%7D%7Bx%7D%2B%5Cfrac%7B2%7D%7Bx%7Ddx%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7B2%7D%7Bx%7D%2B1dx%3D%5Cint_%7B%20%7D%5E%7B%20%7D2%5Ccdot%5Cfrac%7B1%7D%7Bx%7D%2B1dx%3D2%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7B1%7D%7Bx%7D%2B1dx%3D2%5Cln%20x%2Bx%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="\int_{ }^{ }\frac{x+2}{x}dx=\int_{ }^{ }\frac{x}{x}+\frac{2}{x}dx=\int_{ }^{ }\frac{2}{x}+1dx=\int_{ }^{ }2\cdot\frac{1}{x}+1dx=2\int_{ }^{ }\frac{1}{x}+1dx=2\ln x+x+C{,}\ C\in\mathbb{R}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2%5Cln%20x%2Bx-e%7B%2C%7D%5C%20x%3E0" alt="f\left(x\right)=2\ln x+x-e{,}\ x&gt;0"/><!--filtered attribute: style="max-width: 100%; border: none;"--></div> <div>b)</div> <a href="https://peda.net/petajavesi/lukio/Oppiaineet/pitk%C3%A4-matematiikka/ma9p-lops2016/kurssi-2019/itsen%C3%A4inen/luku-1-3/145-147-ja-149:responses/teht%C3%A4v%C3%A4t2/147-b-png" title="147 b.PNG"><img class="inline" src="https://peda.net/petajavesi/lukio/Oppiaineet/pitk%C3%A4-matematiikka/ma9p-lops2016/kurssi-2019/itsen%C3%A4inen/luku-1-3/145-147-ja-149:responses/teht%C3%A4v%C3%A4t2/147-b-png:file/photo/1bc1b4ec9b23d11c7a33da2cf428f54fa26bbfd4/147%20b.PNG" alt="147 b.PNG" title="147 b.PNG"/></a><br/> <div><br/> 149</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7B1%7D%7B2x%5E5%7Ddx%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cfrac%7B1%7D%7Bx%5E5%7Ddx%3D%5Cfrac%7B1%7D%7B2%7D%5Cint_%7B%20%7D%5E%7B%20%7Dx%5E%7B-5%7Ddx%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cfrac%7B1%7D%7B-5%2B1%7Dx%5E%7B-5%2B1%7D%2BC%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cfrac%7B1%7D%7B-4%7D%5Ccdot%5Cfrac%7B1%7D%7Bx%5E4%7D%2BC" alt="\int_{ }^{ }\frac{1}{2x^5}dx=\int_{ }^{ }\frac{1}{2}\cdot\frac{1}{x^5}dx=\frac{1}{2}\int_{ }^{ }x^{-5}dx=\frac{1}{2}\cdot\frac{1}{-5+1}x^{-5+1}+C=\frac{1}{2}\cdot\frac{1}{-4}\cdot\frac{1}{x^4}+C"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Cfrac%7B1%7D%7B8x%5E4%7D%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="=-\frac{1}{8x^4}+C{,}\ C\in\mathbb{R}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7Dx%5E2%5Csqrt%5B%5D%7Bx%7Ddx%3D%5Cint_%7B%20%7D%5E%7B%20%7Dx%5E2x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7Ddx%3D%5Cint_%7B%20%7D%5E%7B%20%7Dx%5E%7B%5Cfrac%7B5%7D%7B2%7D%7Ddx%3D%5Cfrac%7B1%7D%7B%5Cfrac%7B5%7D%7B2%7D%2B1%7Dx%5E%7B%5Cfrac%7B7%7D%7B2%7D%7D%2BC" alt="\int_{ }^{ }x^2\sqrt[]{x}dx=\int_{ }^{ }x^2x^{\frac{1}{2}}dx=\int_{ }^{ }x^{\frac{5}{2}}dx=\frac{1}{\frac{5}{2}+1}x^{\frac{7}{2}}+C"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B2%7D%7B7%7Dx%5E%7B%5Cfrac%7B7%7D%7B2%7D%7D%2BC%3D%5Cfrac%7B2%7D%7B7%7Dx%5E%7B%5Cfrac%7B6%7D%7B2%7D%7Dx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%2BC%3D%5Cfrac%7B2%7D%7B7%7Dx%5E3%5Csqrt%5B%5D%7Bx%7D%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="=\frac{2}{7}x^{\frac{7}{2}}+C=\frac{2}{7}x^{\frac{6}{2}}x^{\frac{1}{2}}+C=\frac{2}{7}x^3\sqrt[]{x}+C{,}\ C\in\mathbb{R}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7Bdx%7D%7B2%5Csqrt%5B%5D%7Bx%7D%7D%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7B1%7D%7B2%5Csqrt%5B%5D%7Bx%7D%7Ddx%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7Bx%7D%7Ddx%3D%5Cfrac%7B1%7D%7B2%7D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cfrac%7B1%7D%7Bx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7Ddx%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cint_%7B%20%7D%5E%7B%20%7Dx%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7Ddx" alt="\int_{ }^{ }\frac{dx}{2\sqrt[]{x}}=\int_{ }^{ }\frac{1}{2\sqrt[]{x}}dx=\int_{ }^{ }\frac{1}{2}\cdot\frac{1}{\sqrt[]{x}}dx=\frac{1}{2}\int_{ }^{ }\frac{1}{x^{\frac{1}{2}}}dx=\frac{1}{2}\cdot\int_{ }^{ }x^{-\frac{1}{2}}dx"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cfrac%7B1%7D%7B-%5Cfrac%7B1%7D%7B2%7D%2B1%7Dx%5E%7B-%5Cfrac%7B1%7D%7B2%7D%2B1%7D%2BC%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot2x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%2BC%3Dx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%2BC%3D%5Csqrt%5B%5D%7Bx%7D%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="=\frac{1}{2}\cdot\frac{1}{-\frac{1}{2}+1}x^{-\frac{1}{2}+1}+C=\frac{1}{2}\cdot2x^{\frac{1}{2}}+C=x^{\frac{1}{2}}+C=\sqrt[]{x}+C{,}\ C\in\mathbb{R}"/></div> </div> 2019-08-21T08:28:10+03:00 https://peda.net/id/71b95df8c3d 2019-09-12T21:30:07+03:00 <span>125</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(6x%5E2-4x%2B1%5Cright)dx%3D6%5Ccdot%5Cfrac%7B1%7D%7B2%2B1%7Dx%5E%7B2%2B1%7D-4%5Ccdot%5Cfrac%7B1%7D%7B1%2B1%7Dx%5E%7B1%2B1%7D%2B1%5Ccdot%20x%2BC%3D%5Cfrac%7B6%7D%7B3%7Dx%5E3-%5Cfrac%7B4%7D%7B2%7Dx%5E2%2Bx" alt="\int_{ }^{ }\left(6x^2-4x+1\right)dx=6\cdot\frac{1}{2+1}x^{2+1}-4\cdot\frac{1}{1+1}x^{1+1}+1\cdot x+C=\frac{6}{3}x^3-\frac{4}{2}x^2+x"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D2x%5E3-2x%5E2%2Bx%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="=2x^3-2x^2+x+C{,}\ C\in\mathbb{R}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(2%5Cright)%3D2%5Ccdot2%5E3-2%5Ccdot2%5E2%2B2%2BC%3D12" alt="F\left(2\right)=2\cdot2^3-2\cdot2^2+2+C=12"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot8-2%5Ccdot4%2B2%2BC%3D12" alt="2\cdot8-2\cdot4+2+C=12"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=16-8%2B2%2BC%3D12" alt="16-8+2+C=12"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=10%2BC%3D12" alt="10+C=12"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=C%3D2" alt="C=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3D2x%5E3-2x%5E2%2Bx%2B2" alt="F\left(x\right)=2x^3-2x^2+x+2"/></div> <div><br/> <div>126</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%5Ccdot%7D%5Cleft(8x%5E5-%5Cfrac%7B3%7D%7B5%7Dx%5E2%2B2%5Cright)dx%3D8%5Ccdot%5Cfrac%7B1%7D%7B5%2B1%7Dx%5E%7B5%2B1%7D-%5Cfrac%7B3%7D%7B5%7D%5Ccdot%5Cfrac%7B1%7D%7B2%2B1%7D%5C%20x%5E%7B2%2B1%7D%2B2%5Ccdot%20x%2BC" alt="\int_{ }^{\cdot}\left(8x^5-\frac{3}{5}x^2+2\right)dx=8\cdot\frac{1}{5+1}x^{5+1}-\frac{3}{5}\cdot\frac{1}{2+1}\ x^{2+1}+2\cdot x+C"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B4%7D%7B3%7Dx%5E6-%5Cfrac%7B1%7D%7B5%7Dx%5E3%2B2x%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="=\frac{4}{3}x^6-\frac{1}{5}x^3+2x+C{,}\ C\in\mathbb{R}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7Dx%5Cleft(3x%2B2%5Cright)dx%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(3x%5E2%2B2x%5Cright)dx%3D3%5Ccdot%5Cfrac%7B1%7D%7B2%2B1%7Dx%5E%7B2%2B1%7D%2B2%5Ccdot%5Cfrac%7B1%7D%7B1%2B1%7Dx%5E%7B1%2B1%7D%2BC" alt="\int_{ }^{ }x\left(3x+2\right)dx=\int_{ }^{ }\left(3x^2+2x\right)dx=3\cdot\frac{1}{2+1}x^{2+1}+2\cdot\frac{1}{1+1}x^{1+1}+C"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3Dx%5E3%2Bx%5E2%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="=x^3+x^2+C{,}\ C\in\mathbb{R}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(x%2B2%5Cright)%5Cleft(3x-4%5Cright)dx%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(3x%5E2%2B2x-8%5Cright)dx%3D3%5Ccdot%5Cfrac%7B1%7D%7B2%2B1%7Dx%5E%7B2%2B1%7D%2B2%5Ccdot%5Cfrac%7B1%7D%7B1%2B1%7Dx%5E%7B1%2B1%7D-8%5Ccdot%5Cfrac%7B1%7D%7B0%2B1%7Dx%5E%7B0%2B1%7D%2BC" alt="\int_{ }^{ }\left(x+2\right)\left(3x-4\right)dx=\int_{ }^{ }\left(3x^2+2x-8\right)dx=3\cdot\frac{1}{2+1}x^{2+1}+2\cdot\frac{1}{1+1}x^{1+1}-8\cdot\frac{1}{0+1}x^{0+1}+C"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3Dx%5E3%2Bx%5E2-8x%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="=x^3+x^2-8x+C{,}\ C\in\mathbb{R}"/><br/> <br/> <div> <div>128</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cfrac%7B1%7D%7B2%7Dx-1" alt="f\left(x\right)=\frac{1}{2}x-1"/></div> b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(%5Cfrac%7B1%7D%7B2%7Dx-1%5Cright)dx%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cfrac%7B1%7D%7B1%2B1%7Dx%5E%7B1%2B1%7D-1%5Ccdot%5Cfrac%7B1%7D%7B0%2B1%7Dx%5E%7B0%2B1%7D%2BC" alt="\int_{ }^{ }\left(\frac{1}{2}x-1\right)dx=\frac{1}{2}\cdot\frac{1}{1+1}x^{1+1}-1\cdot\frac{1}{0+1}x^{0+1}+C"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B1%7D%7B4%7Dx%5E2-x%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="=\frac{1}{4}x^2-x+C{,}\ C\in\mathbb{R}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D2" alt="x=2"/></div> </div> 2019-08-21T08:28:01+03:00 https://peda.net/id/6e159b08c3d 2019-08-21T09:53:48+03:00 <div>105</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F_2" alt="F_2"/>, koska sen muutosnopeus välillä [-1,1] on negatiivinen, vastaavan muutosnopeuksen voidaan nähdä myös kuvaajasta f(x).</div> <div> </div> <div>106</div> <div>Osoitetaan, että</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%27%5Cleft(x%5Cright)%3Df%5Cleft(x%5Cright)" alt="F'\left(x\right)=f\left(x\right)"/></div> <div>Derivoidaan funktio F(x)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%27%5Cleft(x%5Cright)%3DD%5Cleft(%5Cfrac%7B1%7D%7B3%7Dx%5E6-%5Cfrac%7B2%7D%7B5%7Dx%5E5%2B3x%5E2-%5Cfrac%7B2%7D%7B3%7D%5Cright)%3D2x%5E5-2x%5E4%2B6x%3Df%5Cleft(x%5Cright)" alt="F'\left(x\right)=D\left(\frac{1}{3}x^6-\frac{2}{5}x^5+3x^2-\frac{2}{3}\right)=2x^5-2x^4+6x=f\left(x\right)"/></div> Oletetaa, että funktio f(x) toinen integraalifunktio on <img src="https://math-demo.abitti.fi/math.svg?latex=F_2" alt="F_2"/> <div>Jos katsotaan funktio <img src="https://math-demo.abitti.fi/math.svg?latex=F_2" alt="F_2"/>integraalilauseen avulla,  <div><img src="https://math-demo.abitti.fi/math.svg?latex=G%5Cleft(x%5Cright)%3DF%5Cleft(x%5Cright)%2BC" alt="G\left(x\right)=F\left(x\right)+C"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=G%5Cleft(x%5Cright)%3DF_2%5Cleft(x%5Cright)" alt="G\left(x\right)=F_2\left(x\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3Df%5Cleft(x%5Cright)" alt="F\left(x\right)=f\left(x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=C%3D%5Cmathbb%7BR%7D" alt="C=\mathbb{R}"/></div> <div>Tässä tapauksessa C voi olla mikä tahanssa realiluku, </div> <div>Esim.</div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F_2%27%5Cleft(x%5Cright)%3Df%5Cleft(x%5Cright)" alt="F_2'\left(x\right)=f\left(x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=F_2%5Cleft(x%5Cright)%3D%5Cfrac%7B1%7D%7B3%7Dx%5E6-%5Cfrac%7B2%7D%7B5%7Dx%5E5%2B3x%5E2%2B1%7B%2C%7D%5C%20C%3D1" alt="F_2\left(x\right)=\frac{1}{3}x^6-\frac{2}{5}x^5+3x^2+1{,}\ C=1"/></div> <div><br/> <div> <div> <div>108</div> </div> </div> </div> </div> </div> <div>a) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%27%5Cleft(x%5Cright)%3Df%5Cleft(x%5Cright)" alt="F'\left(x\right)=f\left(x\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3Dx%5E4%2BC%7B%2C%7D%5C%20C%3D%5Cmathbb%7BR%7D" alt="F\left(x\right)=x^4+C{,}\ C=\mathbb{R}"/><br/> <div>Koska funktio F(x) kulkee pisteen (0,1/2) kautta, </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(0%5Cright)%3D0%5E4%2BC%3D%5Cfrac%7B1%7D%7B2%7D" alt="F\left(0\right)=0^4+C=\frac{1}{2}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=0%5E4%2BC%3D%5Cfrac%7B1%7D%7B2%7D" alt="0^4+C=\frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=C%3D%5Cfrac%7B1%7D%7B2%7D" alt="C=\frac{1}{2}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3Dx%5E4%2B%5Cfrac%7B1%7D%7B2%7D" alt="F\left(x\right)=x^4+\frac{1}{2}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%27%5Cleft(x%5Cright)%3Df%5Cleft(x%5Cright)" alt="F'\left(x\right)=f\left(x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3Dx%5E5%2BC%7B%2C%7D%5C%20C%3D%5Cmathbb%7BR%7D" alt="F\left(x\right)=x^5+C{,}\ C=\mathbb{R}"/><br/> Koska funktio F(x) kulkee pisteen (0,1/2) kautta, <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(0%5Cright)%3D0%5E5%2BC%3D%5Cfrac%7B1%7D%7B2%7D" alt="F\left(0\right)=0^5+C=\frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=0%5E5%2BC%3D%5Cfrac%7B1%7D%7B2%7D" alt="0^5+C=\frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=C%3D%5Cfrac%7B1%7D%7B2%7D" alt="C=\frac{1}{2}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3Dx%5E5%2B%5Cfrac%7B1%7D%7B2%7D" alt="F\left(x\right)=x^5+\frac{1}{2}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%27%5Cleft(x%5Cright)%3Df%5Cleft(x%5Cright)" alt="F'\left(x\right)=f\left(x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3De%5Ex%2BC%7B%2C%7D%5C%20C%3D%5Cmathbb%7BR%7D" alt="F\left(x\right)=e^x+C{,}\ C=\mathbb{R}"/><br/> Koska funktio F(x) kulkee pisteen (0,1/2) kautta, <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(0%5Cright)%3De%5E0%2BC%3D%5Cfrac%7B1%7D%7B2%7D" alt="F\left(0\right)=e^0+C=\frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5E0%2BC%3D%5Cfrac%7B1%7D%7B2%7D" alt="e^0+C=\frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%2BC%3D%5Cfrac%7B1%7D%7B2%7D" alt="1+C=\frac{1}{2}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=C%3D-%5Cfrac%7B1%7D%7B2%7D" alt="C=-\frac{1}{2}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3De%5Ex-%5Cfrac%7B1%7D%7B2%7D" alt="F\left(x\right)=e^x-\frac{1}{2}"/><br/> <br/> <div>109</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F_1%5Cleft(x%5Cright)%3Dx%5E2%2B1" alt="F_1\left(x\right)=x^2+1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=F_2%5Cleft(x%5Cright)%3Dx%5E2%2B2" alt="F_2\left(x\right)=x^2+2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=F_3%5Cleft(x%5Cright)%3Dx%5E2%2B3" alt="F_3\left(x\right)=x^2+3"/><br/> <a href="https://peda.net/p/kirin_porsti/ma/ma9p/teht%C3%A4v%C3%A4t/1-1/109-a-png#top" title="109 a.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma9p/teht%C3%A4v%C3%A4t/1-1/109-a-png:file/photo/278a116cddb0a8f6918b8f3203392479d365229a/109%20a.PNG" alt="" title="109 a.PNG" class="inline" loading="lazy"/></a><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F_1%5Cleft(3%5Cright)-F_1%5Cleft(2%5Cright)%3D%5Cleft(3%5E2%2B1%5Cright)-%5Cleft(2%5E2%2B1%5Cright)%3D10-5%3D5" alt="F_1\left(3\right)-F_1\left(2\right)=\left(3^2+1\right)-\left(2^2+1\right)=10-5=5"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=F_2%5Cleft(3%5Cright)-F_2%5Cleft(2%5Cright)%3D%5Cleft(3%5E2%2B2%5Cright)-%5Cleft(2%5E2%2B2%5Cright)%3D11-6%3D5" alt="F_2\left(3\right)-F_2\left(2\right)=\left(3^2+2\right)-\left(2^2+2\right)=11-6=5"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=F_3%5Cleft(3%5Cright)-F_3%5Cleft(2%5Cright)%3D%5Cleft(3%5E2%2B3%5Cright)-%5Cleft(2%5E2%2B3%5Cright)%3D12-7%3D5" alt="F_3\left(3\right)-F_3\left(2\right)=\left(3^2+3\right)-\left(2^2+3\right)=12-7=5"/><br/> <span>Huomasin, että funktioiden on ratkaisu on kaikille kolmelle funktiolle 5</span> <div>c) </div> <div>Kyllä, koska lakussa plus ja miinus merkit kumoavat toisensa.</div> </div> 2019-08-21T08:27:55+03:00