https://peda.net/id/ba4ae6ac31e 2020-01-08T08:33:05+02:00 https://peda.net/:static/532/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/b50e03f0426 2020-02-04T00:42:18+02:00 Laatikossa on 7 punaista , 8 sinistä ja 5 mustaa palloa. Millä todennäköisyydellä <div>a) Nostetaan 2 samanvääristä palloa</div> <div>b) Nostetaan vähintään 3 punaista, kun kaikkiaan nostetaan 4 palloa?</div> <div> </div> <div>a) </div> <div>P(2 samanväristä)=P(2p tai 2s tai 2m)</div> <div>(Tapahtumat erillisiä)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(2p%5Cright)%2BP%5Cleft(2s%5Cright)%2BP%5Cleft(2m%5Cright)" alt="=P\left(2p\right)+P\left(2s\right)+P\left(2m\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(1.p%5C%20ja%5C%202.p%5Cright)%2BP%5Cleft(1.s%5C%20ja%5C%202.s%5Cright)%2B%5C%20P%5Cleft(1.m%5C%20ja%5C%202.m%5Cright)" alt="=P\left(1.p\ ja\ 2.p\right)+P\left(1.s\ ja\ 2.s\right)+\ P\left(1.m\ ja\ 2.m\right)"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(1.p%5Cright)%5Ccdot%20P%5Cleft(2.p%5Cright)%2BP%5Cleft(1.s%5Cright)%5Ccdot%20P%5Cleft(2.s%5Cright)%2B%5C%20P%5Cleft(1.m%5Cright)%5Ccdot%20P%5Cleft(2.m%5Cright)" alt="=P\left(1.p\right)\cdot P\left(2.p\right)+P\left(1.s\right)\cdot P\left(2.s\right)+\ P\left(1.m\right)\cdot P\left(2.m\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(1.p%5Cright)%5Ccdot%20P%5Cleft(%5Cfrac%7B2.p%7D%7B1.p%7D%5Cright)%2BP%5Cleft(1.s%5Cright)%5Ccdot%20P%5Cleft(%5Cfrac%7B2.s%7D%7B1.s%7D%5Cright)%2B%5C%20P%5Cleft(1.m%5Cright)%5Ccdot%20P%5Cleft(%5Cfrac%7B2.m%7D%7B1.m%7D%5Cright)" alt="=P\left(1.p\right)\cdot P\left(\frac{2.p}{1.p}\right)+P\left(1.s\right)\cdot P\left(\frac{2.s}{1.s}\right)+\ P\left(1.m\right)\cdot P\left(\frac{2.m}{1.m}\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B7%7D%7B20%7D%5Ccdot%5Cfrac%7B6%7D%7B19%7D%2B%5Cfrac%7B8%7D%7B20%7D%5Ccdot%5Cfrac%7B7%7D%7B19%7D%2B%5Cfrac%7B5%7D%7B20%7D%5Ccdot%5Cfrac%7B4%7D%7B19%7D" alt="=\frac{7}{20}\cdot\frac{6}{19}+\frac{8}{20}\cdot\frac{7}{19}+\frac{5}{20}\cdot\frac{4}{19}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B59%7D%7B190%7D%5Capprox0%7B%2C%7D31" alt="=\frac{59}{190}\approx0{,}31"/></div> <div> </div> <div>b)</div> <div>Kakki alkeistapaukset ovat kaikki mahdolliset 4 pallon joukot eli <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B20%7D%7B4%7D" alt="\binom{20}{4}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(v%C3%A4h%5C%203p%5Cright)%3DP%5Cleft(3p%5C%20tai%5C%204p%5Cright)%3DP%5Cleft(3p%5Cright)%2BP%5Cleft(4p%5Cright)" alt="P\left(väh\ 3p\right)=P\left(3p\ tai\ 4p\right)=P\left(3p\right)+P\left(4p\right)"/></div> Kun 3 punaista nostetaan, nostetaan 1 muuvärinen , eri tapoja on tällöin <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B7%7D%7B3%7D%5Ccdot%5Cbinom%7B13%7D%7B1%7D" alt="\binom{7}{3}\cdot\binom{13}{1}"/> <div>Kun 4 punaista voidaan nostaa <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B7%7D%7B4%7D" alt="\binom{7}{4}"/>eri tavalla.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(3p%5Cright)%2BP%5Cleft(4p%5Cright)%3D%5Cfrac%7B%5Cbinom%7B7%7D%7B3%7D%5Ccdot%5Cbinom%7B17%7D%7B1%7D%7D%7B%5Cbinom%7B20%7D%7B4%7D%7D%2B%5Cfrac%7B%5Cbinom%7B7%7D%7B4%7D%7D%7B%5Cbinom%7B20%7D%7B4%7D%7D%3D%5Cfrac%7B98%7D%7B969%7D%5Capprox0%7B%2C%7D10" alt="P\left(3p\right)+P\left(4p\right)=\frac{\binom{7}{3}\cdot\binom{17}{1}}{\binom{20}{4}}+\frac{\binom{7}{4}}{\binom{20}{4}}=\frac{98}{969}\approx0{,}10"/><br/> <div> <div><br/> 1015<br/> a)<br/> <br/> </div> </div> </div> 2020-01-29T10:16:53+02:00 https://peda.net/id/a85f3e76426 2020-01-29T10:16:32+02:00 802<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(4x%5E6-2x%5E3%2B3x%5Cright)dx%3D%5Cfrac%7B4%7D%7B7%7Dx%5E7-%5Cfrac%7B2%7D%7B4%7Dx%5E4%2B%5Cfrac%7B3%7D%7B2%7Dx%5E2%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="\int_{ }^{ }\left(4x^6-2x^3+3x\right)dx=\frac{4}{7}x^7-\frac{2}{4}x^4+\frac{3}{2}x^2+C{,}\ C\in\mathbb{R}"/><br/> b)<br/> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI4MjIw/1313592/files?id=5e3127380a975a3c4531a579" alt="\int_{ }^{ }\left(x^2-1\right)^2dx=\int_{ }^{ }\left(x^2-1\right)\left(x^2-1\right)=\int_{ }^{ }\left(x^4-2x^2+1\right)"/><!--filtered attribute: data-userfileid="5e3127380a975a3c4531a579"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI4MjIw/1313592/files?id=5e3127490a975a3c4531a634" alt="=\frac{1}{5}x^5-\frac{2}{3}x^3+x+C{,}\ C\in\mathbb{R}"/><!--filtered attribute: data-userfileid="5e3127490a975a3c4531a634"--><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(%5Cfrac%7B1%7D%7Bx%5E3%7D%2B%5Csqrt%5B%5D%7Bx%7D%5Cright)dx%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(x%5E%7B-3%7D%2Bx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%5Cright)dx%3D-%5Cfrac%7B1%7D%7B2%7Dx%5E%7B-2%7D%2B%5Cfrac%7B2%7D%7B3%7Dx%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%3D-%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cfrac%7B1%7D%7Bx%5E2%7D%2B%5Cfrac%7B2%7D%7B3%7Dx%5E%7B%5Cfrac%7B2%7D%7B2%7D%7D%5Ccdot%20x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%3D-%5Cfrac%7B1%7D%7B2x%5E2%7D%2B%5Cfrac%7B2%7D%7B3%7Dx%5Csqrt%5B%5D%7Bx%7D%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="\int_{ }^{ }\left(\frac{1}{x^3}+\sqrt[]{x}\right)dx=\int_{ }^{ }\left(x^{-3}+x^{\frac{1}{2}}\right)dx=-\frac{1}{2}x^{-2}+\frac{2}{3}x^{\frac{3}{2}}=-\frac{1}{2}\cdot\frac{1}{x^2}+\frac{2}{3}x^{\frac{2}{2}}\cdot x^{\frac{1}{2}}=-\frac{1}{2x^2}+\frac{2}{3}x\sqrt[]{x}+C{,}\ C\in\mathbb{R}"/></p> d)<br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI4MjIw/1313592/files?id=5e312b3f0a975a3c4531eff6" alt="\int_{ }^{ }\left(2x+t\right)dx=x^2+tx+C{,}\ C\in\mathbb{R}"/><!--filtered attribute: data-userfileid="5e312b3f0a975a3c4531eff6"--><br/> e)<br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI4MjIw/1313592/files?id=5e312b660a975a3c4531f39e" alt="\int_{ }^{ }\left(2x+t\right)dt=\frac{1}{2}t^2+2xt+C{,}\ C\in\mathbb{R}"/><!--filtered attribute: data-userfileid="5e312b660a975a3c4531f39e"--><br/> f)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(a%2Bb%2Bt%5Cright)dt%3D%5Cfrac%7B1%7D%7B2%7Dt%5E2%2Bat%2Bbt%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="\int_{ }^{ }\left(a+b+t\right)dt=\frac{1}{2}t^2+at+bt+C{,}\ C\in\mathbb{R}"/><br/> <br/> 803<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(%5Csqrt%5B%5D%7Bx%7D%5Cleft(x-2%5Cright)%5Cright)dx%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(x%5Csqrt%5B%5D%7Bx%7D-2%5Csqrt%5B%5D%7Bx%7D%5Cright)dx" alt="\int_{ }^{ }\left(\sqrt[]{x}\left(x-2\right)\right)dx=\int_{ }^{ }\left(x\sqrt[]{x}-2\sqrt[]{x}\right)dx"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(x%5Ccdot%20x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D-2%5Ccdot%20x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%5Cright)dx" alt="=\int_{ }^{ }\left(x\cdot x^{\frac{1}{2}}-2\cdot x^{\frac{1}{2}}\right)dx"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cint_%7B%20%7D%5E%7B%20%7D%5Cleft(x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D-2x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%5Cright)dx" alt="=\int_{ }^{ }\left(x^{\frac{3}{2}}-2x^{\frac{1}{2}}\right)dx"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B2%7D%7B5%7Dx%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D-%5Cfrac%7B4%7D%7B3%7Dx%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%3D%5Cfrac%7B2%7D%7B5%7Dx%5E%7B%5Cfrac%7B4%7D%7B2%7D%7D%5Ccdot%20x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D-%5Cfrac%7B4%7D%7B3%7Dx%5Ccdot%20x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%3D%5Cfrac%7B2%7D%7B5%7Dx%5E2%5Csqrt%5B%5D%7Bx%7D-%5Cfrac%7B4%7D%7B3%7Dx%5Csqrt%5B%5D%7Bx%7D%2BC%7B%2C%7D%5C%20C%5Cin%5Cmathbb%7BR%7D" alt="=\frac{2}{5}x^{\frac{5}{2}}-\frac{4}{3}x^{\frac{3}{2}}=\frac{2}{5}x^{\frac{4}{2}}\cdot x^{\frac{1}{2}}-\frac{4}{3}x\cdot x^{\frac{1}{2}}=\frac{2}{5}x^2\sqrt[]{x}-\frac{4}{3}x\sqrt[]{x}+C{,}\ C\in\mathbb{R}"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3D%5Cfrac%7B2%7D%7B5%7Dx%5E2%5Csqrt%5B%5D%7Bx%7D-%5Cfrac%7B4%7D%7B3%7Dx%5Csqrt%5B%5D%7Bx%7D%2BC" alt="F\left(x\right)=\frac{2}{5}x^2\sqrt[]{x}-\frac{4}{3}x\sqrt[]{x}+C"/> <div>Halutaan, että </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(4%5Cright)%3D7" alt="F\left(4\right)=7"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(4%5Cright)%3D%5Cfrac%7B2%7D%7B5%7D%5Ccdot4%5E2%5Ccdot%5Csqrt%5B%5D%7B4%7D-%5Cfrac%7B4%7D%7B3%7D%5Ccdot4%5Ccdot%5Csqrt%5B%5D%7B4%7D%2BC" alt="F\left(4\right)=\frac{2}{5}\cdot4^2\cdot\sqrt[]{4}-\frac{4}{3}\cdot4\cdot\sqrt[]{4}+C"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(4%5Cright)%3D%5Cfrac%7B64%7D%7B5%7D-%5Cfrac%7B32%7D%7B3%7D%2BC" alt="F\left(4\right)=\frac{64}{5}-\frac{32}{3}+C"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B64%7D%7B5%7D-%5Cfrac%7B32%7D%7B3%7D%2BC%3D7" alt="\frac{64}{5}-\frac{32}{3}+C=7"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=C%3D7-%5Cfrac%7B64%7D%7B5%7D%2B%5Cfrac%7B32%7D%7B3%7D" alt="C=7-\frac{64}{5}+\frac{32}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=C%3D%5Cfrac%7B73%7D%7B15%7D" alt="C=\frac{73}{15}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%5Cleft(x%5Cright)%3D%5Cfrac%7B2%7D%7B5%7Dx%5E2%5Csqrt%5B%5D%7Bx%7D-%5Cfrac%7B4%7D%7B3%7Dx%5Csqrt%5B%5D%7Bx%7D%2B%5Cfrac%7B73%7D%7B15%7D" alt="F\left(x\right)=\frac{2}{5}x^2\sqrt[]{x}-\frac{4}{3}x\sqrt[]{x}+\frac{73}{15}"/></div> <div> </div> 810 <br/> b)<br/> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a70a975a3c45087ab3" alt="y=3x"/><!--filtered attribute: data-userfileid="5e2e97a70a975a3c45087ab3"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a70a975a3c45087abe" alt="y=x^2+2x^2"/><!--filtered attribute: data-userfileid="5e2e97a70a975a3c45087abe"--></p> <div>Ratkaistaan leikkauskohdat</div> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a80a975a3c45087acd" alt="3x^2=x^2+2x^2"/><!--filtered attribute: data-userfileid="5e2e97a80a975a3c45087acd"--><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a80a975a3c45087ad2" alt="x^3+2x-3x=0"/><!--filtered attribute: data-userfileid="5e2e97a80a975a3c45087ad2"--><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a80a975a3c45087acf" alt="x\left(x^2+2x-3\right)=0"/><!--filtered attribute: data-userfileid="5e2e97a80a975a3c45087acf"--><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a80a975a3c45087ac8" alt="x=0"/><!--filtered attribute: data-userfileid="5e2e97a80a975a3c45087ac8"--></div> <div>tai<br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a90a975a3c45087adb" alt="x^2+2x-3=0"/><!--filtered attribute: data-userfileid="5e2e97a90a975a3c45087adb"--><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a90a975a3c45087add" alt="x=\frac{-2\pm\sqrt[]{2^2-4\cdot1\cdot\left(-3\right)}}{2\cdot1}=\frac{-2\pm\sqrt[]{4+12}}{2}=\frac{-2\pm\sqrt[]{16}}{2}=\frac{-2\pm4}{2}"/><!--filtered attribute: data-userfileid="5e2e97a90a975a3c45087add"--><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a90a975a3c45087ae3" alt="x=\frac{-2+4}{2}=1"/><!--filtered attribute: data-userfileid="5e2e97a90a975a3c45087ae3"--></div> <div>tai<br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a90a975a3c45087ad9" alt="x=\frac{-2-4}{2}=-3"/><!--filtered attribute: data-userfileid="5e2e97a90a975a3c45087ad9"-->Koska käyrien järjestys voi muuttua vain leikkauskohdissa, voidaan järjestystä väleillä [-3,0] ja [0,1] tutkia testipisteillä</div> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a90a975a3c45087aef" alt="3\cdot\left(-1\right)=-3"/><!--filtered attribute: data-userfileid="5e2e97a90a975a3c45087aef"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a90a975a3c45087ae1" alt="\left(-1\right)^3+2\cdot\left(-1\right)^2=-1+2=1"/><!--filtered attribute: data-userfileid="5e2e97a90a975a3c45087ae1"--></p> <div>Siis välillä [-3,0] on </div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a90a975a3c45087aea" alt="x^3+2x\ge3x"/><!--filtered attribute: data-userfileid="5e2e97a90a975a3c45087aea"--></p> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a90a975a3c45087af1" alt="3\cdot\left(\frac{1}{2}\right)=\frac{3}{2}"/><!--filtered attribute: data-userfileid="5e2e97a90a975a3c45087af1"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a90a975a3c45087afb" alt="\left(\frac{1}{2}\right)^3+2\cdot\left(\frac{1}{2}\right)=\frac{1}{8}+\frac{2}{4}=\frac{1}{8}+\frac{4}{8}=\frac{5}{8}"/><!--filtered attribute: data-userfileid="5e2e97a90a975a3c45087afb"--></p> <div>Siis välillä [0,1] </div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a90a975a3c45087af3" alt="3x\ge x^3+2x^2"/><!--filtered attribute: data-userfileid="5e2e97a90a975a3c45087af3"--></p> <div>Kysytty pinta-ala on siis </div> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a90a975a3c45087af5" alt="A=\int_{-3}^0x^3+2x^2-3xdx+\int_0^13x-\left(x^3-2x^2\right)dx"/><!--filtered attribute: data-userfileid="5e2e97a90a975a3c45087af5"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97aa0a975a3c45087b06" alt="=\bigg/_{\!\!\!\!\!{-3}}^0\frac{1}{4}x^4+\frac{2}{3}x^3-\frac{3}{2}x^2+\bigg/_{\!\!\!\!\!0}^1\frac{3}{2}x^2-\frac{1}{4}x^4-\frac{2}{3}x^3"/><!--filtered attribute: data-userfileid="5e2e97aa0a975a3c45087b06"--></p> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a90a975a3c45087af8" alt="=\left(0-\left(\frac{1}{4}\left(-3\right)^4+\frac{2}{3}\left(-3\right)^3-\frac{3}{2}\left(-3\right)^2\right)\right)+\left(\left(\frac{3}{2}-\frac{1}{4}-\frac{2}{3}\right)-0\right)"/><!--filtered attribute: data-userfileid="5e2e97a90a975a3c45087af8"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97aa0a975a3c45087b03" alt="=\frac{45}{4}+\frac{7}{12}"/><!--filtered attribute: data-userfileid="5e2e97aa0a975a3c45087b03"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NDM3/1313592/files?id=5e2e97a90a975a3c45087aed" alt="=\frac{71}{6}"/><!--filtered attribute: data-userfileid="5e2e97a90a975a3c45087aed"--></p> 2020-01-29T10:16:32+02:00 https://peda.net/id/c726137e3dd 2020-01-29T08:26:31+02:00 <div>702</div> <div>a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow2%7D%5C%20%5Cfrac%7Bx%5E2-4%7D%7Bx-2%7D%3D%5Clim_%7Bx%5Crightarrow2%7D%5Cfrac%7B%5Cleft(x-2%5Cright)%5Cleft(x%2B2%5Cright)%7D%7B%5Cleft(x-2%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow2%7Dx%2B2%3D2%2B2%3D4" alt="\lim_{x\rightarrow2}\ \frac{x^2-4}{x-2}=\lim_{x\rightarrow2}\frac{\left(x-2\right)\left(x+2\right)}{\left(x-2\right)}=\lim_{x\rightarrow2}x+2=2+2=4"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-7%7D%5Cfrac%7Bx%2B7%7D%7B2x%5E2%2B10x-28%7D" alt="\lim_{x\rightarrow-7}\frac{x+7}{2x^2+10x-28}"/><br/> <div>Lasketaan nimittäjän nollakohdat:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%2B10x-28%3D0" alt="2x^2+10x-28=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-10%5Cpm%5Csqrt%5B%5D%7B10%5E2-4%5Ccdot2%5Ccdot%5Cleft(-28%5Cright)%7D%7D%7B2%5Ccdot2%7D%3D%5Cfrac%7B-10%5Cpm18%7D%7B4%7D" alt="x=\frac{-10\pm\sqrt[]{10^2-4\cdot2\cdot\left(-28\right)}}{2\cdot2}=\frac{-10\pm18}{4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D2" alt="x=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-7" alt="x=-7"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-7%7D%5Cfrac%7Bx%2B7%7D%7B2x%5E2%2B10x-28%7D%3D%5Clim_%7Bx%5Crightarrow-7%7D%5Cfrac%7Bx%2B7%7D%7B2%5Cleft(x-2%5Cright)%5Cleft(x%2B7%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow-7%7D%5Cfrac%7B1%7D%7B2%5Cleft(x-2%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow-7%7D%5Cfrac%7B1%7D%7B2x-4%7D%3D%5Cfrac%7B1%7D%7B2%5Ccdot%5Cleft(-7%5Cright)-4%7D%3D-%5Cfrac%7B1%7D%7B18%7D" alt="\lim_{x\rightarrow-7}\frac{x+7}{2x^2+10x-28}=\lim_{x\rightarrow-7}\frac{x+7}{2\left(x-2\right)\left(x+7\right)}=\lim_{x\rightarrow-7}\frac{1}{2\left(x-2\right)}=\lim_{x\rightarrow-7}\frac{1}{2x-4}=\frac{1}{2\cdot\left(-7\right)-4}=-\frac{1}{18}"/>c)</div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow6%7D%5Cfrac%7Bx%5E2-6x%7D%7Bx%5E3-12x%5E2%2B36x%7D%3D%5Cfrac%7Bx%5Cleft(x-6%5Cright)%7D%7Bx%5Cleft(x%5E2-12x%2B36%5Cright)%7D%3D%5Cfrac%7Bx-6%7D%7B%5Cleft(x-6%5Cright)%5E2%7D%3D%5Cfrac%7B1%7D%7B%5Cleft(x-6%5Cright)%7D%5Crightarrow%5Cinfty%7B%2C%7D%5C%20kun%5C%20x%5Crightarrow6" alt="\lim_{x\rightarrow6}\frac{x^2-6x}{x^3-12x^2+36x}=\frac{x\left(x-6\right)}{x\left(x^2-12x+36\right)}=\frac{x-6}{\left(x-6\right)^2}=\frac{1}{\left(x-6\right)}\rightarrow\infty{,}\ kun\ x\rightarrow6"/></div> <span>Päädytään tilanteeseen ''</span><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B0%7D" alt="\frac{1}{0}"/><span>''</span> <div>Raja-arvoa ei ole.<br/> d)<br/> <img class="fr-fic fr-dii" src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow0%7D%5Cfrac%7Be%5E%7B2x%7D-1%7D%7B1-e%5Ex%7D%3D%5Clim_%7Bx%5Crightarrow0%7D%5Cfrac%7B%5Cleft(e%5Ex%5Cright)%5E2-1%5E2%7D%7B-%5Cleft(e%5Ex-1%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow0%7D%5Cfrac%7B%5Cleft(e%5Ex-1%5Cright)%5Cleft(e%5Ex%2B1%5Cright)%7D%7B-%5Cleft(e%5Ex-1%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow0%7D-e%5Ex-1%3D-1-1%3D-2" alt="\lim_{x\rightarrow0}\frac{e^{2x}-1}{1-e^x}=\lim_{x\rightarrow0}\frac{\left(e^x\right)^2-1^2}{-\left(e^x-1\right)}=\lim_{x\rightarrow0}\frac{\left(e^x-1\right)\left(e^x+1\right)}{-\left(e^x-1\right)}=\lim_{x\rightarrow0}-e^x-1=-1-1=-2"/><!--filtered attribute: data-fr-image-pasted="true"--><br/> <br/> <br/> 703<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%5Cbegin%7Bcases%7D%0Ae%5E%7Bx-1%7D%26%7B%2C%7Dkun%3C1%5C%5C%0A%5Cln%20x%2B1%26%7B%2C%7Dkunx%5Cge1%0A%5Cend%7Bcases%7D" alt="f\left(x\right)\begin{cases}&#10;e^{x-1}&amp;{,}kun&lt;1\\&#10;\ln x+1&amp;{,}kunx\ge1&#10;\end{cases}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1-%7De%5E%7Bx-1%7D%3De%5E%7B1-1%7D%3De%5E0%3D1" alt="\lim_{x\rightarrow1-}e^{x-1}=e^{1-1}=e^0=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1%2B%7D%5Cln%20x%2B1%3D%5Cln1%2B1%3D0%2B1%3D1" alt="\lim_{x\rightarrow1+}\ln x+1=\ln1+1=0+1=1"/><br/> <span> </span><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MTAx/1313592/files?id=5e2e895c0a975a3c4506371b" alt="\lim_{x\rightarrow1}\ f\left(x\right)=1"/><!--filtered attribute: data-userfileid="5e2e895c0a975a3c4506371b"--><br/> <div>Halutaan, että funktio f on jatkuva</div> <div>Joten </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)%3D%5Clim_%7Bx%5Crightarrow1%7Df%5Cleft(x%5Cright)" alt="f\left(1\right)=\lim_{x\rightarrow1}f\left(x\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)%3D%5Cln1%2B1%3D1" alt="f\left(1\right)=\ln1+1=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=1%3D1" alt="1=1"/> <div>Funktio on jakuva</div> b)<br/> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MTAx/1313592/files?id=5e2e898f0a975a3c45063d43" alt="\begin{cases} \frac{x^2-1}{x-1}&amp;{,}\ kun\ x&lt;1\\ 3&amp;{,}\ kun\ x=1\\ \frac{x-1}{\sqrt[]{x}-1}&amp;{,}\ kun\ x&gt;1 \end{cases}"/><!--filtered attribute: data-userfileid="5e2e898f0a975a3c45063d43"--><!--filtered attribute: style="box-sizing: border-box; border: 1px solid #e6f2f8; vertical-align: bottom; position: relative; max-width: 100%; cursor: pointer; display: inline-block; float: none; margin-left: 5px; margin-right: 5px; max-height: 1000px; padding: 3px 10px; background: #edf9ff;"--></div> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MTAx/1313592/files?id=5e2e898f0a975a3c45063d3b" alt="\lim_{x\rightarrow1-}\frac{x^2-1}{x-1}=\lim_{x\rightarrow1-}\frac{\left(x-1\right)\left(x+1\right)}{\left(x-1\right)}=\lim_{x\rightarrow1-}x+1=1+1=2"/><!--filtered attribute: data-userfileid="5e2e898f0a975a3c45063d3b"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MTAx/1313592/files?id=5e2e898f0a975a3c45063d34" alt="\lim_{x\rightarrow1+}\frac{x-1}{\sqrt[]{x}-1}=\lim_{x\rightarrow1+}\frac{\left(\sqrt[]{x}+1\right)\left(\sqrt[]{x}-1\right)}{\sqrt[]{x}-1}=\sqrt[]{x}+1=1+1=2"/><!--filtered attribute: data-userfileid="5e2e898f0a975a3c45063d34"--></p> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MTAx/1313592/files?id=5e2e898f0a975a3c45063d39" alt="\lim_{x\rightarrow1}f\left(x\right)=2"/><!--filtered attribute: data-userfileid="5e2e898f0a975a3c45063d39"--></div> <div>Halutaan, että funktio f on jatkuva<br/> Joten</div> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MTAx/1313592/files?id=5e2e898f0a975a3c45063d4a" alt="f\left(1\right)=\lim_{x\rightarrow1}f\left(x\right)"/><!--filtered attribute: data-userfileid="5e2e898f0a975a3c45063d4a"--></div> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MTAx/1313592/files?id=5e2e898f0a975a3c45063d28" alt="3\ne2"/><!--filtered attribute: data-userfileid="5e2e898f0a975a3c45063d28"--></div> <div>Funktio ei ole jatkuva</div> </div> <div>704</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cbegin%7Bcases%7D%0Aax%5E2%2B3%26%7B%2C%7D%5C%20x%3C-2%5C%5C%0Ax%5E2%2Ba%5E2x-1%26%7B%2C%7D%5C%20x%5Cge-2%0A%5Cend%7Bcases%7D" alt="f\left(x\right)=\begin{cases}&#10;ax^2+3&amp;{,}\ x&lt;-2\\&#10;x^2+a^2x-1&amp;{,}\ x\ge-2&#10;\end{cases}"/></div> <div>Halutaan, että <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-2%5Cright)%3D%5Clim_%7Bx%5Crightarrow-2%7Df%5Cleft(x%5Cright)" alt="f\left(-2\right)=\lim_{x\rightarrow-2}f\left(x\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-2%5Cright)%3D%5Cleft(-2%5Cright)%5E2%2Ba%5E2%5Ccdot%5Cleft(-2%5Cright)-1%3D4-2a%5E2-1" alt="f\left(-2\right)=\left(-2\right)^2+a^2\cdot\left(-2\right)-1=4-2a^2-1"/></div> <div>toispuoleiset raja-arvot</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-2-%7Df%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow-2-%7Dax%5E2%2B3%3Da%5Cleft(-2%5Cright)%5E2%2B3%3D4a%2B3" alt="\lim_{x\rightarrow-2-}f\left(x\right)=\lim_{x\rightarrow-2-}ax^2+3=a\left(-2\right)^2+3=4a+3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-2%2B%7Df%5Cleft(x%5Cright)%3D4-2a%5E2-1" alt="\lim_{x\rightarrow-2+}f\left(x\right)=4-2a^2-1"/></div> <div>Jotta raja-arvo olisi olemassa ja funktio olisi jatkuva, on oltava</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4a%2B3%3D4-2a%5E2-1" alt="4a+3=4-2a^2-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2a%5E2%2B4a%3D0" alt="2a^2+4a=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cleft(2a%2B4%5Cright)%3D0" alt="a\left(2a+4\right)=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D0%5C%20tai%5C%20a%3D-2" alt="a=0\ tai\ a=-2"/><br/> <div>Nyt siis</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-2%7Df%5Cleft(x%5Cright)%3D4%5Ccdot%5Cleft(-2%5Cright)%2B3%3D-5" alt="\lim_{x\rightarrow-2}f\left(x\right)=4\cdot\left(-2\right)+3=-5"/> <div>ja</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-2%5Cright)%3D%5Cleft(-2%5Cright)%5E2%2B%5Cleft(-2%5Cright)%5E2%5Ccdot%5Cleft(-2%5Cright)-1%3D-5" alt="f\left(-2\right)=\left(-2\right)^2+\left(-2\right)^2\cdot\left(-2\right)-1=-5"/><br/> <div>tai </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%7D" alt="\lim_{x\rightarrow}"/></div> <div><br/> 705<br/> a)<br/> <span>Ei mitään. Bolzanon lauseen jatkuvuusehto ei toteudu välillä [1, 3]. Väleillä ]1, 2[ ja ]2, 3[ voi olla nollakohtia, mutta annetut tiedot eivät riitä asian päättelemiseen.</span></div> <div>b)<br/> <span>Tosi. Funktiolla on Bolzanon lauseen perusteella ainakin yksi nollakohta välillä ]3, 4[.<br/> </span>c)<br/> <div class="content-view-wrapper"> <div class="fileupload"> <div class="example-answer-view"> <div class="answered-secondary-data-block"> <div class="example-answer-block"> <div id="example-answer-content" class="example-answer-content"> <div class="media-container"> <div class="media markdown"> <div class="paragraph">Ei mitään. Nollakohtia voi olla enemmänkin kuin yksi – Bolzanon lause tai muu rationaalifunktioon liittyvä tulos ei tätä estä.</div> </div> </div> </div> </div> </div> </div> </div> </div> <br/> <div><b>Esim. Derivoi </b></div> <span><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5Csqrt%5B3%5D%7Bx%5E2%2Bx%7D" alt="f\left(x\right)=x\sqrt[3]{x^2+x}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3Dx%5Cleft(x%5E2%2Bx%5Cright)%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="f'\left(x\right)=x\left(x^2+x\right)^{\frac{1}{3}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D1%5Ccdot%5Csqrt%5B3%5D%7Bx%5E2%2Bx%7D%2Bx%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Cleft(x%5E2%2Bx%5Cright)%5E%7B-%5Cfrac%7B2%7D%7B3%7D%7D%5Ccdot%5Cleft(2x%2B1%5Cright)%5Cright)" alt="f'\left(x\right)=1\cdot\sqrt[3]{x^2+x}+x\cdot\left(\frac{1}{3}\left(x^2+x\right)^{-\frac{2}{3}}\cdot\left(2x+1\right)\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Csqrt%5B3%5D%7Bx%5E2%2Bx%7D%2B%5Cfrac%7B2x%5E2%2Bx%7D%7B3%5Csqrt%5B3%5D%7Bx%5E2%2Bx%7D%5E2%7D" alt="=\sqrt[3]{x^2+x}+\frac{2x^2+x}{3\sqrt[3]{x^2+x}^2}"/></span><br/> <br/> 708<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(3x%5E5-2x%2B%5Cpi%5Cright)%3D15x%5E4-2" alt="D\left(3x^5-2x+\pi\right)=15x^4-2"/><br/> b)<br/> <br/> </div> 712<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E2-2x" alt="f\left(x\right)=x^2-2x"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(-1%5Cright)" alt="f'\left(-1\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(-1%5Cright)%3D%5Clim_%7Bx%5Crightarrow-1%7D%5Cfrac%7Bf%5Cleft(x%5Cright)-f%5Cleft(-1%5Cright)%7D%7Bx-%5Cleft(-1%5Cright)%7D%3D%5Clim_%7Bx%5Crightarrow-1%7D%5Cfrac%7Bx%5E2-3x-%5Cleft(%5Cleft(-1%5Cright)%5E2-3%5Ccdot%5Cleft(-1%5Cright)%5Cright)%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1%7D%5Cfrac%7Bx%5E2-3x-4%7D%7Bx%2B1%7D" alt="f'\left(-1\right)=\lim_{x\rightarrow-1}\frac{f\left(x\right)-f\left(-1\right)}{x-\left(-1\right)}=\lim_{x\rightarrow-1}\frac{x^2-3x-\left(\left(-1\right)^2-3\cdot\left(-1\right)\right)}{x+1}=\lim_{x\rightarrow-1}\frac{x^2-3x-4}{x+1}"/></div> <div>Lasketaan osoittajan nollakohdat: </div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-3x-4%3D0" alt="x^2-3x-4=0"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B3%5Cpm%5Csqrt%5B%5D%7B9%2B16%7D%7D%7B2%7D%3D%5Cfrac%7B3%5Cpm5%7D%7B2%7D%5C%20" alt="x=\frac{3\pm\sqrt[]{9+16}}{2}=\frac{3\pm5}{2}\ "/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D4" alt="x=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1" alt="x=-1"/><br/> <div>Siis</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-3x-4%3D1%5Ccdot%5Cleft(x-4%5Cright)%5Cleft(x-%5Cleft(-1%5Cright)%5Cright)%3D%5Cleft(x-4%5Cright)%5Cleft(x%2B1%5Cright)" alt="x^2-3x-4=1\cdot\left(x-4\right)\left(x-\left(-1\right)\right)=\left(x-4\right)\left(x+1\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-1%7D%5Cfrac%7Bx%5E2-3x-4%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1%7D%3D%5Cfrac%7B%5Cleft(x-4%5Cright)%5Cleft(x%2B1%5Cright)%7D%7Bx%2B1%7D%3D%5Clim_%7Bx%5Crightarrow-1%7Dx-4%3D-1-4%3D-5" alt="\lim_{x\rightarrow-1}\frac{x^2-3x-4}{x+1}=\lim_{x\rightarrow-1}=\frac{\left(x-4\right)\left(x+1\right)}{x+1}=\lim_{x\rightarrow-1}x-4=-1-4=-5"/> 2020-01-23T14:13:34+02:00 https://peda.net/id/36f5b5583c1 2020-01-23T13:46:57+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cleft(2%7B%2C%7D3%7B%2C%7D6%5Cright)" alt="A=\left(2{,}3{,}6\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=B%3D%5Cleft(4%7B%2C%7D-7%7B%2C%7D-3%5Cright)" alt="B=\left(4{,}-7{,}-3\right)"/> <div>Suuntavektori on:<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAB%7D%3D%5Cleft(4-2%5Cright)%5Coverline%7Bi%7D%2B%5Cleft(-7-3%5Cright)%5Coverline%7Bj%7D%2B%5Cleft(-3-6%5Cright)%5Coverline%7Bk%7D%3D2%5Coverline%7Bi%7D-10%5Coverline%7Bj%7D-9%5Coverline%7Bk%7D" alt="\overline{AB}=\left(4-2\right)\overline{i}+\left(-7-3\right)\overline{j}+\left(-3-6\right)\overline{k}=2\overline{i}-10\overline{j}-9\overline{k}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BOB%7D%3D%5Coverline%7BOA%7D%2Bt%5Coverline%7Bv%7D" alt="\overline{OB}=\overline{OA}+t\overline{v}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=4%5Coverline%7Bi%7D-7%5Coverline%7Bj%7D-3%5Coverline%7Bk%7D%3D2%5Coverline%7Bi%7D%2B3%5Coverline%7Bj%7D%2B4%5Coverline%7Bk%7D%2Bt%5Cleft(2%5Coverline%7Bi%7D-10%5Coverline%7Bj%7D-9%5Coverline%7Bk%7D%5Cright)" alt="4\overline{i}-7\overline{j}-3\overline{k}=2\overline{i}+3\overline{j}+4\overline{k}+t\left(2\overline{i}-10\overline{j}-9\overline{k}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4%5Coverline%7Bi%7D-7%5Coverline%7Bj%7D-3%5Coverline%7Bk%7D%3D2%5Coverline%7Bi%7D%2B3%5Coverline%7Bj%7D%2B4%5Coverline%7Bk%7D%2Bt%5Cleft(2%5Coverline%7Bi%7D-10%5Coverline%7Bj%7D-9%5Coverline%7Bk%7D%5Cright)" alt="4\overline{i}-7\overline{j}-3\overline{k}=2\overline{i}+3\overline{j}+4\overline{k}+t\left(2\overline{i}-10\overline{j}-9\overline{k}\right)"/> <div><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%2Bx%7D%7B1-x%7D%3D%5Cfrac%7B1-x%5E2%7D%7B1%2Bx%5E2%7D%7B%2C%7D%5C%20x%5Cne1" alt="\frac{1+x}{1-x}=\frac{1-x^2}{1+x^2}{,}\ x\ne1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5E%7B1%2Bx%5E2%5Ctext%7B)%7D%7D%5Cfrac%7B1%2Bx%7D%7B1-x%7D-%5E%7B1-x%5Ctext%7B)%7D%7D%5Cfrac%7B1-x%5E2%7D%7B1%2Bx%5E2%7D%3D0" alt="^{1+x^2\text{)}}\frac{1+x}{1-x}-^{1-x\text{)}}\frac{1-x^2}{1+x^2}=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft(1%2Bx%5Cright)%5Cleft(1%2Bx%5E2%5Cright)%7D%7B%5Cleft(1-x%5Cright)%5Cleft(1%2Bx%5E2%5Cright)%7D-%5Cfrac%7B%5Cleft(1-x%5Cright)%5Cleft(1-x%5E2%5Cright)%7D%7B%5Cleft(1-x%5Cright)%5Cleft(1%2Bx%5E2%5Cright)%7D%3D0" alt="\frac{\left(1+x\right)\left(1+x^2\right)}{\left(1-x\right)\left(1+x^2\right)}-\frac{\left(1-x\right)\left(1-x^2\right)}{\left(1-x\right)\left(1+x^2\right)}=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%2Bx%5E2%2Bx%2Bx%5E3-%5Cleft(1-x-x%5E2%2Bx%5E3%5Cright)%7D%7B%5Cleft(1-x%5Cright)%5Cleft(1%2Bx%5E2%5Cright)%7D%3D0" alt="\frac{1+x^2+x+x^3-\left(1-x-x^2+x^3\right)}{\left(1-x\right)\left(1+x^2\right)}=0"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%2Bx%5E2%2Bx%2Bx%5E3-1%2Bx%2Bx%5E2-x%5E3%7D%7B%5Cleft(1-x%5Cright)%5Cleft(1%2Bx%5E2%5Cright)%7D%3D0" alt="\frac{1+x^2+x+x^3-1+x+x^2-x^3}{\left(1-x\right)\left(1+x^2\right)}=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2x%5E2%2B2x%7D%7B%5Cleft(1-x%5Cright)%5Cleft(1%2Bx%5E2%5Cright)%7D%3D0" alt="\frac{2x^2+2x}{\left(1-x\right)\left(1+x^2\right)}=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%2B2x%3D0" alt="2x^2+2x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5Cleft(x%2B1%5Cright)%3D0" alt="2x\left(x+1\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%3D0" alt="2x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0" alt="x=0"/> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%2B1%3D0" alt="x+1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1" alt="x=-1"/><br/> <div> </div> <div><br/> 602 <br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%7D%7B3%7D-%5Cfrac%7B3%7D%7Bx%7D%3D0" alt="\frac{x}{3}-\frac{3}{x}=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%7D%7B3%7D%3D%5Cfrac%7B3%7D%7Bx%7D%7B%2C%7D%5C%20x%5Cne0" alt="\frac{x}{3}=\frac{3}{x}{,}\ x\ne0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B9%7D%7Bx%7D" alt="x=\frac{9}{x}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D9" alt="x^2=9"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpm3" alt="x=\pm3"/><br/> b)<br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e27fb500a975a3c45ab2829" alt="1-x=\frac{1}{1-x}{,}\ x\ne1"/><!--filtered attribute: data-userfileid="5e27fb500a975a3c45ab2829"--><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e27fb500a975a3c45ab282c" alt="\left(1-x\right)\left(1-x\right)=1"/><!--filtered attribute: data-userfileid="5e27fb500a975a3c45ab282c"--><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e27fb500a975a3c45ab2842" alt="1-x-x+x^2=1"/><!--filtered attribute: data-userfileid="5e27fb500a975a3c45ab2842"--><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e27fb500a975a3c45ab282e" alt="x^2-2x+1=1"/><!--filtered attribute: data-userfileid="5e27fb500a975a3c45ab282e"--><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e27fb500a975a3c45ab2832" alt="x^2-2x=0"/><!--filtered attribute: data-userfileid="5e27fb500a975a3c45ab2832"--><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-b%5Cpm%5Csqrt%5B%5D%7Bb%5E2-4ac%7D%7D%7B2a%7D%3D%5Cfrac%7B2%5Cpm%5Csqrt%5B%5D%7B%5Cleft(-2%5Cright)%5E2-4%5Ccdot1%5Ccdot0%7D%7D%7B2%5Ccdot1%7D%3D%5Cfrac%7B2%5Cpm%5Csqrt%5B%5D%7B4%7D%7D%7B2%7D%3D%5Cfrac%7B2%5Cpm2%7D%7B2%7D" alt="x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}=\frac{2\pm\sqrt[]{\left(-2\right)^2-4\cdot1\cdot0}}{2\cdot1}=\frac{2\pm\sqrt[]{4}}{2}=\frac{2\pm2}{2}"/><br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e27fb680a975a3c45ab2eef" alt="x=\frac{2+2}{2}=2"/><!--filtered attribute: data-userfileid="5e27fb680a975a3c45ab2eef"--><span>tai</span><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e27fb680a975a3c45ab2efa" alt="x=\frac{2-2}{2}=\frac{0}{2}=0"/><!--filtered attribute: data-userfileid="5e27fb680a975a3c45ab2efa"--><br/> c)<br/> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e255d6b0a975a3c45681299" alt="\frac{1+x}{1-x}=\frac{1-x^2}{1+x^2}{,}\ x\ne1"/><!--filtered attribute: data-userfileid="5e255d6b0a975a3c45681299"--><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e255d6b0a975a3c45681281" alt="^{1+x^2\text{)}}\frac{1+x}{1-x}-^{1-x\text{)}}\frac{1-x^2}{1+x^2}=0"/><!--filtered attribute: data-userfileid="5e255d6b0a975a3c45681281"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e255d6b0a975a3c45681293" alt="\frac{\left(1+x\right)\left(1+x^2\right)}{\left(1-x\right)\left(1+x^2\right)}-\frac{\left(1-x\right)\left(1-x^2\right)}{\left(1-x\right)\left(1+x^2\right)}=0"/><!--filtered attribute: data-userfileid="5e255d6b0a975a3c45681293"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e255d6b0a975a3c4568128d" alt="\frac{1+x^2+x+x^3-\left(1-x-x^2+x^3\right)}{\left(1-x\right)\left(1+x^2\right)}=0"/><!--filtered attribute: data-userfileid="5e255d6b0a975a3c4568128d"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e255d6b0a975a3c4568129f" alt="\frac{1+x^2+x+x^3-1+x+x^2-x^3}{\left(1-x\right)\left(1+x^2\right)}=0"/><!--filtered attribute: data-userfileid="5e255d6b0a975a3c4568129f"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e255d6b0a975a3c456812a6" alt="\frac{2x^2+2x}{\left(1-x\right)\left(1+x^2\right)}=0"/><!--filtered attribute: data-userfileid="5e255d6b0a975a3c456812a6"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e255d6b0a975a3c456812e2" alt="2x^2+2x=0"/><!--filtered attribute: data-userfileid="5e255d6b0a975a3c456812e2"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e255d6b0a975a3c456812da" alt="2x\left(x+1\right)=0"/><!--filtered attribute: data-userfileid="5e255d6b0a975a3c456812da"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e255d6b0a975a3c4568127c" alt="2x=0"/><!--filtered attribute: data-userfileid="5e255d6b0a975a3c4568127c"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e255d6b0a975a3c456812c1" alt="x=0"/><!--filtered attribute: data-userfileid="5e255d6b0a975a3c456812c1"--></p> <p>tai</p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e255d6b0a975a3c456812c7" alt="x+1=0"/><!--filtered attribute: data-userfileid="5e255d6b0a975a3c456812c7"--><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3NjUy/1313592/files?id=5e255d6b0a975a3c456812ba" alt="x=-1"/><!--filtered attribute: data-userfileid="5e255d6b0a975a3c456812ba"--></p> <br/> 605</div> <div>a)</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> Määrittelyehto <div><img src="https://math-demo.abitti.fi/math.svg?latex=3-x%5Cge0" alt="3-x\ge0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cle3" alt="x\le3"/><br/> <div>Juuren arvot ovat ei-negatiivisia</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%2B3%5Cge0" alt="x+3\ge0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cge-3" alt="x\ge-3"/><br/> <div>Siis </div> <img src="https://math-demo.abitti.fi/math.svg?latex=-3%5Cle%20x%5Cle3" alt="-3\le x\le3"/><br/> <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=3-x%3D%5Cleft(x%2B3%5Cright)%5E2" alt="3-x=\left(x+3\right)^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3-x%3Dx%5E2%2B6x%2B9" alt="3-x=x^2+6x+9"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=0%3Dx%5E2%2B7x%2B6" alt="0=x^2+7x+6"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7Bb%5Cpm%5Csqrt%5B%5D%7Bb%5E2-4ac%7D%7D%7B2a%7D" alt="x=\frac{b\pm\sqrt[]{b^2-4ac}}{2a}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-6%5C%20tai%5C%20x%3D-1" alt="x=-6\ tai\ x=-1"/><br/> <div>koska </div> <img src="https://math-demo.abitti.fi/math.svg?latex=-6%3C-3" alt="-6&lt;-3"/>, se ei kelpa vastaukseksi, joten<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3A%5C%20x%3D-1" alt="V:\ x=-1"/></div> <div> </div> <div>610</div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%20e%2B%5Cln1-%5Cln3e%5E2" alt="\ln e+\ln1-\ln3e^2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D1%2B0-%5Cleft(%5Cln3%2B%5Cln%20e%5E2%5Cright)" alt="=1+0-\left(\ln3+\ln e^2\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D1-%5Cln3-2%5Cln%20e" alt="=1-\ln3-2\ln e"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D1-%5Cln3-2" alt="=1-\ln3-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-1-%5Cln3" alt="=-1-\ln3"/><br/> <div>e)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_63%2B%5Clog_612" alt="\log_63+\log_612"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Clog_63%5Ccdot12" alt="=\log_63\cdot12"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Clog_636" alt="=\log_636"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D2" alt="=2"/><br/> <br/> <span>619</span> <div>b) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cfrac%7Bx%7D%7B3%7D%3D%5C%20%5Cfrac%7B%5Csqrt%5B%5D%7B3%7D%7D%7B2%7D" alt="\sin\frac{x}{3}=\ \frac{\sqrt[]{3}}{2}"/></div> <div>Taulukkokirja:</div> <div>Erät ratkaisut ovat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%7D%7B3%7D%3D%5Cfrac%7B%5Csqrt%5B%5D%7B3%7D%7D%7B2%7D" alt="\frac{x}{3}=\frac{\sqrt[]{3}}{2}"/>tai <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%7D%7B3%7D%3D%5Cfrac%7B2%5Cpi%7D%7B3%7D" alt="\frac{x}{3}=\frac{2\pi}{3}"/></div> <div>Siis kaikki ratkaisut</div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%7D%7B3%7D%3D%5Cfrac%7B%5Cpi%7D%7B3%7D%2Bn%5Ccdot2%5Cpi" alt="\frac{x}{3}=\frac{\pi}{3}+n\cdot2\pi"/>tai<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%7D%7B3%7D%3D%5Cfrac%7B2%5Cpi%7D%7B3%7D%2Bn%5Ccdot2%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="\frac{x}{3}=\frac{2\pi}{3}+n\cdot2\pi{,}\ n\in\mathbb{Z}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpi%2Bn%5Ccdot6%5Cpi" alt="x=\pi+n\cdot6\pi"/>  <span> </span><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D2%5Cpi%2Bn%5Ccdot6%5Cpi" alt="x=2\pi+n\cdot6\pi"/></div> <br/> <br/> <span>646</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2%5Ccos3x%2B4" alt="f\left(x\right)=2\cos3x+4"/></div> <div>Perusjakso on <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%5Cpi%7D%7B3%7D" alt="\frac{2\pi}{3}"/></div> <div>Arvojoukko: </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-1%5Cle%5Ccos%20x%5Cle1" alt="-1\le\cos x\le1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-1%5Cle%5Ccos3x%5Cle1" alt="-1\le\cos3x\le1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-2%5Cle%5Ccos3x%5Cle2" alt="-2\le\cos3x\le2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cle2%5Ccos3x%2B4%5Cle6" alt="2\le2\cos3x+4\le6"/></div> <div>Arvojoukko on [2,6]</div> <div> </div> </div> 2020-01-21T08:21:09+02:00 https://peda.net/id/3eb57e90392 2020-01-21T08:20:49+02:00 <div>501<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BDP%7D%3D%5Cfrac%7B1%7D%7B2%7D%5Coverline%7Bu%7D%2B%5Coverline%7Bv%7D" alt="\overline{DP}=\frac{1}{2}\overline{u}+\overline{v}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BDQ%7D%3D%5Coverline%7Bu%7D%2B%5Cfrac%7B3%7D%7B7%7D%5Coverline%7Bv%7D" alt="\overline{DQ}=\overline{u}+\frac{3}{7}\overline{v}"/><br/> <br/> 502<br/> a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cleft(2%7B%2C%7D1%5Cright)" alt="A=\left(2{,}1\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=B%3D%5Cleft(-3%7B%2C%7D-5%5Cright)" alt="B=\left(-3{,}-5\right)"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAB%7D%3D%5Cleft(-3-2%5Cright)%5Coverline%7Bi%7D%2B%5Cleft(-5-2%5Cright)%5Coverline%7Bj%7D%3D-5%5Coverline%7Bi%7D-7%5Coverline%7Bj%7D" alt="\overline{AB}=\left(-3-2\right)\overline{i}+\left(-5-2\right)\overline{j}=-5\overline{i}-7\overline{j}"/><br/> <br/> 503<br/> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3MjY2/1313592/files?id=5e2052c40a975a3c45329cf0" alt="\left(4-2\right)\overline{i}+\left(1+3\right)\overline{j}+\left(-7+5\right)\overline{k}=2\overline{i}+4\overline{j}-2\overline{k}"/><!--filtered attribute: data-userfileid="5e2052c40a975a3c45329cf0"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3MjY2/1313592/files?id=5e2052c40a975a3c45329cf6" alt="\left|\overline{a}-\overline{b}\right|=\sqrt[]{2^2+4^2+\left(-2\right)^2}=2\sqrt[]{6}"/><!--filtered attribute: data-userfileid="5e2052c40a975a3c45329cf6"--><br/> <br/> 504<br/> Pisteen A paikka vektori on </p> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverrightarrow%7BOA%7D%3D%5Coverline%7Bi%7D-%5Coverline%7Bj%7D" alt="\overrightarrow{OA}=\overline{i}-\overline{j}"/></div> <p><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bi%7D-2%5Coverline%7Bj%7D%2B2%5Coverline%7Bk%7D%3D%5Coverline%7Ba%7D" alt="\overline{i}-2\overline{j}+2\overline{k}=\overline{a}"/></p> <div>Määritetään vektorin <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D" alt="\overline{a}"/> yksikkävektoria<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%5E0%3D%5Cfrac%7B%5Coverline%7Ba%7D%7D%7B%5Cleft%7C%5Coverline%7Ba%7D%5Cright%7C%7D%3D%5Cfrac%7B%5Coverline%7Bi%7D-2%5Coverline%7Bj%7D%2B2%5Coverline%7Bk%7D%7D%7B%5Csqrt%5B%5D%7B1%5E2%2B%5Cleft(-2%5Cright)%5E2%2B2%5E2%7D%7D%3D%5Cfrac%7B%5Coverline%7Bi%7D-2%5C%20%5Coverline%7Bj%7D%2B2%5Coverline%7Bk%7D%7D%7B%5Csqrt%5B%5D%7B9%7D%7D%3D%5Cfrac%7B%5Coverline%7Bi%7D-2%5C%20%5Coverline%7Bj%7D%2B2%5Coverline%7Bk%7D%7D%7B3%7D%3D%5Cfrac%7B1%7D%7B3%7D%5Coverline%7Bi%7D-%5Cfrac%7B2%7D%7B3%7D%5Coverline%7Bj%7D%2B%5Cfrac%7B2%7D%7B3%7D%5Coverline%7Bk%7D" alt="\overline{a}^0=\frac{\overline{a}}{\left|\overline{a}\right|}=\frac{\overline{i}-2\overline{j}+2\overline{k}}{\sqrt[]{1^2+\left(-2\right)^2+2^2}}=\frac{\overline{i}-2\ \overline{j}+2\overline{k}}{\sqrt[]{9}}=\frac{\overline{i}-2\ \overline{j}+2\overline{k}}{3}=\frac{1}{3}\overline{i}-\frac{2}{3}\overline{j}+\frac{2}{3}\overline{k}"/><br/> <div>Lasketaan vektoreiden summa</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=OA%2B9%5Coverline%7Ba%7D%5E0%3D%5Cleft(%5Coverline%7Bi%7D-%5Coverline%7Bj%7D%5Cright)%2B9%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Coverline%7Bi%7D-%5Cfrac%7B2%7D%7B3%7D%5Coverline%7Bj%7D%2B%5Cfrac%7B2%7D%7B3%7D%5Coverline%7Bk%7D%5Cright)%3D%5Cleft(%5Coverline%7Bi%7D-%5Coverline%7Bj%7D%5Cright)%2B%5Cleft(3i-6%5Coverline%7Bj%7D%2B6%5Coverline%7Bk%7D%5Cright)" alt="OA+9\overline{a}^0=\left(\overline{i}-\overline{j}\right)+9\left(\frac{1}{3}\overline{i}-\frac{2}{3}\overline{j}+\frac{2}{3}\overline{k}\right)=\left(\overline{i}-\overline{j}\right)+\left(3i-6\overline{j}+6\overline{k}\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(1%2B3%5Cright)%5Coverline%7Bi%7D%2B%5Cleft(-1-6%5Cright)%5Coverline%7Bj%7D%2B6%5Coverline%7Bk%7D" alt="=\left(1+3\right)\overline{i}+\left(-1-6\right)\overline{j}+6\overline{k}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D4%5Coverline%7Bi%7D-7%5Coverline%7Bj%7D%2B6%5C%20%5Coverline%7Bk%7D" alt="=4\overline{i}-7\overline{j}+6\ \overline{k}"/><br/> <div> <div>Määritetään vektori <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bb%7D" alt="\overline{b}"/>yksikkövektoria </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bb%7D%5E0%3D%5Cfrac%7B%5Coverline%7Bb%7D%7D%7B%5Cleft%7C%5Coverline%7Bb%7D%5Cright%7C%7D%3D%5Cfrac%7B3%5Coverline%7Bi%7D-4%5Coverline%7Bk%7D%7D%7B%5Csqrt%5B%5D%7B3%5E2%2B%5Cleft(-4%5Cright)%5E2%7D%7D%3D%5Cfrac%7B3%5Coverline%7Bi%7D-4%5Coverline%7Bk%7D%7D%7B%5Csqrt%5B%5D%7B25%7D%7D%3D%5Cfrac%7B3%7D%7B5%7D%5Coverline%7Bi%7D-%5Cfrac%7B4%7D%7B5%7D%5Coverline%7Bk%7D" alt="\overline{b}^0=\frac{\overline{b}}{\left|\overline{b}\right|}=\frac{3\overline{i}-4\overline{k}}{\sqrt[]{3^2+\left(-4\right)^2}}=\frac{3\overline{i}-4\overline{k}}{\sqrt[]{25}}=\frac{3}{5}\overline{i}-\frac{4}{5}\overline{k}"/></div> <div>Lasketaan edellisien vektoreiden ja <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bb%7D%5E0" alt="\overline{b}^0"/>summa</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(4%5Coverline%7Bi%7D-7%5Coverline%7Bj%7D%2B6%5Coverline%7Bk%7D%5Cright)%2B10%5Cleft(%5Cfrac%7B3%7D%7B5%7D%5Coverline%7Bi%7D-%5Cfrac%7B4%7D%7B5%7D%5Coverline%7Bk%7D%5Cright)%3D%5Cleft(4%5Coverline%7Bi%7D-7%5Coverline%7Bj%7D%2B6%5Coverline%7Bk%7D%5Cright)%2B%5Cleft(6%5Coverline%7Bi%7D-8%5Coverline%7Bk%7D%5Cright)" alt="\left(4\overline{i}-7\overline{j}+6\overline{k}\right)+10\left(\frac{3}{5}\overline{i}-\frac{4}{5}\overline{k}\right)=\left(4\overline{i}-7\overline{j}+6\overline{k}\right)+\left(6\overline{i}-8\overline{k}\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(4%2B6%5Cright)%5Coverline%7Bi%7D-7%5Coverline%7Bj%7D%2B%5Cleft(6-8%5Cright)%5Coverline%7Bk%7D" alt="=\left(4+6\right)\overline{i}-7\overline{j}+\left(6-8\right)\overline{k}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D10%5Coverline%7Bi%7D-7%5Coverline%7Bj%7D%2B-2%5Coverline%7Bk%7D" alt="=10\overline{i}-7\overline{j}+-2\overline{k}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=C%3D%5Cleft(10%7B%2C%7D-7%7B%2C%7D-2%5Cright)" alt="C=\left(10{,}-7{,}-2\right)"/><br/> <br/> 506<br/> <div>Kolmio on tasankylkinen, jos sillä on kaksi samanpituista sivua</div> <div>Lasketaan kolmion sivujen pituudet</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAB%7D%3D%5Cleft(2-3%5Cright)%5Coverline%7Bi%7D%2B%5Cleft(2-4%5Cright)%5Coverline%7Bj%7D%2B%5Cleft(-5-5%5Cright)%5Coverline%7Bk%7D%3D-%5Coverline%7Bi%7D-2j-10%5Coverline%7Bk%7D" alt="\overline{AB}=\left(2-3\right)\overline{i}+\left(2-4\right)\overline{j}+\left(-5-5\right)\overline{k}=-\overline{i}-2j-10\overline{k}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7BAB%7D%5Cright%7C%3D%5Csqrt%5B%5D%7B%5Cleft(-1%5Cright)%5E2%2B%5Cleft(-2%5Cright)%5E2%2B%5Cleft(-10%5Cright)%5E2%7D%3D%5Csqrt%5B%5D%7B105%7D" alt="\left|\overline{AB}\right|=\sqrt[]{\left(-1\right)^2+\left(-2\right)^2+\left(-10\right)^2}=\sqrt[]{105}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BBC%7D%3D%5Cleft(-3-2%5Cright)%5Coverline%7Bi%7D%2B%5Cleft(-2-2%5Cright)%5Coverline%7Bj%7D%2B%5Cleft(3%2B5%5Cright)%5Coverline%7Bk%7D%3D-5%5Coverline%7Bi%7D-4%5Coverline%7Bj%7D%2B8%5Coverline%7Bk%7D" alt="\overline{BC}=\left(-3-2\right)\overline{i}+\left(-2-2\right)\overline{j}+\left(3+5\right)\overline{k}=-5\overline{i}-4\overline{j}+8\overline{k}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7BBC%7D%5Cright%7C%3D%5Csqrt%5B%5D%7B%5Cleft(-5%5Cright)%5E2%2B%5Cleft(-4%5Cright)%5E2%2B8%5E2%7D%3D%5Csqrt%5B%5D%7B105%7D" alt="\left|\overline{BC}\right|=\sqrt[]{\left(-5\right)^2+\left(-4\right)^2+8^2}=\sqrt[]{105}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAC%7D%3D%5Cleft(-3-3%5Cright)%5Coverline%7Bi%7D%2B%5Cleft(-2-4%5Cright)%5Coverline%7Bj%7D%2B%5Cleft(3-5%5Cright)%5Coverline%7Bk%7D%3D-6%5Coverline%7Bi%7D-6%5Coverline%7Bj%7D-2%5Coverline%7Bk%7D" alt="\overline{AC}=\left(-3-3\right)\overline{i}+\left(-2-4\right)\overline{j}+\left(3-5\right)\overline{k}=-6\overline{i}-6\overline{j}-2\overline{k}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7BAC%7D%5Cright%7C%3D%5Csqrt%5B%5D%7B%5Cleft(-6%5Cright)%5E2%2B%5Cleft(-6%5Cright)%5E2%2B%5Cleft(-2%5Cright)%5E2%7D%3D%5Csqrt%5B%5D%7B76%7D%3D2%5Csqrt%5B%5D%7B19%7D" alt="\left|\overline{AC}\right|=\sqrt[]{\left(-6\right)^2+\left(-6\right)^2+\left(-2\right)^2}=\sqrt[]{76}=2\sqrt[]{19}"/><br/> <div>Laskun mukaan <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7B%20%7D" alt="\overline{ }"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7BAB%7D%5Cright%7C%3D%5Cleft%7C%5Coverline%7BBC%7D%5Cright%7C" alt="\left|\overline{AB}\right|=\left|\overline{BC}\right|"/></div> <div>Joten kolmio on tasankylkinen</div> <div>Lasketaan vektorien muodostuma kulman suuruus</div> <div>Huom. koska vektorit eivät lähtevät samasta pisteestä, vektoria <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAB%7D" alt="\overline{AB}"/>on muutettava vektoriksi<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BBA%7D" alt="\overline{BA}"/></div> <div>ja<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BBA%7D%3D-%5Coverline%7BAB%7D" alt="\overline{BA}=-\overline{AB}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csphericalangle%5Cleft(%5Coverline%7BBA%7D%7B%2C%7D%5Coverline%7BBC%7D%5Cright)%3D" alt="\sphericalangle\left(\overline{BA}{,}\overline{BC}\right)="/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5E%7B-1%7D%5Cleft(%5Cfrac%7B%5Coverline%7BBA%7D%5Ccdot%5Coverline%7BBC%7D%7D%7B%5Cleft%7C%5Coverline%7BAB%7D%5Cright%7C%5Cleft%7CBC%5Cright%7C%7D%5Cright)%3D%5Ccos%5E%7B-1%7D%5Cleft(%5Cfrac%7B1%5Ccdot%5Cleft(-5%5Cright)%2B2%5Ccdot%5Cleft(-4%5Cright)%2B10%5Ccdot%5Cleft(8%5Cright)%7D%7B105%7D%5Cright)%3D50%7B%2C%7D35...%5Capprox50%7B%2C%7D4%C2%B0" alt="\cos^{-1}\left(\frac{\overline{BA}\cdot\overline{BC}}{\left|\overline{AB}\right|\left|BC\right|}\right)=\cos^{-1}\left(\frac{1\cdot\left(-5\right)+2\cdot\left(-4\right)+10\cdot\left(8\right)}{105}\right)=50{,}35...\approx50{,}4°"/><br/> <br/> 508<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%3D4%5Coverline%7Bi%7D-2%5Coverline%7Bj%7D" alt="\overline{a}=4\overline{i}-2\overline{j}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bb%7D%3D-3%5Coverline%7Bi%7D%2B%5Coverline%7Bj%7D" alt="\overline{b}=-3\overline{i}+\overline{j}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bc%7D%3Dd%5Coverline%7Bi%7D%2B%5Cleft(d%2B1%5Cright)%5Coverline%7Bj%7D" alt="\overline{c}=d\overline{i}+\left(d+1\right)\overline{j}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%2B%5Coverline%7Bc%7D%3D%5Coverline%7Bb%7D%2B%5Coverline%7Bc%7D" alt="\overline{a}+\overline{c}=\overline{b}+\overline{c}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%2B%5Coverline%7Bc%7D%3D%5Cleft(4%2Bd%5Cright)%5Coverline%7Bi%7D%2B%5Cleft(d%2B1-2%5Cright)%5Coverline%7Bj%7D%3D%5Cleft(4%2Bd%5Cright)%5Coverline%7Bi%7D%2B%5Cleft(d-1%5Cright)%5Coverline%7Bj%7D" alt="\overline{a}+\overline{c}=\left(4+d\right)\overline{i}+\left(d+1-2\right)\overline{j}=\left(4+d\right)\overline{i}+\left(d-1\right)\overline{j}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bb%7D%2B%5Coverline%7Bc%7D%3D%5Cleft(-3%2Bd%5Cright)%5Coverline%7Bi%7D%2B%5Cleft(d%2B1%2B1%5Cright)%5Coverline%7Bj%7D%3D%5Cleft(-3%2Bd%5Cright)%5Coverline%7Bi%7D%2B%5Cleft(d%2B2%5Cright)%5Coverline%7Bj%7D" alt="\overline{b}+\overline{c}=\left(-3+d\right)\overline{i}+\left(d+1+1\right)\overline{j}=\left(-3+d\right)\overline{i}+\left(d+2\right)\overline{j}"/> <div>Oletetaan, että <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%2B%5Coverline%7Bc%7D%3D%5Coverline%7BA%7D" alt="\overline{a}+\overline{c}=\overline{A}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bb%7D%2B%5Coverline%7Bc%7D%3D%5Coverline%7BB%7D" alt="\overline{b}+\overline{c}=\overline{B}"/><br/> <div>Vektorit ovat samansuuntaiset, jos niillä on olemassa sellainen luku t, joka on suurempi kuin 0, ja saa tuloksi </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BB%7D%3Dt%5Coverline%7BA%7D" alt="\overline{B}=t\overline{A}"/></div> <div>Määritetään luku t</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(-3%2Bd%5Cright)%5Coverline%7Bi%7D%2B%5Cleft(d%2B2%5Cright)%5Coverline%7Bj%7D%3D%5Cleft(4%2Bd%5Cright)t%5Coverline%7Bi%7D%2B%5Cleft(d-1%5Cright)t%5Coverline%7Bj%7D" alt="\left(-3+d\right)\overline{i}+\left(d+2\right)\overline{j}=\left(4+d\right)t\overline{i}+\left(d-1\right)t\overline{j}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-3%2Bd%3D%5Cleft(4%2Bd%5Cright)t%5C%5C%0Ad%2B2%3D%5Cleft(d-1%5Cright)t%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;-3+d=\left(4+d\right)t\\&#10;d+2=\left(d-1\right)t&#10;\end{cases}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=t%3D%E2%88%921%5C%20tai%5C%20t%3D-%5Cfrac%7B1%7D%7B2%7D" alt="t=−1\ tai\ t=-\frac{1}{2}"/>(laskin)</div> Koska t:n arvot ovat aidosti negatiivisia, vektorit eivät ole koskaan samansuuntaisia.</div> <div><br/> <div> </div> <div> <div> <div> </div> </div> </div> </div> </div> </div> </div> </div> </div> 2020-01-17T14:16:52+02:00 https://peda.net/id/a4be6f1e32e 2020-02-02T18:58:35+02:00 <span><span>203<br/> a)<br/> Koska eksponenttien potenssi ovat parillisia ja niistä saadaan ainoastaan positiivisia lukuja<br/> b)<br/> ei, koska jos luku x on negatiivinen, silloin g(x) on aidosti pienempi kuin 0<br/> <br/> 205<br/> I B, II C, III A<br/> <br/> 210<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x-2%5Cright)%5Cleft(x-3%5Cright)%3D6" alt="\left(x-2\right)\left(x-3\right)=6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-3x-2x%2B6%3D6" alt="x^2-3x-2x+6=6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-5x%3D0" alt="x^2-5x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B5%5Cpm%5Csqrt%5B%5D%7B%5Cleft(-5%5Cright)%5E2-4%5Ccdot1%5Ccdot0%7D%7D%7B2%5Ccdot1%7D%3D%5Cfrac%7B5%5Cpm%5Csqrt%5B%5D%7B25%7D%7D%7B2%7D%3D%5Cfrac%7B5%5Cpm5%7D%7B2%7D" alt="x=\frac{5\pm\sqrt[]{\left(-5\right)^2-4\cdot1\cdot0}}{2\cdot1}=\frac{5\pm\sqrt[]{25}}{2}=\frac{5\pm5}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B5%2B5%7D%7B2%7D%3D5" alt="x=\frac{5+5}{2}=5"/></span></span> <div>tai<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B5-5%7D%7B2%7D%3D%5Cfrac%7B0%7D%7B2%7D%3D0" alt="x=\frac{5-5}{2}=\frac{0}{2}=0"/><br/> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=7%5Cleft(x-3%5Cright)%2B1%3Dx%5E2-1-%5Cleft(x%5E2-1%5Cright)" alt="7\left(x-3\right)+1=x^2-1-\left(x^2-1\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=7x-21%2B1%3Dx%5E2-1-x%5E2%2B1" alt="7x-21+1=x^2-1-x^2+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=7x-20%3D0" alt="7x-20=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=7x%3D20" alt="7x=20"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B20%7D%7B7%7D" alt="x=\frac{20}{7}"/><br/> <br/> </div> <span>213<br/> 214<br/> 216<br/> 217<br/> 218<br/> 223<br/> 225<br/> 228<br/> 229<br/> 234</span><br/> <span>238<br/> 240<br/> 241<br/> 243<br/> 244<br/> 247<br/> 248<br/> 251<br/> 252<br/> 257<br/> 261<br/> 263<br/> 265<br/> 268<br/> 273</span><br/> <span>274<br/> 277<br/> 278</span> 2020-01-09T16:23:45+02:00 https://peda.net/id/ceb639f831e 2020-01-09T18:17:26+02:00 <span>101<br/> </span>a) <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cfrac%7B1%7D%7B3%7D-1%5Cfrac%7B5%7D%7B7%7D%3D%5Cfrac%7B7%7D%7B3%7D-%5Cfrac%7B12%7D%7B7%7D%3D%5E%7B7%5Ctext%7B)%7D%7D%5Cfrac%7B7%7D%7B3%7D-%5E%7B3%5Ctext%7B)%7D%7D%5Cfrac%7B12%7D%7B7%7D%3D%5Cfrac%7B49%7D%7B21%7D-%5Cfrac%7B36%7D%7B21%7D%3D%5Cfrac%7B13%7D%7B21%7D" alt="2\frac{1}{3}-1\frac{5}{7}=\frac{7}{3}-\frac{12}{7}=^{7\text{)}}\frac{7}{3}-^{3\text{)}}\frac{12}{7}=\frac{49}{21}-\frac{36}{21}=\frac{13}{21}"/><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3%5Cfrac%7B1%7D%7B3%7D%3A1%5Cfrac%7B2%7D%7B3%7D%2B2%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cleft(-1%5Cfrac%7B5%7D%7B9%7D%5Cright)" alt="3\frac{1}{3}:1\frac{2}{3}+2\frac{1}{2}\cdot\left(-1\frac{5}{9}\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B10%7D%7B3%7D%3A%5Cfrac%7B5%7D%7B3%7D%2B%5Cfrac%7B5%7D%7B2%7D%5Ccdot%5Cleft(-%5Cfrac%7B14%7D%7B9%7D%5Cright)" alt="=\frac{10}{3}:\frac{5}{3}+\frac{5}{2}\cdot\left(-\frac{14}{9}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5E%7B2%5Ctext%7B)%7D%7D%5Cfrac%7B10%7D%7B3%7D%3A%5E%7B2%5Ctext%7B)%7D%7D%5Cfrac%7B5%7D%7B3%7D%2B%5Cleft(-%5Cfrac%7B5%7D%7B2%7D%5Ccdot%5Cfrac%7B14%7D%7B9%7D%5Cright)" alt="=^{2\text{)}}\frac{10}{3}:^{2\text{)}}\frac{5}{3}+\left(-\frac{5}{2}\cdot\frac{14}{9}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B20%7D%7B6%7D%3A%5Cfrac%7B10%7D%7B6%7D%2B%5Cleft(-%5Cfrac%7B5%7D%7B1%7D%5Ccdot%5Cfrac%7B7%7D%7B9%7D%5Cright)" alt="=\frac{20}{6}:\frac{10}{6}+\left(-\frac{5}{1}\cdot\frac{7}{9}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D2-%5Cfrac%7B35%7D%7B9%7D%3D%5Cfrac%7B18%7D%7B9%7D-%5Cfrac%7B35%7D%7B9%7D%3D-%5Cfrac%7B17%7D%7B9%7D" alt="=2-\frac{35}{9}=\frac{18}{9}-\frac{35}{9}=-\frac{17}{9}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3%5Ccdot%5Cfrac%7B2-%5Cfrac%7B1%7D%7B3%7D%7D%7B1%2B%5Cfrac%7B1%7D%7B2%7D%7D%3D3%5Ccdot%5Cfrac%7B%5Cfrac%7B6%7D%7B3%7D-%5Cfrac%7B1%7D%7B3%7D%7D%7B%5Cfrac%7B2%7D%7B2%7D%2B%5Cfrac%7B1%7D%7B2%7D%7D%3D3%5Ccdot%5Cfrac%7B%5Cfrac%7B5%7D%7B3%7D%7D%7B%5Cfrac%7B3%7D%7B2%7D%7D%3D3%5Ccdot%5Cleft(%5Cfrac%7B5%7D%7B3%7D%5Ccdot%5Cfrac%7B2%7D%7B3%7D%5Cright)%3D3%5Ccdot%5Cfrac%7B10%7D%7B9%7D%3D%5Cfrac%7B30%7D%7B9%7D%5E%7B%5Ctext%7B(%7D3%7D%3D%5Cfrac%7B10%7D%7B3%7D%3D3%5Cfrac%7B1%7D%7B3%7D" alt="3\cdot\frac{2-\frac{1}{3}}{1+\frac{1}{2}}=3\cdot\frac{\frac{6}{3}-\frac{1}{3}}{\frac{2}{2}+\frac{1}{2}}=3\cdot\frac{\frac{5}{3}}{\frac{3}{2}}=3\cdot\left(\frac{5}{3}\cdot\frac{2}{3}\right)=3\cdot\frac{10}{9}=\frac{30}{9}^{\text{(}3}=\frac{10}{3}=3\frac{1}{3}"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3a%7D%7B4%7D%2B%5Cfrac%7Ba%7D%7B2%7D%3D%5E%7B2%5Ctext%7B)%7D%7D%5Cfrac%7B3a%7D%7B4%7D%2B%5E%7B4%5Ctext%7B)%7D%7D%5Cfrac%7Ba%7D%7B2%7D%3D%5Cfrac%7B6a%7D%7B8%7D%2B%5Cfrac%7B4a%7D%7B8%7D%3D%5Cfrac%7B10a%7D%7B8%7D%5E%7B%5Ctext%7B(%7D2%7D%3D%5Cfrac%7B5a%7D%7B4%7D" alt="\frac{3a}{4}+\frac{a}{2}=^{2\text{)}}\frac{3a}{4}+^{4\text{)}}\frac{a}{2}=\frac{6a}{8}+\frac{4a}{8}=\frac{10a}{8}^{\text{(}2}=\frac{5a}{4}"/></div> <div>e)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B4a%7D%7B5%7D%3A%5Cfrac%7Ba%7D%7B4%7D-%5Cfrac%7Ba%7D%7B2%7D%5Cleft(a-1%5Cright)" alt="\frac{4a}{5}:\frac{a}{4}-\frac{a}{2}\left(a-1\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B16a%7D%7B5a%7D-%5Cfrac%7Ba%5E2%7D%7B2%7D%2B%5Cfrac%7Ba%7D%7B2%7D" alt="=\frac{16a}{5a}-\frac{a^2}{2}+\frac{a}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Cfrac%7Ba%5E2%7D%7B2%7D%2B%5Cfrac%7Ba%7D%7B2%7D%2B%5Cfrac%7B16%7D%7B5%7D" alt="=-\frac{a^2}{2}+\frac{a}{2}+\frac{16}{5}"/> <div>f)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-a%5Ccdot%5Cleft(%5Cfrac%7B2a-3%7D%7B3%7D%5Cright)%3A3" alt="-a\cdot\left(\frac{2a-3}{3}\right):3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B-2a%5E2%2B3a%7D%7B3%7D%3A%5Cfrac%7B3%7D%7B1%7D" alt="=\frac{-2a^2+3a}{3}:\frac{3}{1}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B-2a%5E2%2B3a%7D%7B9%7D%3D%5Cfrac%7B-2a%5E2%7D%7B9%7D%2B%5Cfrac%7B3a%7D%7B9%7D%5E%7B%5Ctext%7B(%7D3%7D%3D%5Cfrac%7B-2a%5E2%7D%7B9%7D%2B%5Cfrac%7Ba%7D%7B3%7D" alt="=\frac{-2a^2+3a}{9}=\frac{-2a^2}{9}+\frac{3a}{9}^{\text{(}3}=\frac{-2a^2}{9}+\frac{a}{3}"/></div> <br/> <div>103<br/> <div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4%5Ccdot2%5E%7B-3%7D-2%5E%7B-1%7D%2B2%5E0" alt="4\cdot2^{-3}-2^{-1}+2^0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D4%5Ccdot%5Cfrac%7B1%7D%7B2%5E3%7D-%5Cfrac%7B1%7D%7B2%7D%2B1" alt="=4\cdot\frac{1}{2^3}-\frac{1}{2}+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D4%5Ccdot%5Cfrac%7B1%7D%7B8%7D-%5Cfrac%7B1%7D%7B2%7D%2B1" alt="=4\cdot\frac{1}{8}-\frac{1}{2}+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B4%7D%7B8%7D%5E%7B%5Ctext%7B(%7D4%7D-%5Cfrac%7B1%7D%7B2%7D%2B1" alt="=\frac{4}{8}^{\text{(}4}-\frac{1}{2}+1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7B2%7D%2B1%3D1" alt="=\frac{1}{2}-\frac{1}{2}+1=1"/><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3%5E0%2B%5Cleft(%5Cleft(-1%5Cright)%5E3%5Cright)%5E7" alt="3^0+\left(\left(-1\right)^3\right)^7"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D1%2B%5Cleft(-1%5Cright)%5E7" alt="=1+\left(-1\right)^7"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D1-1%3D0" alt="=1-1=0"/><br/> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%5Ccdot3%5E2%7D%7B27%7D-%5Cfrac%7B1%7D%7B3%5E2%7D" alt="\frac{2\cdot3^2}{27}-\frac{1}{3^2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B18%7D%7B27%7D-%5E%7B3%5Ctext%7B)%7D%7D%5Cfrac%7B1%7D%7B9%7D" alt="=\frac{18}{27}-^{3\text{)}}\frac{1}{9}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B18%7D%7B27%7D-%5Cfrac%7B3%7D%7B27%7D" alt="=\frac{18}{27}-\frac{3}{27}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B15%7D%7B27%7D%5E%7B%5Ctext%7B(%7D3%7D" alt="=\frac{15}{27}^{\text{(}3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B5%7D%7B9%7D" alt="=\frac{5}{9}"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=7%5E4%5Ccdot7%5E%7B-4%7D-7%5E%7B-2%7D%2B%5Cleft(-7%5Cright)%5E2" alt="7^4\cdot7^{-4}-7^{-2}+\left(-7\right)^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D7%5E%7B4-4%7D-%5Cfrac%7B1%7D%7B7%5E2%7D%2B49" alt="=7^{4-4}-\frac{1}{7^2}+49"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D1-%5Cfrac%7B1%7D%7B49%7D%2B49" alt="=1-\frac{1}{49}+49"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D50-%5Cfrac%7B1%7D%7B49%7D%3D%5Cfrac%7B50%5Ccdot49-1%7D%7B49%7D" alt="=50-\frac{1}{49}=\frac{50\cdot49-1}{49}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B2449%7D%7B49%7D" alt="=\frac{2449}{49}"/></div> <div>e)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cfrac%7B3%7D%7B4%7D%5Cright)%5E%7B-2%7D-%5Cfrac%7B2%5E4%7D%7B3%5E2%7D" alt="\left(\frac{3}{4}\right)^{-2}-\frac{2^4}{3^2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Cfrac%7B1%7D%7B3%5E2%7D%3A%5Cfrac%7B1%7D%7B4%5E2%7D%5Cright)-%5Cfrac%7B16%7D%7B9%7D" alt="=\left(\frac{1}{3^2}:\frac{1}{4^2}\right)-\frac{16}{9}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Cfrac%7B1%7D%7B9%7D%5Ccdot16%5Cright)-%5Cfrac%7B16%7D%7B9%7D" alt="=\left(\frac{1}{9}\cdot16\right)-\frac{16}{9}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D0" alt="=0"/></div> <div>f)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(5%5Ccdot10%5E%7B-4%7D%5Cright)%5Ccdot%5Cleft(3%5Ccdot10%5E6%5Cright)" alt="\left(5\cdot10^{-4}\right)\cdot\left(3\cdot10^6\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D5%5Ccdot%5Cfrac%7B1%7D%7B10%5E4%7D%5Ccdot%5Cleft(3%5Ccdot10%5E6%5Cright)" alt="=5\cdot\frac{1}{10^4}\cdot\left(3\cdot10^6\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B5%7D%7B10000%7D%5Ccdot3%5Ccdot1000000" alt="=\frac{5}{10000}\cdot3\cdot1000000"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B15000000%7D%7B10000%7D%3D1500" alt="=\frac{15000000}{10000}=1500"/><br/> <br/> </div> </div> </div> <div>105 <div><span>a)</span><br/> <span>Luvut ovat toistensa käänteisluvut, kun niiden tulo on 1</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Csqrt%5B%5D%7B6%7D%7D%7B3%7D%5Ccdot%5Cfrac%7B%5Csqrt%5B%5D%7B6%7D%7D%7B2%7D%3D%5Cfrac%7B%5Cleft(%5Csqrt%5B%5D%7B6%7D%5Cright)%5E2%7D%7B6%7D%3D%5Cfrac%7B6%7D%7B6%7D%3D1" alt="\frac{\sqrt[]{6}}{3}\cdot\frac{\sqrt[]{6}}{2}=\frac{\left(\sqrt[]{6}\right)^2}{6}=\frac{6}{6}=1"/><br/> <span>Luvut ovat toistensa kääteislukuja.</span><br/> <span>b)</span><br/> <span>Luvut ovat toistensa käänteisluvut, kun niiden tulo on 1 </span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2%2B%5Csqrt%5B%5D%7B3%7D%5Cright)%5Cleft(2-%5Csqrt%5B%5D%7B3%7D%5Cright)%3D2%5E2-%5Cleft(%5Csqrt%5B%5D%7B3%7D%5Cright)%5E2%3D4-3%3D1" alt="\left(2+\sqrt[]{3}\right)\left(2-\sqrt[]{3}\right)=2^2-\left(\sqrt[]{3}\right)^2=4-3=1"/><br/> <span>Luvut ovat toistensa käänteislukuja</span><br/> <span>Luvut ovat toistensa vastaluvut, kun niiden summa on 0</span><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2%2B%5Csqrt%5B%5D%7B3%7D%5Cright)%5Cleft(2-%5Csqrt%5B%5D%7B3%7D%5Cright)%3D2%2B2%2B%5Csqrt%5B%5D%7B3%7D-%5Csqrt%5B%5D%7B3%7D%3D4%5Cne0" alt="\left(2+\sqrt[]{3}\right)\left(2-\sqrt[]{3}\right)=2+2+\sqrt[]{3}-\sqrt[]{3}=4\ne0"/></div> <span>Luvut eivät olet toistensa vastaluvut.</span></div> </div> <div><br/> 107<br/> a)<br/> <div> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzEx/1313592/files?id=5e15f9200a975a3c455a63bd" alt="\left(a+3\right)^2-\left(a-3\right)^2"/><!--filtered attribute: data-userfileid="5e15f9200a975a3c455a63bd"--></div> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzEx/1313592/files?id=5e15f9200a975a3c455a63bb" alt="=\left(a^2+6a+3^2\right)-\left(a^2-6a+3^2\right)"/><!--filtered attribute: data-userfileid="5e15f9200a975a3c455a63bb"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzEx/1313592/files?id=5e15f9210a975a3c455a63c1" alt="=a^2+6a+9-a^2+6a-9"/><!--filtered attribute: data-userfileid="5e15f9210a975a3c455a63c1"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzEx/1313592/files?id=5e15f9200a975a3c455a63bf" alt="=12a"/><!--filtered attribute: data-userfileid="5e15f9200a975a3c455a63bf"--></p> </div> b)<br/> <div> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzEx/1313592/files?id=5e15fa150a975a3c455a70d9" alt="\left(\left(a+3\right)\left(a-3\right)\right)^2"/><!--filtered attribute: data-userfileid="5e15fa150a975a3c455a70d9"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzEx/1313592/files?id=5e15fa150a975a3c455a70dd" alt="=\left(a^2-3^2\right)^2"/><!--filtered attribute: data-userfileid="5e15fa150a975a3c455a70dd"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzEx/1313592/files?id=5e15fa150a975a3c455a70df" alt="=\left(a^2-9\right)^2"/><!--filtered attribute: data-userfileid="5e15fa150a975a3c455a70df"--></p> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzEx/1313592/files?id=5e15fa150a975a3c455a70d6" alt="=\left(a^2\right)^2-18a^2+\left(-9\right)^2"/><!--filtered attribute: data-userfileid="5e15fa150a975a3c455a70d6"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzEx/1313592/files?id=5e15fa150a975a3c455a70db" alt="=a^4-18a^2+81"/><!--filtered attribute: data-userfileid="5e15fa150a975a3c455a70db"--></p> </div> c)<br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzEx/1313592/files?id=5e15fa900a975a3c455a791f" alt="2+2\left(\sqrt[]{a}+1\right)\left(\sqrt[]{a}-1\right)"/><!--filtered attribute: data-userfileid="5e15fa900a975a3c455a791f"--><br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzEx/1313592/files?id=5e15fa900a975a3c455a7921" alt="=2+2\cdot\left(\left(\sqrt[]{a}\right)^2-1^2\right)"/><!--filtered attribute: data-userfileid="5e15fa900a975a3c455a7921"--><br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzEx/1313592/files?id=5e15fa900a975a3c455a7928" alt="=2+2a-2"/><!--filtered attribute: data-userfileid="5e15fa900a975a3c455a7928"--><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzEx/1313592/files?id=5e15fa900a975a3c455a7923" alt="=2a"/><!--filtered attribute: data-userfileid="5e15fa900a975a3c455a7923"--><br/> <br/> </div> <div>109<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba%5E2%7D%7B3%7D-%5Cleft(%5Cfrac%7B-a%7D%7B3%7D%5Cright)%5E2" alt="\frac{a^2}{3}-\left(\frac{-a}{3}\right)^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7Ba%5E2%7D%7B3%7D-%5Cleft(%5Cfrac%7B%5Cleft(-a%5Cright)%5E2%7D%7B3%5E2%7D%5Cright)" alt="=\frac{a^2}{3}-\left(\frac{\left(-a\right)^2}{3^2}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7Ba%5E2%7D%7B3%7D-%5Cfrac%7Ba%5E2%7D%7B9%7D" alt="=\frac{a^2}{3}-\frac{a^2}{9}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5E%7B3%5Ctext%7B)%7D%7D%5Cfrac%7Ba%5E2%7D%7B3%7D-%5Cfrac%7Ba%5E2%7D%7B9%7D" alt="=^{3\text{)}}\frac{a^2}{3}-\frac{a^2}{9}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B3a%5E2%7D%7B9%7D-%5Cfrac%7Ba%5E2%7D%7B9%7D" alt="=\frac{3a^2}{9}-\frac{a^2}{9}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B2a%5E2%7D%7B9%7D" alt="=\frac{2a^2}{9}"/><br/> b)<br/> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzE1/1313592/files?id=5e1602d50a975a3c455affd0" alt="\frac{a^2-b^2}{a-b}+\frac{a^2-b^2}{a+b}"/><!--filtered attribute: data-userfileid="5e1602d50a975a3c455affd0"--></p> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzE1/1313592/files?id=5e1602d50a975a3c455affce" alt="=^{\left(a+b\right)\text{)}}\frac{\left(a^2-b^2\right)\left(a+b\right)}{\left(a-b\right)\left(a+b\right)}+^{\left(a-b\right)\text{)}}\frac{\left(a^2-b^2\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}"/><!--filtered attribute: data-userfileid="5e1602d50a975a3c455affce"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzE1/1313592/files?id=5e1602d50a975a3c455affcc" alt="=\frac{\left(a^2-b^2\right)\left(a+b\right)}{\left(a^2-b^2\right)}+\frac{\left(a^2-b^2\right)\left(a-b\right)}{\left(a^2-b^2\right)}"/><!--filtered attribute: data-userfileid="5e1602d50a975a3c455affcc"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2MzE1/1313592/files?id=5e1602d50a975a3c455affda" alt="=\left(a+b\right)+\left(a-b\right)"/><!--filtered attribute: data-userfileid="5e1602d50a975a3c455affda"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D2a%2Bb-b" alt="=2a+b-b"/></p> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D2a" alt="=2a"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(a%2Bb%5Cright)%5E2%5Ccdot%5Cleft(a-b%5Cright)%5E2-%5Cleft(a%5E4%2Bb%5E4%5Cright)" alt="\left(a+b\right)^2\cdot\left(a-b\right)^2-\left(a^4+b^4\right)"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(a%2Bb%5Cright)%5E2%5Ccdot%5Cleft(a-b%5Cright)%5E2-%5Cleft(a%5E4%2Bb%5E4%5Cright)" alt="=\left(a+b\right)^2\cdot\left(a-b\right)^2-\left(a^4+b^4\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(a%5E2-b%5E2%5Cright)%5Cleft(a%5E2-b%5E2%5Cright)-%5Cleft(a%5E4%2Bb%5E4%5Cright)" alt="=\left(a^2-b^2\right)\left(a^2-b^2\right)-\left(a^4+b^4\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(a%5E4-2a%5E2b%5E2-b%5E4%5Cright)-%5Cleft(a%5E4%2Bb%5E4%5Cright)" alt="=\left(a^4-2a^2b^2-b^4\right)-\left(a^4+b^4\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3Da%5E4-2a%5E2b%5E2-b%5E4-a%5E4-b%5E4" alt="=a^4-2a^2b^2-b^4-a^4-b^4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-2a%5E2b%5E2" alt="=-2a^2b^2"/><br/> <br/> </div> </div> <div>110<br/> a) 1,15a<br/> b) 0,65a<br/> c) 3,3a<br/> d) 2,25a<br/> <br/> </div> <div>113</div> <div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2NTgy/1313592/files?id=5e16e4f40a975a3c456a2e36" alt="m_{alku}\left(suola\right)=500g\cdot0{,}06=30g"/><!--filtered attribute: data-userfileid="5e16e4f40a975a3c456a2e36"--></p> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2NTgy/1313592/files?id=5e16e4f40a975a3c456a2e3e" alt="m_2\left(suola\right)=300g\cdot0{,}05=15g"/><!--filtered attribute: data-userfileid="5e16e4f40a975a3c456a2e3e"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2NTgy/1313592/files?id=5e16e4f40a975a3c456a2e32" alt="m_{loppu}\left(suola\right)=45g"/><!--filtered attribute: data-userfileid="5e16e4f40a975a3c456a2e32"--></p> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2NTgy/1313592/files?id=5e16e4f40a975a3c456a2e30" alt="m_{loppu}\left(liuos\right)=800g"/><!--filtered attribute: data-userfileid="5e16e4f40a975a3c456a2e30"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2NTgy/1313592/files?id=5e16e5180a975a3c456a449f" alt="suolapitoisuus=\frac{45g}{800g}=0{,}05625\approx0{,}056=5{,}6\%"/><!--filtered attribute: data-userfileid="5e16e5180a975a3c456a449f"--></p> <br/> 115<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=Osake%3D35%7B%2C%7D50%E2%82%AC" alt="Osake=35{,}50€"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=Osakkeen%5C%20arvo%5C%20lopussa%3D35%7B%2C%7D50%5Ccdot1%7B%2C%7D12%5Ccdot0%7B%2C%7D90%3D35%7B%2C%7D784%E2%82%AC" alt="Osakkeen\ arvo\ lopussa=35{,}50\cdot1{,}12\cdot0{,}90=35{,}784€"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B35%7B%2C%7D784%7D%7B35%7B%2C%7D50%7D%3D1%7B%2C%7D008%3D%2B0%7B%2C%7D8%5C%25" alt="\frac{35{,}784}{35{,}50}=1{,}008=+0{,}8\%"/></div> <div>Arvo kasvoi 0,8%</div> </div> <div>116<br/> A3 B5 C2 D6 E1 F4<br/> <br/> </div> <div>117<br/> a) <br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7B2%7D%3D1%7B%2C%7D414...%3E1" alt="\sqrt[]{2}=1{,}414...&gt;1"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <div>Luku on negatiivinen<br/> b)<br/> <p><span>Luvun itseisarvo on sen vastaluku, kun sen on negatiivinen</span></p> <div>Näin ollen</div> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI2NjE2/1313592/files?id=5e16e8400a975a3c456bcd83" alt="\left|1-\sqrt[]{2}\right|=-\left(1-\sqrt[]{2}\right)=\sqrt[]{2}-1"/><!--filtered attribute: data-userfileid="5e16e8400a975a3c456bcd83"--></div> </div> </div> <div>119<br/> a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Csqrt%5B%5D%7B3%7D%2B2%5Cright%7C-%5Cleft%7C%5Csqrt%5B%5D%7B3%7D-2%5Cright%7C%3D%5Cleft(%5Csqrt%5B%5D%7B3%7D%2B2%5Cright)-%5Cleft(-%5Cleft(%5Csqrt%5B%5D%7B3%7D-2%5Cright)%5Cright)" alt="\left|\sqrt[]{3}+2\right|-\left|\sqrt[]{3}-2\right|=\left(\sqrt[]{3}+2\right)-\left(-\left(\sqrt[]{3}-2\right)\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Csqrt%5B%5D%7B3%7D%2B2%5Cright)-%5Cleft(-%5Csqrt%5B%5D%7B3%7D%2B2%5Cright)" alt="=\left(\sqrt[]{3}+2\right)-\left(-\sqrt[]{3}+2\right)"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Csqrt%5B%5D%7B3%7D%2B2%5Cright)%2B%5Csqrt%5B%5D%7B3%7D-2" alt="=\left(\sqrt[]{3}+2\right)+\sqrt[]{3}-2"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D2%5Csqrt%5B%5D%7B3%7D" alt="=2\sqrt[]{3}"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft%7C1-%5Csqrt%5B%5D%7B2%7D%5Cright%7C%7D%7B%5Csqrt%5B%5D%7B2%7D-1%7D%3D%5Cfrac%7B-%5Cleft(1-%5Csqrt%5B%5D%7B2%7D%5Cright)%7D%7B%5Csqrt%5B%5D%7B2%7D-1%7D%3D%5Cfrac%7B%5Csqrt%5B%5D%7B2%7D-1%7D%7B%5Csqrt%5B%5D%7B2%7D-1%7D%3D1" alt="\frac{\left|1-\sqrt[]{2}\right|}{\sqrt[]{2}-1}=\frac{-\left(1-\sqrt[]{2}\right)}{\sqrt[]{2}-1}=\frac{\sqrt[]{2}-1}{\sqrt[]{2}-1}=1"/><br/> c)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Csqrt%5B%5D%7B5%7D-5%5Cright%7C-%5Cleft%7C5-%5Csqrt%5B%5D%7B5%7D%5Cright%7C%3D-%5Cleft(%5Csqrt%5B%5D%7B5%7D-5%5Cright)-%5Cleft(5-%5Csqrt%5B%5D%7B5%7D%5Cright)" alt="\left|\sqrt[]{5}-5\right|-\left|5-\sqrt[]{5}\right|=-\left(\sqrt[]{5}-5\right)-\left(5-\sqrt[]{5}\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Csqrt%5B%5D%7B5%7D%2B5-5%2B%5Csqrt%5B%5D%7B5%7D%3D0" alt="=-\sqrt[]{5}+5-5+\sqrt[]{5}=0"/></div> </div> <div>123<br/> a) Epätosi<br/> b) Epätosi<br/> c) Tosi<br/> d) Epätosi<br/> e) Epätosi<br/> f) Tosi<br/> g) Tosi<br/> h) Tosi<br/> <br/> </div> <div>124<br/> a)<br/> <div>Suurin:<span> </span><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODEz/1313592/files?id=5e16ef300a975a3c456e69a6" alt="\frac{2}{2}=1"/><!--filtered attribute: data-userfileid="5e16ef300a975a3c456e69a6"--></div> <div><span>Pienin </span><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODEz/1313592/files?id=5e16ef540a975a3c456e7144" alt="-\frac{1}{2}"/><!--filtered attribute: data-userfileid="5e16ef540a975a3c456e7144"--></div> <br/> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1%7B%2C%7D5%3D%5Cfrac%7B3%7D%7B2%7D" alt="1{,}5=\frac{3}{2}"/></div> <div>Käänteisluvun vastaluku: <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B2%7D%7B3%7D" alt="-\frac{2}{3}"/></div> <br/> <span>Vastaluvun käänteisluku: </span><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B2%7D%7B3%7D" alt="-\frac{2}{3}"/><br/> <br/> </div> <div>126<br/> a)<br/> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODE0/1313592/files?id=5e16f1390a975a3c456ed692" alt="^{\text{35)}}\frac{1}{2}{,}\ ^{14\text{)}}\frac{3}{5}{,}\ ^{10\text{)}}\frac{4}{7}"/><!--filtered attribute: data-userfileid="5e16f1390a975a3c456ed692"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODE0/1313592/files?id=5e16f1390a975a3c456ed69d" alt="\frac{35}{70}{,}\ \frac{42}{70}{,}\ \frac{40}{70}"/><!--filtered attribute: data-userfileid="5e16f1390a975a3c456ed69d"--></p> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODE0/1313592/files?id=5e16f1390a975a3c456ed698" alt="\frac{1}{2}&lt;\frac{4}{7}&lt;\frac{3}{5}"/><!--filtered attribute: data-userfileid="5e16f1390a975a3c456ed698"--><br/> b)<br/> <span>Koska</span><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODE0/1313592/files?id=5e16f2140a975a3c456efb40" alt="\sqrt[]{a+b}"/><!--filtered attribute: data-userfileid="5e16f2140a975a3c456efb40"--><span>:n ratkaisu pienenee, kun </span><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODE0/1313592/files?id=5e16f2140a975a3c456efb3c" alt="b&lt;0"/><!--filtered attribute: data-userfileid="5e16f2140a975a3c456efb3c"--></div> </div> <div>128<br/> <span>Oletetaan, että </span><img src="https://math-demo.abitti.fi/math.svg?latex=a%3D1" alt="a=1"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%3D0%7B%2C%7D75b" alt="a=0{,}75b"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D75b%3D1" alt="0{,}75b=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=b%3D%5Cfrac%7B4%7D%7B3%7D%3D1%7B%2C%7D33333..." alt="b=\frac{4}{3}=1{,}33333..."/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%7B%2C%7D33333...-1%3D0%7B%2C%7D33333...%5Capprox0%7B%2C%7D33%3D33%5C%25" alt="1{,}33333...-1=0{,}33333...\approx0{,}33=33\%"/></div> </div> <div>132<br/> a)<br/> <div>Luvut ovat toistensa vastalukuja, kun niiden summa on 0</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(a-b%5Cright)%2B%5Cleft(b-a%5Cright)%3Da-a-b%2Bb%3D0" alt="\left(a-b\right)+\left(b-a\right)=a-a-b+b=0"/></div> <div>Luvut ovat toistensa vastalukuja </div> b)<br/> <div>Luvut ovat toistensa käänteisluvut, kun niiden tulo on 1</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Csqrt%5B%5D%7Ba%7D%7D%7Bb%7D%5Ccdot%5Cfrac%7Bb%5Csqrt%5B%5D%7Ba%7D%7D%7Ba%7D" alt="\frac{\sqrt[]{a}}{b}\cdot\frac{b\sqrt[]{a}}{a}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Csqrt%5B%5D%7Ba%7D%7D%7Bb%7D%5Ccdot%5Cfrac%7Bb%5Csqrt%5B%5D%7Ba%7D%7D%7Ba%7D%3D%5Cfrac%7Bb%5Ccdot%5Cleft(%5Csqrt%5B%5D%7Ba%7D%5Cright)%5E2%7D%7Bab%7D%3D%5Cfrac%7Bab%7D%7Bab%7D%3D1" alt="\frac{\sqrt[]{a}}{b}\cdot\frac{b\sqrt[]{a}}{a}=\frac{b\cdot\left(\sqrt[]{a}\right)^2}{ab}=\frac{ab}{ab}=1"/></div> <div>Luvut ovat toistensa käänteislukuja.</div> </div> <div>134<br/> a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7B2%7D%7D%2B%5Cfrac%7B1%7D%7B2%2B%5Csqrt%5B%5D%7B2%7D%7D%3D%5E%7B2%2B%5Csqrt%5B%5D%7B2%7D%5Ctext%7B)%7D%7D%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7B2%7D%7D%2B%5E%7B%5Csqrt%5B%5D%7B2%7D%5Ctext%7B)%7D%7D%5Cfrac%7B1%7D%7B2%2B%5Csqrt%5B%5D%7B2%7D%7D" alt="\frac{1}{\sqrt[]{2}}+\frac{1}{2+\sqrt[]{2}}=^{2+\sqrt[]{2}\text{)}}\frac{1}{\sqrt[]{2}}+^{\sqrt[]{2}\text{)}}\frac{1}{2+\sqrt[]{2}}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B2%2B%5Csqrt%5B%5D%7B2%7D%7D%7B2%5Csqrt%5B%5D%7B2%7D%2B2%7D%2B%5E%7B%5Csqrt%5B%5D%7B2%7D%5Ctext%7B)%7D%7D%5Cfrac%7B%5Csqrt%5B%5D%7B2%7D%7D%7B2%5Csqrt%5B%5D%7B2%7D%2B2%7D%3D%5Cfrac%7B2%2B2%5Csqrt%5B%5D%7B2%7D%7D%7B2%2B2%5Csqrt%5B%5D%7B2%7D%7D%3D1" alt="=\frac{2+\sqrt[]{2}}{2\sqrt[]{2}+2}+^{\sqrt[]{2}\text{)}}\frac{\sqrt[]{2}}{2\sqrt[]{2}+2}=\frac{2+2\sqrt[]{2}}{2+2\sqrt[]{2}}=1"/></div> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7B3%5Cfrac%7B3%7D%7B4%7D%7D%3A%5Csqrt%5B%5D%7B1%5Cfrac%7B2%7D%7B3%7D%7D" alt="\sqrt[]{3\frac{3}{4}}:\sqrt[]{1\frac{2}{3}}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Csqrt%5B%5D%7B%5Cfrac%7B15%7D%7B4%7D%7D%3A%5Csqrt%5B%5D%7B%5Cfrac%7B5%7D%7B3%7D%7D%3D%5Cfrac%7B%5Csqrt%5B%5D%7B15%7D%7D%7B%5Csqrt%5B%5D%7B4%7D%7D%3A%5Cfrac%7B%5Csqrt%5B%5D%7B5%7D%7D%7B%5Csqrt%5B%5D%7B3%7D%7D%3D%5Cfrac%7B%5Csqrt%5B%5D%7B3%5Ccdot15%7D%7D%7B%5Csqrt%5B%5D%7B4%5Ccdot5%7D%7D%3D%5Cfrac%7B%5Csqrt%5B%5D%7B45%7D%7D%7B%5Csqrt%5B%5D%7B20%7D%7D%3D%5Cfrac%7B%5Csqrt%5B%5D%7B5%5Ccdot9%7D%7D%7B%5Csqrt%5B%5D%7B5%5Ccdot4%7D%7D%3D%5Cfrac%7B%5Csqrt%5B%5D%7B5%5Ccdot3%5E2%7D%7D%7B%5Csqrt%5B%5D%7B5%5Ccdot2%5E2%7D%7D%3D%5Cfrac%7B3%5Csqrt%5B%5D%7B5%7D%7D%7B2%5Csqrt%5B%5D%7B5%7D%7D%3D%5Cfrac%7B3%7D%7B2%7D" alt="=\sqrt[]{\frac{15}{4}}:\sqrt[]{\frac{5}{3}}=\frac{\sqrt[]{15}}{\sqrt[]{4}}:\frac{\sqrt[]{5}}{\sqrt[]{3}}=\frac{\sqrt[]{3\cdot15}}{\sqrt[]{4\cdot5}}=\frac{\sqrt[]{45}}{\sqrt[]{20}}=\frac{\sqrt[]{5\cdot9}}{\sqrt[]{5\cdot4}}=\frac{\sqrt[]{5\cdot3^2}}{\sqrt[]{5\cdot2^2}}=\frac{3\sqrt[]{5}}{2\sqrt[]{5}}=\frac{3}{2}"/><br/> <br/> </div> <div>137<br/> a)<br/> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e1700370a975a3c45736b92" alt="\sqrt[]{27-10\sqrt[]{2}}=5-\sqrt[]{2}"/><!--filtered attribute: data-userfileid="5e1700370a975a3c45736b92"--></div> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e1700370a975a3c45736b88" alt="=\sqrt[]{27-10\sqrt[]{2}+\sqrt[]{2}^2-2}"/><!--filtered attribute: data-userfileid="5e1700370a975a3c45736b88"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e1700370a975a3c45736b8b" alt="=\sqrt[]{\sqrt[]{2}^2-10\sqrt[]{2}+25}"/><!--filtered attribute: data-userfileid="5e1700370a975a3c45736b8b"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e1700370a975a3c45736b8f" alt="=\sqrt[]{\sqrt[]{2}^2-10\sqrt[]{2}+5^2}=\sqrt[]{\sqrt[]{2}^2-2\cdot5\sqrt[]{2}+5^2}"/><!--filtered attribute: data-userfileid="5e1700370a975a3c45736b8f"--> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e1700370a975a3c45736b8d" alt="\left|\right|\left(a-b\right)^2{,}\ a=\sqrt[]{2}{,}\ b=5"/><!--filtered attribute: data-userfileid="5e1700370a975a3c45736b8d"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e1700370a975a3c45736b84" alt="=\sqrt[]{\left(\sqrt[]{2}-5\right)^2}=\sqrt[]{\left(5-\sqrt[]{2}\right)^2}"/><!--filtered attribute: data-userfileid="5e1700370a975a3c45736b84"--> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e1700370a975a3c45736bb2" alt="\left|\right|a^2=\left(-a\right)^2"/><!--filtered attribute: data-userfileid="5e1700370a975a3c45736bb2"--></p> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e1700370a975a3c45736b97" alt="=5-\sqrt[]{2}"/><!--filtered attribute: data-userfileid="5e1700370a975a3c45736b97"--><br/> <b>b)*****</b><br/> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e17014b0a975a3c4573c4b7" alt="\sqrt[]{9-4\sqrt[]{5}}=2-\sqrt[]{5}"/><!--filtered attribute: data-userfileid="5e17014b0a975a3c4573c4b7"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e17014b0a975a3c4573c4b2" alt="=\sqrt[]{9-4\sqrt[]{5}+\sqrt[]{5}^2-5}"/><!--filtered attribute: data-userfileid="5e17014b0a975a3c4573c4b2"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e17014b0a975a3c4573c4be" alt="=\sqrt[]{\sqrt[]{5}^2-4\cdot\sqrt[]{5}+4}=\sqrt[]{\sqrt[]{5}^2-2\cdot2\cdot\sqrt[]{5}+2^2}"/><!--filtered attribute: data-userfileid="5e17014b0a975a3c4573c4be"--> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e17014b0a975a3c4573c4b5" alt="\left|\right|\left(a-b\right)^2{,}\ a=\sqrt[]{5}{,}\ b=2"/><!--filtered attribute: data-userfileid="5e17014b0a975a3c4573c4b5"--><br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e17014b0a975a3c4573c4ac" alt="=\sqrt[]{\left(\sqrt[]{5}-2\right)^2}=\sqrt[]{\left(2-\sqrt[]{5}\right)^2}"/><!--filtered attribute: data-userfileid="5e17014b0a975a3c4573c4ac"--> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e17014b0a975a3c4573c4b9" alt="\left|\right|a^2=\left(-a\right)^2"/><!--filtered attribute: data-userfileid="5e17014b0a975a3c4573c4b9"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI3ODQx/1313592/files?id=5e17014c0a975a3c4573c4d3" alt="=2-\sqrt[]{5}"/><!--filtered attribute: data-userfileid="5e17014c0a975a3c4573c4d3"--></p> <p>On<br/> c)</p> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B3%5D%7Ba%5Csqrt%5B%5D%7Ba%7D%7D" alt="\sqrt[3]{a\sqrt[]{a}}"/></div> <p><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Csqrt%5B3%5D%7Ba%5E1a%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%3D%5Csqrt%5B3%5D%7Ba%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%3Da%5E%7B%5Cfrac%7B3%7D%7B2%7D%5Ccdot%5Cfrac%7B1%7D%7B3%7D%7D%3Da%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%3D%5Csqrt%5B%5D%7Ba%7D" alt="=\sqrt[3]{a^1a^{\frac{1}{2}}}=\sqrt[3]{a^{\frac{3}{2}}}=a^{\frac{3}{2}\cdot\frac{1}{3}}=a^{\frac{1}{2}}=\sqrt[]{a}"/><br/> <br/> </p> </div> </div> <div>139<br/> a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=5%5Csqrt%5B%5D%7B3%7D%3E6%5Csqrt%5B%5D%7B2%7D" alt="5\sqrt[]{3}&gt;6\sqrt[]{2}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--><br/> b)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B3%5D%7B3%7D%3E%5Csqrt%5B4%5D%7B4%7D" alt="\sqrt[3]{3}&gt;\sqrt[4]{4}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff; color: #333333; font-family: 'Times New Roman'; font-size: 17px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"--><br/> <div>c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B3%5D%7B3%7D%3E%5Csqrt%5B6%5D%7B6%7D" alt="\sqrt[3]{3}&gt;\sqrt[6]{6}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> </div> <div>141<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7B%2C%7D25x-x%7D%7B1%7B%2C%7D25x%7D%3D%3D%5Cfrac%7B0%7B%2C%7D25x%7D%7B1%7B%2C%7D25x%7D%3D%5Cfrac%7B0%7B%2C%7D25%7D%7B1%7B%2C%7D25%7D%3D0%7B%2C%7D2%3D20%5C%25" alt="\frac{1{,}25x-x}{1{,}25x}==\frac{0{,}25x}{1{,}25x}=\frac{0{,}25}{1{,}25}=0{,}2=20\%"/><br/> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x-0%7B%2C%7D25x%3D1" alt="x-0{,}25x=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D75x%3D1" alt="0{,}75x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1%7D%7B0%7B%2C%7D75%7D%3D1%7B%2C%7D333...%5Capprox1%7B%2C%7D33%3D%2B33%5C%25" alt="x=\frac{1}{0{,}75}=1{,}333...\approx1{,}33=+33\%"/><br/> <br/> </div> <div>144<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%5E%7Bn-3%7D%5Ccdot4%5En%7D%7B8%5E%7Bn-1%7D%7D%3D%5Cfrac%7B%5Cfrac%7B2%5En%7D%7B2%5E3%7D%5Ccdot4%5En%7D%7B%5Cfrac%7B8%5En%7D%7B8%7D%7D%3D%5Cfrac%7B%5Cfrac%7B8%5En%7D%7B8%7D%7D%7B%5Cfrac%7B8%5En%7D%7B8%7D%7D%3D1" alt="\frac{2^{n-3}\cdot4^n}{8^{n-1}}=\frac{\frac{2^n}{2^3}\cdot4^n}{\frac{8^n}{8}}=\frac{\frac{8^n}{8}}{\frac{8^n}{8}}=1"/><br/> <br/> </div> <div>148<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=m-%5C%25%5Cleft(suola%5Cright)%3D4%7B%2C%7D0%5C%25" alt="m-\%\left(suola\right)=4{,}0\%"/></div> <div>Oletetaan, että meriveden massa on x</div> <div>suolan määrä alkutilanteessa oli<img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D04x" alt="0{,}04x"/></div> <div>Meriveden massa vähenetään 28%, mutta suolan määrä ei muutu</div> <div>näin ollen </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B0%7B%2C%7D04x%7D%7Bx-0%7B%2C%7D28x%7D%3D%5Cfrac%7B0%7B%2C%7D04x%7D%7B0%7B%2C%7D72x%7D%3D%5Cfrac%7B1%7D%7B18%7D%3D0%7B%2C%7D0555...%5Capprox0%7B%2C%7D056%3D5%7B%2C%7D6%5C%25" alt="\frac{0{,}04x}{x-0{,}28x}=\frac{0{,}04x}{0{,}72x}=\frac{1}{18}=0{,}0555...\approx0{,}056=5{,}6\%"/></div> </div> <div>150<br/> a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%3D4" alt="a^{\frac{1}{3}}=4"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_a4%3D%5Cfrac%7B1%7D%7B3%7D" alt="\log_a4=\frac{1}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D64" alt="a=64"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%3D64%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%3D16" alt="a^{\frac{2}{3}}=64^{\frac{2}{3}}=16"/></div> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Em%3D3" alt="2^m=3"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_23%3Dm" alt="\log_23=m"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=8%5Em%3D8%5E%7B%5Clog_23%7D%3D27" alt="8^m=8^{\log_23}=27"/><br/> c)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4%5Ek%3D6" alt="4^k=6"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_46%3Dk" alt="\log_46=k"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5E%7B4k%7D%3D2%5E%7B4%5Ccdot%5Clog_46%7D%3D36" alt="2^{4k}=2^{4\cdot\log_46}=36"/></div> </div> <div>151<br/> <div> <div>vettä=80%</div> <div>sokeri=4%</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=vett%C3%A4_%7Bloppu%7D%3D20%5C%25" alt="vettä_{loppu}=20\%"/></div> <div>Omenan messasta on poistettu <img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D6x" alt="0{,}6x"/></div> <div>Oletetaan, että y on omenan loppu sokeriprosentti</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B0%7B%2C%7D04x%7D%7Bx%7D%3D%5Cfrac%7By%7D%7Bx-0%7B%2C%7D6x%7D" alt="\frac{0{,}04x}{x}=\frac{y}{x-0{,}6x}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B0%7B%2C%7D04x%7D%7Bx%7D%3D%5Cfrac%7By%7D%7B0%7B%2C%7D4x%7D" alt="\frac{0{,}04x}{x}=\frac{y}{0{,}4x}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B0%7B%2C%7D04x%5Ccdot0%7B%2C%7D4x%7D%7Bx%7D%3Dy" alt="\frac{0{,}04x\cdot0{,}4x}{x}=y"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D04%5Ccdot0%7B%2C%7D4%3Dy" alt="0{,}04\cdot0{,}4=y"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D0%7B%2C%7D016%3D1.6%5C%25" alt="y=0{,}016=1.6\%"/></div> </div> <div>153<br/> a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cge0%5C%20ja%5C%20b%5Cge0" alt="a\ge0\ ja\ b\ge0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cle0%5C%20ja%5C%20b%5Cle0" alt="a\le0\ ja\ b\le0"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3C0%5C%20ja%5C%20b%3E0" alt="a&lt;0\ ja\ b&gt;0"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff; color: #333333; font-family: 'Times New Roman'; font-size: 17px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3E0%5C%20ja%5C%20b%3C0" alt="a&gt;0\ ja\ b&lt;0"/><br/> <div> <div> <div> <div> <div> <div> <div> </div> </div> </div> </div> </div> </div> </div> </div> <div>156<br/> a)<br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI4MTAz/1313592/files?id=5e1749f40a975a3c4581679e" alt="\left|x\right|\begin{cases} x{,}&amp;kun\ x\ge0\\ -x{,}&amp;kun\ x&lt;0 \end{cases}"/><!--filtered attribute: data-userfileid="5e1749f40a975a3c4581679e"--><br/> b)<br/> <p>Esim. </p> <p>Kun x on negatiivinen, luvun itseisarvo on aidosti suurempi kuin x itse</p> <p>Kun x on positiivinen, luvun itseisarvo on aidosti yhtä suuri kuin x itse</p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI4MTAz/1313592/files?id=5e174a7c0a975a3c458173da" alt="x=1{,}\ \left|x\right|=1=x"/><!--filtered attribute: data-userfileid="5e174a7c0a975a3c458173da"--></p> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI4MTAz/1313592/files?id=5e174a7d0a975a3c458173dd" alt="x=-1{,}\ \left|x\right|=1&gt;x"/><!--filtered attribute: data-userfileid="5e174a7d0a975a3c458173dd"--><!--filtered attribute: style="box-sizing: border-box; border: 1px solid #e6f2f8; vertical-align: bottom; position: relative; max-width: 100%; cursor: pointer; display: inline-block; float: none; margin-left: 5px; margin-right: 5px; max-height: 1000px; padding: 3px 10px; background: #edf9ff;"--><br/> c)</p> <div> <p>Esim. </p> <p>Kun x tai y on negatiivinen, luvun itseisarvo on aidosti suurempi kuin luku itse</p> <p>Kun x tai y on positiivinen, luvun itseisarvo on aidosti yhtä suuri kuin luku itse</p> </div> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI4MTAz/1313592/files?id=5e174b220a975a3c45817e2f" alt="x=1{,}\ y=2"/><!--filtered attribute: data-userfileid="5e174b220a975a3c45817e2f"--></div> <p><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI4MTAz/1313592/files?id=5e174b220a975a3c45817e2b" alt="\left|x\right|+\left|y\right|=1+2=3=x+y"/><!--filtered attribute: data-userfileid="5e174b220a975a3c45817e2b"--></p> <div><img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI4MTAz/1313592/files?id=5e174b220a975a3c45817e28" alt="x=-1{,}\ y=2"/><!--filtered attribute: data-userfileid="5e174b220a975a3c45817e28"--><br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI4MTAz/1313592/files?id=5e174b220a975a3c45817e2d" alt="\left|x\right|+\left|y\right|=1+2=3&gt;-1+2=1"/><!--filtered attribute: data-userfileid="5e174b220a975a3c45817e2d"--><!--filtered attribute: style="box-sizing: border-box; border: 1px solid #e6f2f8; vertical-align: bottom; position: relative; max-width: 100%; cursor: pointer; display: inline-block; float: none; margin-left: 5px; margin-right: 5px; max-height: 1000px; padding: 3px 10px; background: #edf9ff;"--><br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI4MTAz/1313592/files?id=5e174b220a975a3c45817e32" alt="x=-1{,}\ y=-2"/><!--filtered attribute: data-userfileid="5e174b220a975a3c45817e32"--><br/> <img class="fr-fic fr-dii" src="https://c2-julkaisu.otava.fi/o/task-container/49a34b72/YXNzZW1ibHkvNDlhMzRiNzIvMzI4MTAz/1313592/files?id=5e174b220a975a3c45817e34" alt="\left|x\right|+\left|y\right|=1+2=3&gt;-1-2=-3"/><!--filtered attribute: data-userfileid="5e174b220a975a3c45817e34"--><!--filtered attribute: style="box-sizing: border-box; border: 1px solid #e6f2f8; vertical-align: bottom; position: relative; max-width: 100%; cursor: pointer; display: inline-block; float: none; margin-left: 5px; margin-right: 5px; max-height: 1000px; padding: 3px 10px; background: #edf9ff;"--><br/> d)<br/> <div>Esim. jos x on negatiivinen, luvun itseisarvo on aidosti suurempi kuin x itse</div> <div>taas kun x on positiivinen, luvun itseisarvo on aidosti yhtä suuri kuin x itse</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1%7B%2C%7D%5C%20y%3D2" alt="x=1{,}\ y=2"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7Cx%5Cright%7C%2B%5Cleft%7Cy%5Cright%7C%3D1%2B2%3D3%3D%5Cleft%7C1%2B2%5Cright%7C%3D%5Cleft%7Cx%2By%5Cright%7C" alt="\left|x\right|+\left|y\right|=1+2=3=\left|1+2\right|=\left|x+y\right|"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%7B%2C%7D%5C%20y%3D2" alt="x=-1{,}\ y=2"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7Cx%5Cright%7C%2B%5Cleft%7Cy%5Cright%7C%3D1%2B2%3D3%3E%5Cleft%7C-1%2B2%5Cright%7C%3D%5Cleft%7Cx%2By%5Cright%7C" alt="\left|x\right|+\left|y\right|=1+2=3&gt;\left|-1+2\right|=\left|x+y\right|"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%7B%2C%7D%5C%20y%3D-2" alt="x=-1{,}\ y=-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7Cx%5Cright%7C%2B%5Cleft%7Cy%5Cright%7C%3D1%2B2%3D3%3D%5Cleft%7C-1-2%5Cright%7C%3D%5Cleft%7Cx%2By%5Cright%7C" alt="\left|x\right|+\left|y\right|=1+2=3=\left|-1-2\right|=\left|x+y\right|"/></div> e)<br/> <div>Esim. jos x on negatiivinen, luvun itseisarvo on aidosti suurempi kuin x itse</div> <div>taas kun x on positiivinen, luvun itseisarvo on aidosti yhtä suuri kuin x itse</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1%7B%2C%7D%5C%20y%3D2" alt="x=1{,}\ y=2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Cleft%7Cx%5Cright%7C%2B%5Cleft%7Cy%5Cright%7C%5Cright%7C%3D1%2B2%3D3%3D%5Cleft%7C1%5Cright%7C%2B%5Cleft%7C2%5Cright%7C%3D%5Cleft%7Cx%5Cright%7C%2B%5Cleft%7Cy%5Cright%7C" alt="\left|\left|x\right|+\left|y\right|\right|=1+2=3=\left|1\right|+\left|2\right|=\left|x\right|+\left|y\right|"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%7B%2C%7D%5C%20y%3D2" alt="x=-1{,}\ y=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Cleft%7Cx%5Cright%7C%2B%5Cleft%7Cy%5Cright%7C%5Cright%7C%3D1%2B2%3D3%3D%5Cleft%7C-1%5Cright%7C%2B%5Cleft%7C2%5Cright%7C%3D%5Cleft%7Cx%5Cright%7C%2B%5Cleft%7Cy%5Cright%7C" alt="\left|\left|x\right|+\left|y\right|\right|=1+2=3=\left|-1\right|+\left|2\right|=\left|x\right|+\left|y\right|"/><br/> <div> <div> <div> <div> <div> <div> <div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%7B%2C%7D%5C%20y%3D-2" alt="x=-1{,}\ y=-2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Cleft%7Cx%5Cright%7C%2B%5Cleft%7Cy%5Cright%7C%5Cright%7C%3D1%2B2%3D3%3D%5Cleft%7C-1%5Cright%7C%2B%5Cleft%7C-2%5Cright%7C%3D%5Cleft%7Cx%5Cright%7C%2B%5Cleft%7Cy%5Cright%7C" alt="\left|\left|x\right|+\left|y\right|\right|=1+2=3=\left|-1\right|+\left|-2\right|=\left|x\right|+\left|y\right|"/></div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> <div>157</div> </div> 2020-01-08T10:13:52+02:00