https://peda.net/id/490b259e3f3 2019-03-05T13:39:07+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/5a42fe28597 2019-04-08T00:16:04+03:00 <div>439</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=28" alt="28"/></div> b) <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D%5Cfrac%7B4-%5Cleft(-2%5Cright)%7D%7B3%7D%3D2" alt="h=\frac{4-\left(-2\right)}{3}=2"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-2%7D%5E4%5Cleft(x%5E2-3x%2B3%5Cright)dx%3D2%5Cleft(%5Cfrac%7B1%7D%7B2%7Df%5Cleft(-2%5Cright)%2Bf%5Cleft(0%5Cright)%2Bf%5Cleft(2%5Cright)%2B%5Cfrac%7B1%7D%7B2%7Df%5Cleft(4%5Cright)%5Cright)%3D2%5Cleft(6%7B%2C%7D5%2B3%2B1%2B3%7B%2C%7D5%5Cright)%3D28" alt="\int_{-2}^4\left(x^2-3x+3\right)dx=2\left(\frac{1}{2}f\left(-2\right)+f\left(0\right)+f\left(2\right)+\frac{1}{2}f\left(4\right)\right)=2\left(6{,}5+3+1+3{,}5\right)=28"/></div> <div> </div> <div>442</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5E2f%5Cleft(x%5Cright)dx%3D0%7B%2C%7D5%5Cleft(%5Cfrac%7B1%7D%7B2%7Df%5Cleft(0%5Cright)%2Bf%5Cleft(0%7B%2C%7D5%5Cright)%2Bf%5Cleft(1%5Cright)%2Bf%5Cleft(1%7B%2C%7D5%5Cright)%2B%5Cfrac%7B1%7D%7B2%7Df%5Cleft(2%5Cright)%5Cright)%3D%5Cfrac%7B%5Cleft(1%2B1-3-5-3%5Cright)%7D%7B2%7D%3D-4%7B%2C%7D5" alt="\int_0^2f\left(x\right)dx=0{,}5\left(\frac{1}{2}f\left(0\right)+f\left(0{,}5\right)+f\left(1\right)+f\left(1{,}5\right)+\frac{1}{2}f\left(2\right)\right)=\frac{\left(1+1-3-5-3\right)}{2}=-4{,}5"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5E2f%5Cleft(x%5Cright)dx%3D%5Cfrac%7B0%7B%2C%7D5%7D%7B3%7D%5Cleft(f%5Cleft(0%5Cright)%2B4f%5Cleft(0%7B%2C%7D5%5Cright)%2B2f%5Cleft(1%5Cright)%2B4f%5Cleft(1%7B%2C%7D5%5Cright)%2Bf%5Cleft(2%5Cright)%5Cright)%3D%5Cfrac%7B0%7B%2C%7D5%7D%7B3%7D%5Cleft(2%2B4-6-20-6%5Cright)%3D-4%7B%2C%7D33333...%5Capprox-4%7B%2C%7D3" alt="\int_0^2f\left(x\right)dx=\frac{0{,}5}{3}\left(f\left(0\right)+4f\left(0{,}5\right)+2f\left(1\right)+4f\left(1{,}5\right)+f\left(2\right)\right)=\frac{0{,}5}{3}\left(2+4-6-20-6\right)=-4{,}33333...\approx-4{,}3"/><br/> <div> </div> <div> </div> <div> </div> <div> </div> </div> </div> 2019-04-08T00:16:04+03:00 https://peda.net/id/2301fab2578 2019-04-08T22:17:50+03:00 <div>401</div> <div>a) 13,23</div> <div>b) Pienempi</div> <div> </div> <div>403</div> <div>3.1, vähittäin 91 osaväliä<br/> <br/> <div>408</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%5Capprox21%7B%2C%7D5" alt="A\approx21{,}5"/></div> <div> </div> <div>411</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%5Capprox16%7B%2C%7D5" alt="A\approx16{,}5"/><br/> <br/> </div> <div> <div>405. </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D-x%5E2%2B8x-7" alt="f\left(x\right)=-x^2+8x-7"/></div> <div>Lasketaan kuvaajan ja x-akselin leikkauskodat.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-x%5E2%2B8x-7%3D0" alt="-x^2+8x-7=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1%5C%20tai%5C%20x%3D7%5C%20%5Cleft(laskin%5Cright)" alt="x=1\ tai\ x=7\ \left(laskin\right)"/><br/> <div>Suorakulmioita on 3 kpl eli n=3</div> <div>Yhden osavälin pituus eli suorakulmion leveys on</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D%5Cfrac%7B7-1%7D%7B3%7D%3D%5Cfrac%7B6%7D%7B3%7D%3D2" alt="h=\frac{7-1}{3}=\frac{6}{3}=2"/></div> Osavälit ovat [1,3] [3,5] [5,7]. <div>Osavälien keskipisteet ovat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%2B3%7D%7B2%7D%3D%5Cfrac%7B4%7D%7B2%7D%3D2" alt="\frac{1+3}{2}=\frac{4}{2}=2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3%2B5%7D%7B2%7D%3D%5Cfrac%7B8%7D%7B2%7D%3D4" alt="\frac{3+5}{2}=\frac{8}{2}=4"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B5%2B7%7D%7B2%7D%3D6" alt="\frac{5+7}{2}=6"/></div> <div>Suorakulmion korkeudet ovat fubktiob arvot välien keskpisteissä eli </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(2%5Cright)%7B%2C%7D%5C%20f%5Cleft(4%5Cright)%5C%20ja%5C%20f%5Cleft(6%5Cright)" alt="f\left(2\right){,}\ f\left(4\right)\ ja\ f\left(6\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A%5Capprox2f%5Cleft(2%5Cright)%2B2f%5Cleft(4%5Cright)%2B2f%5Cleft(6%5Cright)%3D38" alt="A\approx2f\left(2\right)+2f\left(4\right)+2f\left(6\right)=38"/><br/> <div> </div> 421 <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(0%5Cright)%3D-5%7B%2C%7D%5C%20f%5Cleft(2%5Cright)%3D-9%7B%2C%7D%5C%20f%5Cleft(4%5Cright)%3D-5" alt="f\left(0\right)=-5{,}\ f\left(2\right)=-9{,}\ f\left(4\right)=-5"/></div> a) Kuvaaja on x.akselin alapuolella, joten suorakulmioiden korkeudet ovat funktion arvojen vastalukuja. Suorakulmioiden leveys on 2. <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%5Capprox2%5Cleft(-f%5Cleft(0%5Cright)-f%5Cleft(2%5Cright)-f%5Cleft(4%5Cright)%5Cright)%3D2%5Cleft(5%2B9%2B5%5Cright)%3D38" alt="A\approx2\left(-f\left(0\right)-f\left(2\right)-f\left(4\right)\right)=2\left(5+9+5\right)=38"/></div> <div>b) Määrätty integraali on negatiivinen, koska kuvaaja on x-akselin alapuolella.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-1%7D%5E5x%5Cleft(dx%5Cright)%5Capprox-A%3D-38" alt="\int_{-1}^5x\left(dx\right)\approx-A=-38"/></div> <div> </div> <div>423</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_0%5E2f%5Cleft(x%5Cright)dx%3D-A2%3D-2" alt="\int_0^2f\left(x\right)dx=-A2=-2"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-2%7D%5E3f%5Cleft(x%5Cright)dx%3DA1-A2%2BA3%3D4%5Cfrac%7B1%7D%7B3%7D-2%2B%5Cfrac%7B1%7D%7B2%7D%3D2%5Cfrac%7B5%7D%7B6%7D" alt="\int_{-2}^3f\left(x\right)dx=A1-A2+A3=4\frac{1}{3}-2+\frac{1}{2}=2\frac{5}{6}"/></div> <div>c)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-2%7D%5E3%5Cleft%7Cf%5Cleft(x%5Cright)%5Cright%7Cdx%3DA1%2BA2%2BA3%3D6%5Cfrac%7B5%7D%7B6%7D" alt="\int_{-2}^3\left|f\left(x\right)\right|dx=A1+A2+A3=6\frac{5}{6}"/><br/> <br/> </div> 426 <div><img src="https://math-demo.abitti.fi/math.svg?latex=-0%7B%2C%7D52" alt="-0{,}52"/></div> <div> </div> <div>427</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft%5B0%7B%2C%7D%5Cpi%5Cright%5D%5Capprox3%7B%2C%7D28987" alt="A\left[0{,}\pi\right]\approx3{,}28987"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A%5Cleft%5B%5Cpi%7B%2C%7D2%5Cpi%5Cright%5D%5Capprox-9.67144" alt="A\left[\pi{,}2\pi\right]\approx-9.67144"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h_1%3D%5Cfrac%7B%5Cpi-0%7D%7B2%7D%3D%5Cfrac%7B%5Cpi%7D%7B2%7D" alt="h_1=\frac{\pi-0}{2}=\frac{\pi}{2}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A_1%5Capprox%5Cfrac%7B%5Cpi%7D%7B2%7Df%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%5Capprox2.467" alt="A_1\approx\frac{\pi}{2}f\left(\frac{\pi}{2}\right)\approx2.467"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=h_2%3D%5Cfrac%7B%5C%202%5Cpi-%5Cpi%7D%7B2%7D%3D%5Cpi" alt="h_2=\frac{\ 2\pi-\pi}{2}=\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A_2%5Capprox%5Cpi%20f%5Cleft(%5Cfrac%7B3%5Cpi%7D%7B2%7D%5Cright)%3D" alt="A_2\approx\pi f\left(\frac{3\pi}{2}\right)="/> <div> </div> </div> </div> 2019-04-05T12:42:33+03:00 https://peda.net/id/dc5f4ff6540 2019-04-01T11:08:30+03:00 <div>322</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D-0%7B%2C%7D5%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(1%2B%5Cleft(-0%7B%2C%7D5%5Cright)%5Cright)-f%5Cleft(1%5Cright)%7D%7B-0%7B%2C%7D5%7D%3D%5Cfrac%7B65-55%7D%7B-0%7B%2C%7D5%7D%3D-20%5C%20%5Cfrac%7B%C2%B0C%7D%7B%5Cmin%7D%3D20%5C%20%5Cfrac%7B%C2%B0C%7D%7B%5Cmin%7D" alt="h=-0{,}5;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(1+\left(-0{,}5\right)\right)-f\left(1\right)}{-0{,}5}=\frac{65-55}{-0{,}5}=-20\ \frac{°C}{\min}=20\ \frac{°C}{\min}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D5%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(1%2B%5Cleft(0%7B%2C%7D5%5Cright)%5Cright)-f%5Cleft(1%5Cright)%7D%7B0%7B%2C%7D5%7D%3D%5Cfrac%7B47-55%7D%7B0%7B%2C%7D5%7D%3D%5Cfrac%7B-8%7D%7B0%7B%2C%7D5%7D%3D-16%5C%20%5Cfrac%7B%C2%B0C%7D%7B%5Cmin%7D%3D16%5C%20%5Cfrac%7B%C2%B0C%7D%7B%5Cmin%7D" alt="h=0{,}5;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(1+\left(0{,}5\right)\right)-f\left(1\right)}{0{,}5}=\frac{47-55}{0{,}5}=\frac{-8}{0{,}5}=-16\ \frac{°C}{\min}=16\ \frac{°C}{\min}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D-0%7B%2C%7D5%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(2%2B%5Cleft(-0%7B%2C%7D5%5Cright)%5Cright)-f%5Cleft(2%5Cright)%7D%7B-0%7B%2C%7D5%7D%3D%5Cfrac%7B47-40%7D%7B-0%7B%2C%7D5%7D%3D-14%5C%20%5Cfrac%7B%C2%B0C%7D%7B%5Cmin%7D%3D14%5C%20%5Cfrac%7B%C2%B0C%7D%7B%5Cmin%7D" alt="h=-0{,}5;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+\left(-0{,}5\right)\right)-f\left(2\right)}{-0{,}5}=\frac{47-40}{-0{,}5}=-14\ \frac{°C}{\min}=14\ \frac{°C}{\min}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D5%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(2%2B%5Cleft(0%7B%2C%7D5%5Cright)%5Cright)-f%5Cleft(2%5Cright)%7D%7B0%7B%2C%7D5%7D%3D%5Cfrac%7B36-40%7D%7B0%7B%2C%7D5%7D%3D%5Cfrac%7B-4%7D%7B0%7B%2C%7D5%7D%3D-8%5C%20%5Cfrac%7B%C2%B0C%7D%7B%5Cmin%7D%3D8%5C%20%5Cfrac%7B%C2%B0C%7D%7B%5Cmin%7D" alt="h=0{,}5;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+\left(0{,}5\right)\right)-f\left(2\right)}{0{,}5}=\frac{36-40}{0{,}5}=\frac{-4}{0{,}5}=-8\ \frac{°C}{\min}=8\ \frac{°C}{\min}"/><br/> c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D-0%7B%2C%7D5%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(4%2B%5Cleft(-0%7B%2C%7D5%5Cright)%5Cright)-f%5Cleft(4%5Cright)%7D%7B-0%7B%2C%7D5%7D%3D%5Cfrac%7B30-28%7D%7B-0%7B%2C%7D5%7D%3D-4%5C%20%5Cfrac%7B%C2%B0C%7D%7B%5Cmin%7D%3D4%5C%20%5Cfrac%7B%C2%B0C%7D%7B%5Cmin%7D" alt="h=-0{,}5;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(4+\left(-0{,}5\right)\right)-f\left(4\right)}{-0{,}5}=\frac{30-28}{-0{,}5}=-4\ \frac{°C}{\min}=4\ \frac{°C}{\min}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D5%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(4%2B%5Cleft(0%7B%2C%7D5%5Cright)%5Cright)-f%5Cleft(4%5Cright)%7D%7B0%7B%2C%7D5%7D%3D%5Cfrac%7B27-28%7D%7B0%7B%2C%7D5%7D%3D%5Cfrac%7B-1%7D%7B0%7B%2C%7D5%7D%3D-2%5C%20%5Cfrac%7B%C2%B0C%7D%7B%5Cmin%7D%3D2%5C%20%5Cfrac%7B%C2%B0C%7D%7B%5Cmin%7D" alt="h=0{,}5;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(4+\left(0{,}5\right)\right)-f\left(4\right)}{0{,}5}=\frac{27-28}{0{,}5}=\frac{-1}{0{,}5}=-2\ \frac{°C}{\min}=2\ \frac{°C}{\min}"/></div> <div> </div> <div>324</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D1%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a-h%5Cright)%7D%7B2h%7D%5Capprox0.66816" alt="h=0{,}1;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx0.66816"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D01%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a-h%5Cright)%7D%7B2h%7D%5Capprox0.67104" alt="h=0{,}01;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx0.67104"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D001%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a-h%5Cright)%7D%7B2h%7D%5Capprox0.67107" alt="h=0{,}001;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx0.67107"/><br/> <span> </span> <div>325</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D5%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a-h%5Cright)%7D%7B2h%7D%3D%5Cfrac%7B3.5-2.4%7D%7B1%7D%3D1.1%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="h=0{,}5;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}=\frac{3.5-2.4}{1}=1.1\ \frac{m}{s}"/></div> <div> </div> <div> <div>326</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D1%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(2%2B0%7B%2C%7D1%5Cright)-f%5Cleft(2%5Cright)%7D%7B0%7B%2C%7D1%7D%3D7.49638...%5Capprox7.4964" alt="h=0{,}1;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+0{,}1\right)-f\left(2\right)}{0{,}1}=7.49638...\approx7.4964"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D01%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(2%2B0%7B%2C%7D01%5Cright)-f%5Cleft(2%5Cright)%7D%7B0%7B%2C%7D01%7D%3D6.840402...%5Capprox6.8404" alt="h=0{,}01;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+0{,}01\right)-f\left(2\right)}{0{,}01}=6.840402...\approx6.8404"/></div> <div> <img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D001%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(2%2B0%7B%2C%7D001%5Cright)-f%5Cleft(2%5Cright)%7D%7B0%7B%2C%7D001%7D%3D6.77932...%5Capprox6.7793" alt="h=0{,}001;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+0{,}001\right)-f\left(2\right)}{0{,}001}=6.77932...\approx6.7793"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D-0%7B%2C%7D1%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(2%2B%5Cleft(-0%7B%2C%7D1%5Cright)%5Cright)-f%5Cleft(2%5Cright)%7D%7B-0%7B%2C%7D1%7D%3D6.14429...%5Capprox6.1443" alt="h=-0{,}1;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+\left(-0{,}1\right)\right)-f\left(2\right)}{-0{,}1}=6.14429...\approx6.1443"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=h%3D-0%7B%2C%7D01%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(2%2B%5Cleft(-0%7B%2C%7D01%5Cright)%5Cright)-f%5Cleft(2%5Cright)%7D%7B-0%7B%2C%7D01%7D%3D6.70572...%5Capprox6.7057" alt="h=-0{,}01;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+\left(-0{,}01\right)\right)-f\left(2\right)}{-0{,}01}=6.70572...\approx6.7057"/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=h%3D-0%7B%2C%7D001%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(2%2B%5Cleft(-0%7B%2C%7D001%5Cright)%5Cright)-f%5Cleft(2%5Cright)%7D%7B-0%7B%2C%7D001%7D%3D6.76585...%5Capprox6.7659" alt="h=-0{,}001;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(2+\left(-0{,}001\right)\right)-f\left(2\right)}{-0{,}001}=6.76585...\approx6.7659"/> </div> <span>c)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D1%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a-h%5Cright)%7D%7B2h%7D%5Capprox6.8203" alt="h=0{,}1;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx6.8203"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D01%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a-h%5Cright)%7D%7B2h%7D%5Capprox6.7731" alt="h=0{,}01;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx6.7731"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D001%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a-h%5Cright)%7D%7B2h%7D%5Capprox6.7726" alt="h=0{,}001;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx6.7726"/>¨</div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(2%5Cright)%5Capprox6.77259" alt="f'\left(2\right)\approx6.77259"/><br/> <br/> </div> </div> <div>329 <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D1%3B%5C%20%5Capprox-0.3466" alt="h=0{,}1;\ \approx-0.3466"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D01%3B%5C%20%5Capprox-0.3466" alt="h=0{,}01;\ \approx-0.3466"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D001%3B%5C%20%5Capprox-0.3466" alt="h=0{,}001;\ \approx-0.3466"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D2%5E%7B-x%7D%5Ccdot%5Cleft(-%5Cln2%5Cright)" alt="f'\left(x\right)=2^{-x}\cdot\left(-\ln2\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(1%5Cright)%3D2%5E%7B-1%7D%5Ccdot%5Cleft(-%5Cln2%5Cright)%3D%E2%88%920.34657...%5Capprox-0%7B%2C%7D3466" alt="f'\left(1\right)=2^{-1}\cdot\left(-\ln2\right)=−0.34657...\approx-0{,}3466"/></div> <div> <div><br/> 331</div> <div>Raja-arvo ei ole</div> <div> </div> <div>332</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3De%5Ex" alt="f\left(x\right)=e^x"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3De%5Ex" alt="f'\left(x\right)=e^x"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(2%5Cright)%3De%5E2" alt="f'\left(2\right)=e^2"/></div> <br/> <div>333<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D-0%7B%2C%7D1%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(0%2B%5Cleft(-0%7B%2C%7D1%5Cright)%5Cright)-f%5Cleft(0%5Cright)%7D%7B-0%7B%2C%7D1%7D%5Capprox%E2%88%921.0488" alt="h=-0{,}1;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(0+\left(-0{,}1\right)\right)-f\left(0\right)}{-0{,}1}\approx−1.0488"/> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D1%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a%5Cright)%7D%7Bh%7D%3D%5Cfrac%7Bf%5Cleft(0%2B0%7B%2C%7D1%5Cright)-f%5Cleft(0%5Cright)%7D%7B0%7B%2C%7D1%7D%5Capprox0.9487" alt="h=0{,}1;\ \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{f\left(0+0{,}1\right)-f\left(0\right)}{0{,}1}\approx0.9487"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%3D0%7B%2C%7D1%3B%5C%20%5Cfrac%7Bf%5Cleft(a%2Bh%5Cright)-f%5Cleft(a-h%5Cright)%7D%7B2h%7D%5Capprox-0.0501" alt="h=0{,}1;\ \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}\approx-0.0501"/> </div> <div>Ei näytä oleva derivoituva<br/> <br/> </div> </div> <div>340<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(0.5%5Cright)%5Capprox0.877586521890373" alt="f'\left(0.5\right)\approx0.877586521890373"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=p%3D3%3B%5C%20%5Capprox0.877582415626633" alt="p=3;\ \approx0.877582415626633"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=p%3D4%3B%5C%20%5Capprox0.877582560427638" alt="p=4;\ \approx0.877582560427638"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=p%3D5%3B%5C%20%5Capprox0.87758256187287" alt="p=5;\ \approx0.87758256187287"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=p%3D6%3B%5C%20%5Capprox0.87758256189785" alt="p=6;\ \approx0.87758256189785"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=p%3D7%3B%5C%20%5Capprox0.877582561620295" alt="p=7;\ \approx0.877582561620295"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=p%3D8%3B%5C%20%5Capprox0.877582562175406" alt="p=8;\ \approx0.877582562175406"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=p%3D9%3B%5C%20%5Capprox0.877582562175405" alt="p=9;\ \approx0.877582562175405"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=p%3D10%3B%5C%20%5Capprox0.877582451153103" alt="p=10;\ \approx0.877582451153103"/></div> <div><span>8 </span><br/> <br/> </div> </div> </div> 2019-04-01T02:29:42+03:00 https://peda.net/id/e3cabea6500 2019-03-27T22:18:12+02:00 <div>301</div> <div>a) 1</div> <div>b) 0,25</div> <div>c) -2</div> <div>d) 0</div> <div><br/> 302<br/> I, IV<br/> <br/> </div> <div>303</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D4x-1" alt="f'\left(x\right)=4x-1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(3%5Cright)%3D4%5Ccdot3-1%3D11" alt="f'\left(3\right)=4\cdot3-1=11"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%7D%5Cleft(%5Cfrac%7Bf%5Cleft(3%2Bh%5Cright)-f%5Cleft(3%5Cright)%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%7D%5Cleft(%5Cfrac%7B2%5Ccdot%5Cleft(3%2Bh%5Cright)%5E2-%5Cleft(3%2Bh%5Cright)-15%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%7D%5Cleft(%5Cfrac%7B2%5Ccdot%5Cleft(9%2B5h%2Bh%5E2%5Cright)-3%2Bh-15%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%7D%5Cleft(%5Cfrac%7B2h%5E2%2B11h%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%7D%5Cleft(2h%2B11%5Cright)%3D2%5Ccdot0%2B11%3D11" alt="f'\left(x\right)=\lim_{h\rightarrow0}\left(\frac{f\left(3+h\right)-f\left(3\right)}{h}\right)=\lim_{h\rightarrow0}\left(\frac{2\cdot\left(3+h\right)^2-\left(3+h\right)-15}{h}\right)=\lim_{h\rightarrow0}\left(\frac{2\cdot\left(9+5h+h^2\right)-3+h-15}{h}\right)=\lim_{h\rightarrow0}\left(\frac{2h^2+11h}{h}\right)=\lim_{h\rightarrow0}\left(2h+11\right)=2\cdot0+11=11"/></div> <div> </div> <div>304</div> <div>I Ei</div> <div>II Ei</div> <div>III Kyllä</div> <div>IV Ei</div> <div> <br/> 305  <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=5h%5E2-30h%2B65" alt="5h^2-30h+65"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(1%5Cright)%3D65" alt="f'\left(1\right)=65"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(t%5Cright)%3D15t%5E2-90t%2B140" alt="f'\left(t\right)=15t^2-90t+140"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(1%5Cright)%3D15-90%2B140%3D65" alt="f'\left(1\right)=15-90+140=65"/><br/> <br/> 306 <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D-%5Cleft(x-3%5Cright)%5E2" alt="f\left(x\right)=-\left(x-3\right)^2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D-x%5E2%2B6x-9%3D-2x%2B6" alt="f'\left(x\right)=-x^2+6x-9=-2x+6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(-2%5Cright)%3D-2%5Ccdot%5Cleft(-2%5Cright)%2B6%3D10" alt="f'\left(-2\right)=-2\cdot\left(-2\right)+6=10"/><br/> <br/> </div> <div>307</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D-2x" alt="f'\left(x\right)=-2x"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D-2%5Cleft(-2%5Cright)%3D4" alt="f'\left(x\right)=-2\left(-2\right)=4"/><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%7D%5Cleft(%5Cfrac%7Bf%5Cleft(-2%2Bh%5Cright)-f%5Cleft(-2%5Cright)%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%7D%5Cleft(%5Cfrac%7B%5Cleft(1-%5Cleft(-2%2Bh%5Cright)%5E2-%5Cleft(1-4%5Cright)%5Cright)%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%7D%5Cleft(%5Cfrac%7B%5Cleft(1-%5Cleft(h%5E2-4h%2B4%5Cright)%5Cright)%2B3%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%7D%5Cleft(%5Cfrac%7B4-%5Cleft(h%5E2-4h%2B4%5Cright)%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%7D%5Cleft(4-h-4%2B4%5Cright)%3D4-0-4%2B4%3D4" alt="f'\left(x\right)=\lim_{h\rightarrow0}\left(\frac{f\left(-2+h\right)-f\left(-2\right)}{h}\right)=\lim_{h\rightarrow0}\left(\frac{\left(1-\left(-2+h\right)^2-\left(1-4\right)\right)}{h}\right)=\lim_{h\rightarrow0}\left(\frac{\left(1-\left(h^2-4h+4\right)\right)+3}{h}\right)=\lim_{h\rightarrow0}\left(\frac{4-\left(h^2-4h+4\right)}{h}\right)=\lim_{h\rightarrow0}\left(4-h-4+4\right)=4-0-4+4=4"/><br/> <div> </div> <div>308</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E2%2Bx-1" alt="f\left(x\right)=x^2+x-1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E-%7D%5Cleft(%5Cfrac%7Bf%5Cleft(1%2Bh%5Cright)-f%5Cleft(1%5Cright)%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E-%7D%5Cleft(%5Cfrac%7B%5Cleft(%5Cleft(1%2Bh%5Cright)%5E2%2B%5Cleft(1%2Bh%5Cright)-1%5Cright)-1%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E-%7D%5Cleft(%5Cfrac%7B%5Cleft(%5Cleft(h%5E2%2B2h%2B1%5Cright)%2B%5Cleft(1%2Bh%5Cright)-1%5Cright)-1%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E-%7D%5Cleft(%5Cfrac%7Bh%5E2%2B3h%2B2-1-1%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E-%7D%5Cleft(h%2B3%2B2-1-1%5Cright)%3D0%2B3%2B2-2%3D3" alt="f'\left(x\right)=\lim_{h\rightarrow0^-}\left(\frac{f\left(1+h\right)-f\left(1\right)}{h}\right)=\lim_{h\rightarrow0^-}\left(\frac{\left(\left(1+h\right)^2+\left(1+h\right)-1\right)-1}{h}\right)=\lim_{h\rightarrow0^-}\left(\frac{\left(\left(h^2+2h+1\right)+\left(1+h\right)-1\right)-1}{h}\right)=\lim_{h\rightarrow0^-}\left(\frac{h^2+3h+2-1-1}{h}\right)=\lim_{h\rightarrow0^-}\left(h+3+2-1-1\right)=0+3+2-2=3"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D-x%5E2%2B5x-3" alt="f\left(x\right)=-x^2+5x-3"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bh%5Crightarrow0%5E%2B%7D%5Cleft(%5Cfrac%7Bf%5Cleft(1%2Bh%5Cright)-f%5Cleft(1%5Cright)%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E%2B%7D%5Cleft(%5Cfrac%7B%5Cleft(-%5Cleft(1%2Bh%5Cright)%5E2%2B5%5Ccdot%5Cleft(1%2Bh%5Cright)-3%5Cright)-%5Cleft(-1%2B5-3%5Cright)%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E%2B%7D%5Cleft(%5Cfrac%7B%5Cleft(-%5Cleft(h%5E2%2B2h%2B1%5Cright)%2B5%2B5h-3%5Cright)-1%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E%2B%7D%5Cleft(%5Cfrac%7B-h%5E2-2h%2B5h-1%2B5-3-1%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E%2B%7D%5Cleft(%5Cfrac%7B-h%5E2%2B3h%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E%2B%7D%5Cleft(-h%2B3%5Cright)%3D-0%2B3%3D3" alt="\lim_{h\rightarrow0^+}\left(\frac{f\left(1+h\right)-f\left(1\right)}{h}\right)=\lim_{h\rightarrow0^+}\left(\frac{\left(-\left(1+h\right)^2+5\cdot\left(1+h\right)-3\right)-\left(-1+5-3\right)}{h}\right)=\lim_{h\rightarrow0^+}\left(\frac{\left(-\left(h^2+2h+1\right)+5+5h-3\right)-1}{h}\right)=\lim_{h\rightarrow0^+}\left(\frac{-h^2-2h+5h-1+5-3-1}{h}\right)=\lim_{h\rightarrow0^+}\left(\frac{-h^2+3h}{h}\right)=\lim_{h\rightarrow0^+}\left(-h+3\right)=-0+3=3"/><br/> <div>c)</div> <div>On</div> <div>d) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=k%3D3" alt="k=3"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=y_0%3D1" alt="y_0=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x_0%3D1" alt="x_0=1"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(y-y_0%5Cright)%3Dk%5Cleft(x-x_0%5Cright)" alt="\left(y-y_0\right)=k\left(x-x_0\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=y-1%3D3x-3" alt="y-1=3x-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D3x-2" alt="y=3x-2"/><br/> <br/> <div>309</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2x%2B1" alt="f\left(x\right)=2x+1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bh%5Crightarrow0%5E-%7D%5E%7B%20%7D%5Cleft(%5Cfrac%7Bf%5Cleft(-1%2Bh%5Cright)-f%5Cleft(-1%5Cright)%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E-%7D%5E%7B%20%7D%5Cleft(%5Cfrac%7B%5Cleft(2%5Ccdot%5Cleft(-1%2Bh%5Cright)%2B1%5Cright)%2B1%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E-%7D%5E%7B%20%7D%5Cleft(%5Cfrac%7B-2%2B2h%2B1%2B1%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E-%7D%5E%7B%20%7D%5Cleft(%5Cfrac%7B2h%7D%7Bh%7D%5Cright)%3D2" alt="\lim_{h\rightarrow0^-}^{ }\left(\frac{f\left(-1+h\right)-f\left(-1\right)}{h}\right)=\lim_{h\rightarrow0^-}^{ }\left(\frac{\left(2\cdot\left(-1+h\right)+1\right)+1}{h}\right)=\lim_{h\rightarrow0^-}^{ }\left(\frac{-2+2h+1+1}{h}\right)=\lim_{h\rightarrow0^-}^{ }\left(\frac{2h}{h}\right)=2"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E2%2B3x%2B1" alt="f\left(x\right)=x^2+3x+1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bh%5Crightarrow0%5E%2B%7D%5Cleft(%5Cfrac%7Bf%5Cleft(-1%2Bh%5Cright)-f%5Cleft(-1%5Cright)%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E%2B%7D%5Cleft(%5Cfrac%7B%5Cleft(%5Cleft(-1%2Bh%5Cright)%5Cright)%5E2%2B3%5Ccdot%5Cleft(-1%2Bh%5Cright)%2B1-%5Cleft(-1-3%2B1%5Cright)%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E%2B%7D%5Cleft(%5Cfrac%7B%5Cleft(h%5E2-2h%2B1%5Cright)%2B3h-3%2B1%2B1%2B3-1%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E%2B%7D%5Cleft(%5Cfrac%7Bh%5E2-2h%2B3h-2%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E%2B%7D%5Cleft(%5Cfrac%7Bh%5E2%2Bh-2%7D%7Bh%7D%5Cright)%3D%5Clim_%7Bh%5Crightarrow0%5E%2B%7D%5Cleft(h%2B1-2%5Cright)%3D0%2B1-2%3D-1" alt="\lim_{h\rightarrow0^+}\left(\frac{f\left(-1+h\right)-f\left(-1\right)}{h}\right)=\lim_{h\rightarrow0^+}\left(\frac{\left(\left(-1+h\right)\right)^2+3\cdot\left(-1+h\right)+1-\left(-1-3+1\right)}{h}\right)=\lim_{h\rightarrow0^+}\left(\frac{\left(h^2-2h+1\right)+3h-3+1+1+3-1}{h}\right)=\lim_{h\rightarrow0^+}\left(\frac{h^2-2h+3h-2}{h}\right)=\lim_{h\rightarrow0^+}\left(\frac{h^2+h-2}{h}\right)=\lim_{h\rightarrow0^+}\left(h+1-2\right)=0+1-2=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-1%5Cne2" alt="-1\ne2"/><br/> <div>Ei ole<br/> <br/> 310<br/> <a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/3djdt/310-png#top" title="310.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/3djdt/310-png:file/photo/9964b59c838fbac38acb6ff8ae88872f71a8ea1f/310.PNG" alt="" title="310.PNG" class="inline" loading="lazy"/></a><br/> <br/> <br/> </div> </div> 2019-03-26T23:12:40+02:00 https://peda.net/id/644b80a44ed 2019-03-26T12:55:46+02:00 <div>251</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox1%7B%2C%7D15447" alt="x\approx1{,}15447"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox7.38906" alt="x\approx7.38906"/></div> <div> </div> <div>252</div> <div>C</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox1%7B%2C%7D17951" alt="x\approx1{,}17951"/></div> <div> </div> <div>254</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox2%7B%2C%7D83809" alt="x\approx2{,}83809"/></div> <br/> <div> </div> <div>257</div> <div>a)<br/> <a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2kji/257-a-png#top" title="257 a.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2kji/257-a-png:file/photo/0c6317a77d8e88b3d528cf4bb8baa133696d4c2b/257%20a.PNG" alt="" title="257 a.PNG" class="inline" loading="lazy"/></a></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D-x-3" alt="g\left(x\right)=-x-3"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-x-3" alt="x=-x-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%3D-3" alt="2x=-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B3%7D%7B2%7D" alt="x=-\frac{3}{2}"/><br/> <div> <div> </div> <div>264</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D3x%5E2-2" alt="f'\left(x\right)=3x^2-2"/></div> <div>b)</div> c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3-2x-5%3D0" alt="x^3-2x-5=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3%3D2x%2B5" alt="x^3=2x+5"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Csqrt%5B3%5D%7B2x%2B5%7D" alt="x=\sqrt[3]{2x+5}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox2%7B%2C%7D0946" alt="x\approx2{,}0946"/><br/> <br/> <a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2kji/264-b-png#top" title="264 b.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2kji/264-b-png:file/photo/568be3cb3c22199ce5883bb286a3ea4c1d24b9b8/264%20b.PNG" alt="" title="264 b.PNG" class="inline" loading="lazy"/></a></div> <div> </div> <div>265</div> <div>a) Funktio on kasvava eli sillä on korkeitaan yksi ratkaisu</div> <div><a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2kji/265-a-png#top" title="265 a.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2kji/265-a-png:file/photo/dd0b043c8117e79d744ed0cd5668ef220891a0cc/265%20a.PNG" alt="" title="265 a.PNG" class="inline" loading="lazy"/></a><br/> <a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2kji/265-a2-png2#top" title="265 a2.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2kji/265-a2-png2:file/photo/b7ec19b666784d69f4ba67e70ace1ef7b5fb46d2/265%20a2.PNG" alt="" title="265 a2.PNG" class="inline" loading="lazy"/></a><br/> Bolzanon lauseen nojalla voidaan oleta, että välillä ]0,2[ on ainakin yksi nollakohta.<br/> b)<br/> Neljäs<br/> <a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2kji/265-b-png#top" title="265 b.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2kji/265-b-png:file/photo/b9d8304f6a7972bfd788d4e5f2f292bf5402eb3a/265%20b.PNG" alt="" title="265 b.PNG" class="inline" loading="lazy"/></a><br/> <br/> <div> <div>267</div> <div>a)</div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox3%7B%2C%7D146" alt="x\approx3{,}146"/></div> <span>b)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox1%7B%2C%7D343467" alt="x\approx1{,}343467"/></div> <div> </div> <div> <div>272</div> <div>yhdistetään funktiot, saadan funktio h(x)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%5Cleft(x%5Cright)%3D3%5Ccos%20x-1%2Bx" alt="h\left(x\right)=3\cos x-1+x"/></div> <div>Kuvan kautta voidaan ottaa 3 alkuarvausta, ja niiden avulla voidaan laskea nollakohtien x-koordnaatit </div> <div>Kun x-koordinaatit on laskettu, voidaan niitä sijoita funktioon f(x) tai g(x), saadaan niiden y-koortit</div> <div>Leikkauspisteet ovat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox-0%7B%2C%7D8894704%7B%2C%7D%5C%20y%3D1.8894704" alt="x\approx-0{,}8894704{,}\ y=1.8894704"/><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(-0%7B%2C%7D89%3B1%7B%2C%7D89%5Cright)" alt="\left(-0{,}89;1{,}89\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox1%7B%2C%7D8623649%7B%2C%7D%5C%20y%3D%E2%88%920.8623649" alt="x\approx1{,}8623649{,}\ y=−0.8623649"/><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(1%7B%2C%7D86%3B-0%7B%2C%7D86%5Cright)" alt="\left(1{,}86;-0{,}86\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox3%7B%2C%7D6379580%7B%2C%7D%5C%20y%3D%E2%88%922.637958" alt="x\approx3{,}6379580{,}\ y=−2.637958"/><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3%7B%2C%7D64%3B-2%7B%2C%7D64%5Cright)" alt="\left(3{,}64;-2{,}64\right)"/></div> </div> 2019-03-25T11:07:18+02:00 https://peda.net/id/609c34fc4a6 2019-03-22T01:48:02+02:00 <div>229<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox-0%7B%2C%7D70347" alt="x\approx-0{,}70347"/><br/> <br/> 230<br/> <a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2nm/230-png#top" title="230.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2nm/230-png:file/photo/c73cdcc04a202b47b8909eea1dc8fc5afaeba90c/230.PNG" alt="" title="230.PNG" class="inline" loading="lazy"/></a><br/> <div><br/> 231<br/> x=1.109956 or x=3.35302 or x=5.50089<br/> <br/> 232<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox2%7B%2C%7D0" alt="x\approx2{,}0"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox2%7B%2C%7D0" alt="x\approx2{,}0"/><br/> c)<br/> ei mikään<br/> <br/> 233</div> <div>a)</div> <div>Annettu alkuarvo ei toimi, otetaan uudeksi alkuarvoksi luku 3</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox2%7B%2C%7D76929" alt="x\approx2{,}76929"/></div> <div>b)</div> <div>Annettu alkuarvo ei toimi, otetaan uudeksi alkuarvoksi luku </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox4%7B%2C%7D02752" alt="x\approx4{,}02752"/><br/> <br/> </div> 234<br/> <span>Funktio on kasvava eli sillä on korkeitaan yksi ratkaisu</span> <div>Lasketaan funktion ratkaisut välin pätepisteessä 0 ja 1</div> <div>jos tuloksien merkit vaihtuu, funktio noudataa Bolzanon lausetta, tällöin funktiolla on nollakohta avoimella välillä ]0,1[</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(0%5Cright)%3D-2" alt="f\left(0\right)=-2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)%3D1" alt="f\left(1\right)=1"/><br/> <div>Kun alkuarvaus1, tulos on 0,83512, ei muita ratkaisuja.<br/> <br/> </div> 235<br/> a) 3<br/> <a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2nm/235-a-png#top" title="235 a.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2nm/235-a-png:file/photo/b0bc4eb103569e4fb6319baa291bf23ee20a6876/235%20a.PNG" alt="" title="235 a.PNG" class="inline" loading="lazy"/></a><br/> b) 5<br/> <a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2nm/235-b-png#top" title="235 b.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2nm/235-b-png:file/photo/cdc1556bf9bed16c51f5b303a479b3ddeb9dc131/235%20b.PNG" alt="" title="235 b.PNG" class="inline" loading="lazy"/></a><br/> <br/> 236<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Capprox-0%7B%2C%7D4425" alt="\approx-0{,}4425"/><br/> <a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2nm/236-png#top" title="236.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2nm/236-png:file/photo/dca4e25b5a84be6dddc97fd4c6032c5f5bfd99e2/236.PNG" alt="" title="236.PNG" class="inline" loading="lazy"/></a><br/> <br/> 237</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cleft(x%5Cright)%2B2x%3D3" alt="\sin\left(x\right)+2x=3"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cleft(x%5Cright)%2B2x-3%3D0" alt="\sin\left(x\right)+2x-3=0"/><br/> <div> </div> <div>Merkitään</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Csin%20x%2B2x-3" alt="f\left(x\right)=\sin x+2x-3"/></div> <span>Tutkitaan onko funktiolla f enemmän kuin yksi nollakohta</span> <div>Funktio f on jatkuva kaikkialla</div> <div>Tutkitaan funktion f kulkua deravaatan avulla</div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Ccos%20x%2B2" alt="f'\left(x\right)=\cos x+2"/></div> <div> </div> <span>Tiedetään, että </span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-1%5Cle%5Ccos%20x%5Cle1%5C%20%5Cparallel%2B2" alt="-1\le\cos x\le1\ \parallel+2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=1%5Cle%5Ccos%20x%2B2%5Cle3" alt="1\le\cos x+2\le3"/><br/> <div> </div> <div>Derivaattafunktio <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%5Cge1" alt="f'\left(x\right)\ge1"/>, jolloin funktio f on kasvava. Siten funktiolla f voi korkeintaan olla yksi nollakohta</div> <div>Newtonin menetelmällä saatava likiarvo on siten funktion f ainoan nollakohdan likiarvo.</div> <div>Newtonin menetelmän rekursiokaava</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x_%7Bn%2B1%7D%3Dx_n-%5Cfrac%7Bf%5Cleft(x_n%5Cright)%7D%7Bf%27%5Cleft(x_n%5Cright)%7D" alt="x_{n+1}=x_n-\frac{f\left(x_n\right)}{f'\left(x_n\right)}"/></div> <br/> <div>Käytetään alkuarvoa</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x_1%3D1" alt="x_1=1"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x_2%3Dx_1-%5Cfrac%7Bf%5Cleft(x_1%5Cright)%7D%7Bf%27%5Cleft(x_1%5Cright)%7D%3D1-%5Cfrac%7Bf%5Cleft(1%5Cright)%7D%7Bf%27%5Cleft(1%5Cright)%7D%3D1-%5Cfrac%7B%5Csin1%2B2%5Ccdot1-3%7D%7B%5Ccos1%2B2%7D%3D1%7B%2C%7D062405" alt="x_2=x_1-\frac{f\left(x_1\right)}{f'\left(x_1\right)}=1-\frac{f\left(1\right)}{f'\left(1\right)}=1-\frac{\sin1+2\cdot1-3}{\cos1+2}=1{,}062405"/><br/> <div> </div> <div>Lasketaan nollakohdalle likiarvot taulukkolaskentaohjelmalla</div> <div> </div> <div>Nollakohta 5 desimaalin tarkkuudella on <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox1%7B%2C%7D06307" alt="x\approx1{,}06307"/></div> <div> </div> <div>Tarkistetaan, että näin on</div> <div>Välillä ]1,063065;1,063075[ olevat luvut pyöristyvät 5 desimaalin tarkkuudella luvuksi 1,06307</div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%7B%2C%7D063065%5Cright)%3C0" alt="f\left(1{,}063065\right)&lt;0"/> ja</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%7B%2C%7D063075%5Cright)%3E0" alt="f\left(1{,}063075\right)&gt;0"/><!--filtered attribute: style="max-width: 100%; border: none;"--></div> <div> </div> <span>Bolzanon lauseen nojalla funktiolla on nollakohta kysesellä välillä ja se on </span><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox1%7B%2C%7D06307" alt="x\approx1{,}06307"/><br/> <br/> 238<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x_0%3D2" alt="x_0=2"/> <span>on funktion </span><img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E3-4x%5E2-8x%2B1" alt="2x^3-4x^2-8x+1"/><span> derivaatan nollakohta</span><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0%7B%2C%7D1184" alt="x=0{,}1184"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D3%7B%2C%7D2009" alt="x=3{,}2009"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-1%7B%2C%7D3193" alt="x=-1{,}3193"/><br/> <br/> 239<br/> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex%2Bex" alt="e^x+ex"/><br/> yksi nollakohta</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox-0%7B%2C%7D278465" alt="x\approx-0{,}278465"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%20x-1" alt="\ln x-1"/><br/> yksi nollakohta</div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox2%7B%2C%7D718281" alt="x\approx2{,}718281"/></div> <br/> 241<br/> <div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox0%7B%2C%7D1612058" alt="x\approx0{,}1612058"/></div> </div> </div> <a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2nm/240-1-png#top" title="240 1.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2nm/240-1-png:file/photo/d0c4ef55807f583ecd4fb09ce524f76dd10cd926/240%201.PNG" alt="" title="240 1.PNG" class="inline" loading="lazy"/></a><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox1%7B%2C%7D1418905" alt="x\approx1{,}1418905"/><br/> <a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2nm/240-2-png#top" title="240 2.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2nm/240-2-png:file/photo/da6c245bc80e4aca5fabde13b682b9ffd2027509/240%202.PNG" alt="" title="240 2.PNG" class="inline" loading="lazy"/></a><br/> <br/> 242 2019-03-19T19:49:43+02:00 https://peda.net/id/3f0fc37c49b 2019-03-19T13:49:28+02:00 <span>201</span> <div>1,8; 1,9; 2,0</div> <div> </div> <div>202</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Capprox2%7B%2C%7D4" alt="\approx2{,}4"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Capprox2%7B%2C%7D6" alt="\approx2{,}6"/><br/> <br/> <br/> 203<br/> <div>a)</div> <div>Lasketaan funktion derivaatta nollakohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D3x%5E2%2B3" alt="f'\left(x\right)=3x^2+3"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E2%2B3%3D0" alt="3x^2+3=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E2%3D-3" alt="3x^2=-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D-1" alt="x^2=-1"/> <div> Ei ratkaisua  <br/> <div>Tämä tarkoittaa sitä että funktiolla ei ole missään paikassa maksimipistettä</div> <div>Lasketaan funktion nollakohdat</div> <div><a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2hjrl/203-a-png#top" title="203 a.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2hjrl/203-a-png:file/photo/8e452d4e9a5cb6b8c37ec2e2d0f3df8c76ab5203/203%20a.PNG" alt="" title="203 a.PNG" class="inline" loading="lazy"/></a><br/> <div> <div>Bolaznon lauseen nojalla voidaan testaa testipisteiden avulla, että joko funktio on kasvava tai laskeva<br/> <a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2hjrl/203-a2-png#top" title="203 a2.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2hjrl/203-a2-png:file/photo/6d6fc72b83477740ca1b14f57a8123e0f723389a/203%20a2.PNG" alt="" title="203 a2.PNG" class="inline" loading="lazy"/></a></div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26%26-0%7B%2C%7D817...%26%5C%5C%0A%5Chline%0Af%5Cleft(x%5Cright)%26-%26%26%2B%5C%5C%0A%26%26%5C%20%5C%20%5C%20%5C%20%5Cnearrow%26%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;&amp;-0{,}817...&amp;\\&#10;\hline&#10;f\left(x\right)&amp;-&amp;&amp;+\\&#10;&amp;&amp;\ \ \ \ \nearrow&amp;&#10;\end{array}"/></div> <div>Näin voidaan oleta, että funktio on kasvava</div> <div>b)</div> <div>Laskettiin entisessä vaiheessa funktion nollakota</div> <div>eli -1&lt;-0,817732&lt;0</div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Capprox0%7B%2C%7D82" alt="\approx0{,}82"/></div> <a href="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2hjrl/203-c-png2#top" title="203 c.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma12s/teht%C3%A4v%C3%A4t/2hjrl/203-c-png2:file/photo/239c210faca4423aba0dc69793a82e1f815e26d1/203%20c.PNG" alt="" title="203 c.PNG" class="inline" loading="lazy"/></a><br/> <br/> 204<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox0%7B%2C%7D16" alt="x\approx0{,}16"/><br/> <br/> 205<br/> <br/> <div> </div> <div> </div> </div> </div> </div> </div> 2019-03-18T21:01:33+02:00 https://peda.net/id/0aecd03044b 2019-03-18T10:10:18+02:00 <div> <div>162</div> <div>Koska f(4)=0, niin olynomi on jaollinen binomilla x-4<br/> Lasketaa jakolasku <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bf%5Cleft(x%5Cright)%7D%7Bx-4%7D" alt="\frac{f\left(x\right)}{x-4}"/> jakoalgoritmilla</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%263x%5E2%2Bx-2%5C%5C%0A%5Chline%0Ax-4%263x%5E3-11x%5E2-6x%2B8%5C%5C%0A-%26%5Cleft(3x%5E3-12x%5E2%5Cright)%5C%5C%0A%26--------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20x%5E2-6x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(x%5E2-4x%5Cright)%5C%5C%0A%26----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-2x%2B8%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-2x%2B8%5Cright)%5C%5C%0A%26------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;3x^2+x-2\\&#10;\hline&#10;x-4&amp;3x^3-11x^2-6x+8\\&#10;-&amp;\left(3x^3-12x^2\right)\\&#10;&amp;--------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ x^2-6x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \left(x^2-4x\right)\\&#10;&amp;----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -2x+8\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(-2x+8\right)\\&#10;&amp;------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/> <div>Lasketaan nollakohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E2%2Bx-2%3D0" alt="3x^2+x-2=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-1%5Cpm%5Csqrt%5B%5D%7B1%5E2-4%5Ccdot3%5Ccdot%5Cleft(-2%5Cright)%7D%7D%7B2%5Ccdot3%7D%3D%5Cfrac%7B-1%5Cpm%5Csqrt%5B%5D%7B25%7D%7D%7B6%7D%3D%5Cfrac%7B-1%5Cpm5%7D%7B6%7D" alt="x=\frac{-1\pm\sqrt[]{1^2-4\cdot3\cdot\left(-2\right)}}{2\cdot3}=\frac{-1\pm\sqrt[]{25}}{6}=\frac{-1\pm5}{6}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-1%2B5%7D%7B6%7D%3D%5Cfrac%7B4%7D%7B6%7D%3D%5Cfrac%7B2%7D%7B3%7D" alt="x=\frac{-1+5}{6}=\frac{4}{6}=\frac{2}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=tai" alt="tai"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-1-5%7D%7B6%7D%3D-1" alt="x=\frac{-1-5}{6}=-1"/><br/> <br/> <div> <div>163</div> <div>Koska f(3)=0, niin olynomi on jaollinen binomilla x-3</div> <div>Lasketaa jakolasku <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bf%5Cleft(x%5Cright)%7D%7Bx-3%7D" alt="\frac{f\left(x\right)}{x-3}"/> jakoalgoritmilla </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26-x%5E2-x%2B2%5C%5C%0A%5Chline%0Ax-3%26-x%5E3%2B2x%5E2%2B5x-6%5C%5C%0A-%26%5Cleft(-x%5E3%2B3x%5E2%5Cright)%5C%5C%0A%26-------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-x%5E2%2B5x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-x%5E2%2B3x%5Cright)%5C%5C%0A%26----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%202x-6%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(2x-6%5Cright)%5C%5C%0A%26------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;-x^2-x+2\\&#10;\hline&#10;x-3&amp;-x^3+2x^2+5x-6\\&#10;-&amp;\left(-x^3+3x^2\right)\\&#10;&amp;-------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ -x^2+5x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \left(-x^2+3x\right)\\&#10;&amp;----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x-6\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(2x-6\right)\\&#10;&amp;------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <span>Lasketaan nollakohdat</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-x%5E2-x%2B2%3D0" alt="-x^2-x+2=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1%5Cpm%5Csqrt%5B%5D%7B%5Cleft(-1%5Cright)%5E2-4%5Ccdot%5Cleft(-1%5Cright)%5Ccdot2%7D%7D%7B2%5Ccdot%5Cleft(-1%5Cright)%7D%3D%5Cfrac%7B1%5Cpm%5Csqrt%5B%5D%7B9%7D%7D%7B-2%7D%3D%5Cfrac%7B1%5Cpm3%7D%7B-2%7D" alt="x=\frac{1\pm\sqrt[]{\left(-1\right)^2-4\cdot\left(-1\right)\cdot2}}{2\cdot\left(-1\right)}=\frac{1\pm\sqrt[]{9}}{-2}=\frac{1\pm3}{-2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1%2B3%7D%7B-2%7D%3D%5Cfrac%7B4%7D%7B-2%7D%3D-2" alt="x=\frac{1+3}{-2}=\frac{4}{-2}=-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=tai" alt="tai"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1-3%7D%7B-2%7D%3D%5Cfrac%7B-2%7D%7B-2%7D%3D1" alt="x=\frac{1-3}{-2}=\frac{-2}{-2}=1"/><br/> <br/> <b>164</b><br/> a) kolme<br/> b)  <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cpm1%7B%2C%7D%5Cpm2%7B%2C%7D%5Cpm4%7B%2C%7D%5Cpm8" alt="\pm1{,}\pm2{,}\pm4{,}\pm8"/></div> <span>c)</span><br/> <br/> <br/> </div> </div> 171<br/> a) <div><img src="https://math-demo.abitti.fi/math.svg?latex=6x%5E3-13x%5E2%2B4%3D0" alt="6x^3-13x^2+4=0"/></div> <div> Nimitään <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D6x%5E3-13x%5E2%2B4%3D0" alt="P\left(x\right)=6x^3-13x^2+4=0"/><br/> Etsitään yhtälölle P(x)=0 jokin kokonaislukuratkaisu. Jos yhtälöllä on kokonaislukuratkaisuja, ne löydetään sijoittamalla yhtälöön vakiotermin -6 tekijöitä ±1 ±2 ±4</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(1%5Cright)%3D6%5Ccdot1%5E3-13%5Ccdot1%5E2%2B4%3D6-13%2B4%3D-3" alt="P\left(1\right)=6\cdot1^3-13\cdot1^2+4=6-13+4=-3"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(-1%5Cright)%3D6%5Ccdot%5Cleft(-1%5Cright)%5E3-13%5Ccdot%5Cleft(-1%5Cright)%5E2%2B4%3D-6-13%2B4%3D-15" alt="P\left(-1\right)=6\cdot\left(-1\right)^3-13\cdot\left(-1\right)^2+4=-6-13+4=-15"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(2%5Cright)%3D6%5Ccdot2%5E3-13%5Ccdot2%5E2%2B4%3D48-52%2B4%3D0" alt="P\left(2\right)=6\cdot2^3-13\cdot2^2+4=48-52+4=0"/><br/> <div>Koska P(2)=0, niin olynomi on jaollinen binomilla x-2<br/> Lasketaa jakolasku <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7BP%5Cleft(x%5Cright)%7D%7Bx-2%7D" alt="\frac{P\left(x\right)}{x-2}"/> jakologaritmilla</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%266x%5E2%2Bx-2%5C%5C%0A%5Chline%0Ax-2%266x%5E3-13x%5E2%2B0x%2B4%5C%5C%0A-%26%5Cleft(6x%5E3-12x%5E2%5Cright)%5C%5C%0A%26--------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20x%5E2%2B0x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(x%5E2-2x%5Cright)%5C%5C%0A%26----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-2x%2B4%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-2x%2B4%5Cright)%5C%5C%0A%26------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;6x^2+x-2\\&#10;\hline&#10;x-2&amp;6x^3-13x^2+0x+4\\&#10;-&amp;\left(6x^3-12x^2\right)\\&#10;&amp;--------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^2+0x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \left(x^2-2x\right)\\&#10;&amp;----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -2x+4\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(-2x+4\right)\\&#10;&amp;------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D%5Cleft(x-2%5Cright)%5Cleft(6x%5E2%2Bx-2%5Cright)" alt="P\left(x\right)=\left(x-2\right)\left(6x^2+x-2\right)"/><br/> Nimitään <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D6x%5E2%2Bx-2" alt="s\left(x\right)=6x^2+x-2"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%3D6%7B%2C%7D%5C%20b%3D1%7B%2C%7D%5C%20x%3D-2" alt="a=6{,}\ b=1{,}\ x=-2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Ccdot%20v%3D-12%5C%20ja%5C%20u%2Bv%3D1" alt="u\cdot v=-12\ ja\ u+v=1"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%3D-3%5C%20ja%5C%20v%3D4" alt="u=-3\ ja\ v=4"/></div> <div>Jaetaan polynomi g(x) tekijöihin ryhmittelemällä.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D%5Cleft(6x%5E2-3x%5Cright)%2B%5Cleft(4x-2%5Cright)%3D3x%5Cleft(2x-1%5Cright)%2B2%5Cleft(2x-1%5Cright)%3D%5Cleft(3x%2B2%5Cright)%5Cleft(2x-1%5Cright)%3D0" alt="s\left(x\right)=\left(6x^2-3x\right)+\left(4x-2\right)=3x\left(2x-1\right)+2\left(2x-1\right)=\left(3x+2\right)\left(2x-1\right)=0"/><br/> <div>Ratkaistaan yhtälö P(x)=0 tulon nollasäännöllä<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D%5Cleft(3x%2B2%5Cright)%5Cleft(2x-1%5Cright)" alt="P\left(x\right)=\left(3x+2\right)\left(2x-1\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%2B2%3D0" alt="3x+2=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D-2" alt="3x=-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B2%7D%7B3%7D" alt="x=-\frac{2}{3}"/> <div>tai<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x-1%3D0" alt="2x-1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%3D1" alt="2x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1%7D%7B2%7D" alt="x=\frac{1}{2}"/></div> <div>V: <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1%7D%7B2%7D" alt="x=\frac{1}{2}"/>,<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B2%7D%7B3%7D" alt="x=-\frac{2}{3}"/>.<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D2" alt="x=2"/></div> </div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E5-2x%5E4-x%5E3-3x%5E2-6x%3D0" alt="2x^5-2x^4-x^3-3x^2-6x=0"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cleft(2x%5E4-2x%5E3-x%5E2-3x-6%5Cright)%3D0" alt="x\left(2x^4-2x^3-x^2-3x-6\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0" alt="x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=tai" alt="tai"/> <div>Nimitään<span> </span><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D2x%5E4-2x%5E3-x%5E2-3x-6" alt="P\left(x\right)=2x^4-2x^3-x^2-3x-6"/></div> <div>Etsitään yhtälölle P(x)=0 jokin kokonaislukuratkaisu. Jos yhtälöllä on kokonaislukuratkaisuja, ne löydetään sijoittamalla yhtälöön vakiotermin -6 tekijöitä ±1 ±2 ±3 ±6<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(1%5Cright)%3D2%5Ccdot1%5E4-2%5Ccdot1%5E3-1%5E2-3%5Ccdot1-6%3D-10" alt="P\left(1\right)=2\cdot1^4-2\cdot1^3-1^2-3\cdot1-6=-10"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(-1%5Cright)%3D2%2B2-1%2B3-6%3D0" alt="P\left(-1\right)=2+2-1+3-6=0"/><br/> <div>Koska P(-1)=0, niin olynomi on jaollinen binomilla x+1</div> <div>Lasketaa jakolasku <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7BP%5Cleft(x%5Cright)%7D%7Bx%2B1%7D" alt="\frac{P\left(x\right)}{x+1}"/> jakologaritmilla</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%262x%5E3-4x%5E2%2B3x-6%5C%5C%0A%5Chline%0Ax%2B1%262x%5E4-2x%5E3-x%5E2-3x-6%5C%5C%0A-%26%5Cleft(2x%5E4%2B2x%5E3%5Cright)%5C%5C%0A%26-------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-4x%5E3-x%5E2%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-4x%5E3-4x%5E2%5Cright)%5C%5C%0A%26------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%203x%5E2-3x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(3x%5E2%2B3x%5Cright)%5C%5C%0A%26---------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-6x-6%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-6x-6%5Cright)%5C%5C%0A%26-----------------%5C%5C%0A%5C%20%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;2x^3-4x^2+3x-6\\&#10;\hline&#10;x+1&amp;2x^4-2x^3-x^2-3x-6\\&#10;-&amp;\left(2x^4+2x^3\right)\\&#10;&amp;-------\\&#10;&amp;\ \ \ \ \ \ \ \ \ -4x^3-x^2\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \left(-4x^3-4x^2\right)\\&#10;&amp;------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3x^2-3x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(3x^2+3x\right)\\&#10;&amp;---------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -6x-6\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(-6x-6\right)\\&#10;&amp;-----------------\\&#10;\ &amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D%5Cleft(x%2B1%5Cright)%5Cleft(2x%5E3-4x%5E2%2B3x-6%5Cright)" alt="P\left(x\right)=\left(x+1\right)\left(2x^3-4x^2+3x-6\right)"/></div> <div><span> Nimitään </span><img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D%5Cleft(2x%5E3-4x%5E2%5Cright)%2B%5Cleft(3x-6%5Cright)" alt="s\left(x\right)=\left(2x^3-4x^2\right)+\left(3x-6\right)"/><br/> <span>Jaetaan polynomi g(x) tekijöihin ryhmittelemällä.</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D2x%5E2%5Cleft(x-2%5Cright)%2B3%5Cleft(x-2%5Cright)%3D%5Cleft(x-2%5Cright)%5Cleft(2x%5E2%2B3%5Cright)" alt="s\left(x\right)=2x^2\left(x-2\right)+3\left(x-2\right)=\left(x-2\right)\left(2x^2+3\right)"/></div> <span>Ratkaistaan yhtälö P(x)=0 tulon nollasäännöllä</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D%5Cleft(x%2B1%5Cright)%5Cleft(x-2%5Cright)%5Cleft(2x%5E2%2B3%5Cright)" alt="P\left(x\right)=\left(x+1\right)\left(x-2\right)\left(2x^2+3\right)"/></div> <br/> <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/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-2%3D0" alt="x-2=0"/><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=2x%5E2%2B3%3D0" alt="2x^2+3=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%3D-3" alt="2x^2=-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D-%5Cfrac%7B3%7D%7B2%7D" alt="x^2=-\frac{3}{2}"/><br/> <div>Ei ratkaisua</div> <div> </div> <div>V: x=0, x=-1, x=2</div> <div><span> </span></div> 2019-03-12T14:04:52+02:00 https://peda.net/id/22842e4843e 2019-03-18T11:20:47+02:00 <span>141</span> <div>a)</div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%263x%5E2%2B3x%2B1%5C%5C%0A%5Chline%0Ax%2B5%263x%5E3%2B18x%5E2%2B16x%2B5%5C%5C%0A-%26%5Cleft(3x%5E3%2B15x%5E2%5Cright)%5C%5C%0A%26--------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%203x%5E2%2B16x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(3x%5E2%2B15x%5Cright)%5C%5C%0A%26-----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20x%2B5%5C%5C%0A-%5C%20%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(x%2B5%5Cright)%5C%20%5C%5C%0A%26-------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%5C%5C%0A%26%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;3x^2+3x+1\\&#10;\hline&#10;x+5&amp;3x^3+18x^2+16x+5\\&#10;-&amp;\left(3x^3+15x^2\right)\\&#10;&amp;--------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ 3x^2+16x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \left(3x^2+15x\right)\\&#10;&amp;-----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x+5\\&#10;-\ &amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(x+5\right)\ \\&#10;&amp;-------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\\&#10;&amp;&#10;\end{array}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26x%5E2-2x%2B1%5C%5C%0A%5Chline%0Ax-2%26x%5E3%2B0x%5E2-3x-2%5C%5C%0A-%26%5Cleft(x%5E3-2x%5E2%5Cright)%5C%5C%0A%26-------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-2x%5E2-3x%5C%5C%0A-%5C%20%5C%20%5C%20%5C%20%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-2x%5E2%2B4x%5Cright)%5C%5C%0A%26----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20x-2%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(x-2%5Cright)%5C%5C%0A%26------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;x^2-2x+1\\&#10;\hline&#10;x-2&amp;x^3+0x^2-3x-2\\&#10;-&amp;\left(x^3-2x^2\right)\\&#10;&amp;-------\\&#10;&amp;\ \ \ \ \ \ -2x^2-3x\\&#10;-\ \ \ \ &amp;\ \ \ \ \ \ \left(-2x^2+4x\right)\\&#10;&amp;----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x-2\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(x-2\right)\\&#10;&amp;------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26x%5E2-x%2B2%5C%5C%0A%5Chline%0A3x%5E2-2x%263x%5E4-5x%5E3%2B8x%5E2-4x%2B2%5C%5C%0A-%26%5Cleft(3x%5E4-2x%5E3%5Cright)%5C%5C%0A%26-------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-3x%5E3%2B8x%5E2%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-3x%5E3%2B2x%5E2%5Cright)%5C%5C%0A%26-----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%206x%5E2-4x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(6x%5E2-4x%5Cright)%5C%5C%0A%26--------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%202%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;x^2-x+2\\&#10;\hline&#10;3x^2-2x&amp;3x^4-5x^3+8x^2-4x+2\\&#10;-&amp;\left(3x^4-2x^3\right)\\&#10;&amp;-------\\&#10;&amp;\ \ \ \ \ \ \ \ -3x^3+8x^2\\&#10;-&amp;\ \ \ \ \ \ \ \left(-3x^3+2x^2\right)\\&#10;&amp;-----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6x^2-4x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(6x^2-4x\right)\\&#10;&amp;--------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2&#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-x%2B2%2B%5Cfrac%7B2%7D%7B3x%5E2-2x%7D" alt="x^2-x+2+\frac{2}{3x^2-2x}"/></div> </div> <div>142</div> <div>I, II, IV</div> <div> </div> <div>143</div> <div>A II</div> <div>B I</div> <div>C Ei mikään</div> <div>D III</div> <div><br/> <div>144</div> </div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(3%5Cright)%3D-2%5Ccdot3%5E3%2B5%5Ccdot3%5E2%2B6%5Ccdot3-9%3D-54%2B45%2B18-9%3D-6%5Ccdot9%2B5%5Ccdot9%2B2%5Ccdot9-1%5Ccdot9%3D9%5Cleft(-6%2B5%2B2-1%5Cright)%3D0" alt="P\left(3\right)=-2\cdot3^3+5\cdot3^2+6\cdot3-9=-54+45+18-9=-6\cdot9+5\cdot9+2\cdot9-1\cdot9=9\left(-6+5+2-1\right)=0"/></div> <span>On jaollinen</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26-2x%5E2-x%2B3%5C%5C%0A%5Chline%0Ax-3%26-2x%5E3%2B5x%5E2%2B6x-9%5C%5C%0A-%26%5Cleft(-2x%5E3%2B6x%5E2%5Cright)%5C%5C%0A%26--------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-x%5E2%2B6x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-x%5E2%2B3x%5Cright)%5C%5C%0A%26----------%5C%5C%0A%5C%20%5C%20%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%203x-9%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(3x-9%5Cright)%5C%5C%0A%26-------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;-2x^2-x+3\\&#10;\hline&#10;x-3&amp;-2x^3+5x^2+6x-9\\&#10;-&amp;\left(-2x^3+6x^2\right)\\&#10;&amp;--------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ -x^2+6x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \left(-x^2+3x\right)\\&#10;&amp;----------\\&#10;\ \ &amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3x-9\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(3x-9\right)\\&#10;&amp;-------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D%5Cleft(x-3%5Cright)%5Cleft(-2x%5E2-x%2B3%5Cright)" alt="P\left(x\right)=\left(x-3\right)\left(-2x^2-x+3\right)"/></div> <div> </div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26x-3%5C%5C%0A%5Chline%0A-2x%5E2-x%2B3%26-2x%5E3%2B5x%5E2%2B6x-9%5C%5C%0A-%26%5Cleft(-2x%5E3-x%5E2%2B3x%5Cright)%5C%5C%0A%26----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%206x%5E2%2B3x-9%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(6x%5E2%2B3x-9%5Cright)%5C%20%5C%20%5C%5C%0A%26------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;x-3\\&#10;\hline&#10;-2x^2-x+3&amp;-2x^3+5x^2+6x-9\\&#10;-&amp;\left(-2x^3-x^2+3x\right)\\&#10;&amp;----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ 6x^2+3x-9\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \left(6x^2+3x-9\right)\ \ \\&#10;&amp;------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D%5Cleft(x-3%5Cright)%5Cleft(-2x%5E2-x%2B3%5Cright)" alt="P\left(x\right)=\left(x-3\right)\left(-2x^2-x+3\right)"/></div> <div>  <div>145</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26x%5E2-x%2B2%5C%5C%0A%5Chline%0A-3x%2B2%26-3x%5E3%2B5x%5E2-8x%2B4%5C%5C%0A-%26%5Cleft(-3x%5E3%2B2x%5E2%5Cright)%5C%5C%0A%26--------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%203x%5E2-8x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(3x%5E2-2x%5Cright)%5C%5C%0A%26-----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-6x%2B4%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-6x%2B4%5Cright)%5C%20%5C%5C%0A%26-------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;x^2-x+2\\&#10;\hline&#10;-3x+2&amp;-3x^3+5x^2-8x+4\\&#10;-&amp;\left(-3x^3+2x^2\right)\\&#10;&amp;--------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3x^2-8x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(3x^2-2x\right)\\&#10;&amp;-----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -6x+4\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(-6x+4\right)\ \\&#10;&amp;-------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-3x%5E3%2B5x%5E2-8x%2B4%3D%5Cleft(-3x%2B2%5Cright)%5Cleft(x%5E2-x%2B2%5Cright)" alt="-3x^3+5x^2-8x+4=\left(-3x+2\right)\left(x^2-x+2\right)"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26x%5E2-%5Cfrac%7B5%7D%7B3%7Dx%2B%5Cfrac%7B8%7D%7B3%7D%5C%5C%0A%5Chline%0A-3x%26-3x%5E3%2B5x%5E2-8x%2B4%5C%5C%0A-%26-3x%5E3%5C%5C%0A%26-----%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%205x%5E2%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%205x%5E2%5C%5C%0A%26-------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-8x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-8x%5C%5C%0A%26----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%204%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;x^2-\frac{5}{3}x+\frac{8}{3}\\&#10;\hline&#10;-3x&amp;-3x^3+5x^2-8x+4\\&#10;-&amp;-3x^3\\&#10;&amp;-----\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ 5x^2\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ 5x^2\\&#10;&amp;-------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -8x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -8x\\&#10;&amp;----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4&#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-3x%5E3%2B5x%5E2-8x%2B4%3D%5Cleft(-3x%5Cright)%5Cleft(x%5E2-%5Cfrac%7B5%7D%7B3%7Dx%2B%5Cfrac%7B8%7D%7B3%7D%5Cright)%2B4" alt="-3x^3+5x^2-8x+4=\left(-3x\right)\left(x^2-\frac{5}{3}x+\frac{8}{3}\right)+4"/><br/> <br/> </div> </div> <div>146<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26-x%5E2%2B2x%2B3%5C%5C%0A%5Chline%0A2x%2B3%26-2x%5E3%2Bx%5E2%2B12x%2B9%5C%5C%0A-%26%5Cleft(-2x%5E3-3x%5E2%5Cright)%5C%5C%0A%26--------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%204x%5E2%2B12x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(4x%5E2%2B6x%5Cright)%5C%5C%0A%26-----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%206x%2B9%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(6x%2B9%5Cright)%5C%5C%0A%26-------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;-x^2+2x+3\\&#10;\hline&#10;2x+3&amp;-2x^3+x^2+12x+9\\&#10;-&amp;\left(-2x^3-3x^2\right)\\&#10;&amp;--------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4x^2+12x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \left(4x^2+6x\right)\\&#10;&amp;-----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6x+9\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(6x+9\right)\\&#10;&amp;-------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=V%3A%5C%20-x%5E2%2B2x%2B3" alt="V:\ -x^2+2x+3"/></div> <br/> 147<br/> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D2x%5E3-5x%5E2%2B3x-2" alt="P\left(x\right)=2x^3-5x^2+3x-2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(-1%5Cright)%3D2%5Ccdot%5Cleft(-1%5Cright)%5E3-5%5Ccdot%5Cleft(-1%5Cright)%5E2%2B3%5Ccdot%5Cleft(-1%5Cright)-2%3D-2-5-3-2%3D-12" alt="P\left(-1\right)=2\cdot\left(-1\right)^3-5\cdot\left(-1\right)^2+3\cdot\left(-1\right)-2=-2-5-3-2=-12"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(2%5Cright)%3D2%5Ccdot2%5E3-5%5Ccdot2%5E2%2B3%5Ccdot2-2%3D16-20%2B6-2%3D0" alt="P\left(2\right)=2\cdot2^3-5\cdot2^2+3\cdot2-2=16-20+6-2=0"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=On%5C%20jaollinen%5C%20%5Cleft(x-2%5Cright)%3Alla" alt="On\ jaollinen\ \left(x-2\right):lla"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%262x%5E2-x%2B1%5C%5C%0A%5Chline%0Ax-2%262x%5E3-5x%5E2%2B3x-2%5C%5C%0A-%26%5Cleft(2x%5E3-4x%5E2%5Cright)%5C%5C%0A%26-------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-x%5E2%2B3x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-x%5E2%2B2x%5Cright)%5C%5C%0A%26----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20x-2%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(x-2%5Cright)%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%5C%0A%26------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;2x^2-x+1\\&#10;\hline&#10;x-2&amp;2x^3-5x^2+3x-2\\&#10;-&amp;\left(2x^3-4x^2\right)\\&#10;&amp;-------\\&#10;&amp;\ \ \ \ \ \ \ \ -x^2+3x\\&#10;-&amp;\ \ \ \ \ \ \ \left(-x^2+2x\right)\\&#10;&amp;----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x-2\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(x-2\right)\ \ \ \ \ \ \ \ \ \\&#10;&amp;------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D%5Cleft(x-2%5Cright)%5Cleft(2x%5E2-x%2B1%5Cright)" alt="P\left(x\right)=\left(x-2\right)\left(2x^2-x+1\right)"/></div> <div> </div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D6x%5E3%2B7x%5E2-1" alt="P\left(x\right)=6x^3+7x^2-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(-1%5Cright)%3D6%5Ccdot%5Cleft(-1%5Cright)%5E3%2B7%5Ccdot%5Cleft(-1%5Cright)%5E2-1%3D-6%2B7-1%3D0" alt="P\left(-1\right)=6\cdot\left(-1\right)^3+7\cdot\left(-1\right)^2-1=-6+7-1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(2%5Cright)%3D6%5Ccdot2%5E3%2B7%5Ccdot2%5E2-1%3D48%2B28-1%3D76-1%3D75" alt="P\left(2\right)=6\cdot2^3+7\cdot2^2-1=48+28-1=76-1=75"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=On%5C%20jaollinen%5C%20%5Cleft(x%2B1%5Cright)%3All%C3%A4" alt="On\ jaollinen\ \left(x+1\right):llä"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%266x%5E2%2Bx-1%5C%5C%0A%5Chline%0Ax%2B1%266x%5E3%2B7x%5E2%2B0x-1%5C%5C%0A-%26%5Cleft(6x%5E3%2B6x%5E2%5Cright)%5C%5C%0A%26-------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20x%5E2%2B0x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(x%5E2%2Bx%5Cright)%5C%5C%0A%26---------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-x-1%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-x-1%5Cright)%5C%5C%0A%26-----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;6x^2+x-1\\&#10;\hline&#10;x+1&amp;6x^3+7x^2+0x-1\\&#10;-&amp;\left(6x^3+6x^2\right)\\&#10;&amp;-------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ x^2+0x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \left(x^2+x\right)\\&#10;&amp;---------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -x-1\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(-x-1\right)\\&#10;&amp;-----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/><br/> <br/> <div>148</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D2x%5E3-x%5E2%2B4x%2Bk" alt="P\left(x\right)=2x^3-x^2+4x+k"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(-2%5Cright)%3D2%5Ccdot%5Cleft(-2%5Cright)%5E3-%5Cleft(-2%5Cright)%5E2%2B4%5Ccdot%5Cleft(-2%5Cright)%2Bk" alt="P\left(-2\right)=2\cdot\left(-2\right)^3-\left(-2\right)^2+4\cdot\left(-2\right)+k"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot%5Cleft(-2%5Cright)%5E3-%5Cleft(-2%5Cright)%5E2%2B4%5Ccdot%5Cleft(-2%5Cright)%2Bk%3D0" alt="2\cdot\left(-2\right)^3-\left(-2\right)^2+4\cdot\left(-2\right)+k=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-16-4-8%2Bk%3D0" alt="-16-4-8+k=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-28%2Bk%3D0" alt="-28+k=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D28" alt="k=28"/><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%262x%2B1%5E%7B%20%7D%5C%5C%0A%5Chline%0Ax%5E2%2B2%262x%5E3-x%5E2%2B4x%2Bk%5C%5C%0A-%26%5Cleft(2x%5E3%2B0x%5E2%2B4x%5Cright)%5C%5C%0A%26----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-x%5E2%2Bk%5C%5C%0A-%5C%20%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(x%5E2%2B2%5Cright)%5C%5C%0A%26----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;2x+1^{ }\\&#10;\hline&#10;x^2+2&amp;2x^3-x^2+4x+k\\&#10;-&amp;\left(2x^3+0x^2+4x\right)\\&#10;&amp;----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ -x^2+k\\&#10;-\ &amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(x^2+2\right)\\&#10;&amp;----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D-2" alt="k=-2"/><br/> <br/> <div>149</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26x%5E2%2B2%5C%5C%0A%5Chline%0Ax%5E2-x%26x%5E4-x%5E3%2B2x%5E2-2x%2B1%5C%5C%0A-%26%5Cleft(x%5E4-x%5E3%5Cright)%5C%5C%0A%26------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%202x%5E2-2x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(2x%5E2-2x%5Cright)%5C%5C%0A%26------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%201%5C%20%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;x^2+2\\&#10;\hline&#10;x^2-x&amp;x^4-x^3+2x^2-2x+1\\&#10;-&amp;\left(x^4-x^3\right)\\&#10;&amp;------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^2-2x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(2x^2-2x\right)\\&#10;&amp;------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\ &#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E4-x%5E3%2B2x%5E2-2x%2B1%3D%5Cleft(x%5E2-x%5Cright)%5Cleft(x%5E2%2B2%5Cright)%2B1" alt="x^4-x^3+2x^2-2x+1=\left(x^2-x\right)\left(x^2+2\right)+1"/><br/> <br/> <div>150</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%5E2%2B1%5Cright)%5Cleft(3x%5E2-1%5Cright)%2B2x%3D3x%5E4-x%5E2%2B3x%5E2-1%2B2x%3D3x%5E4%2B2x%5E2%2B2x-1" alt="\left(x^2+1\right)\left(3x^2-1\right)+2x=3x^4-x^2+3x^2-1+2x=3x^4+2x^2+2x-1"/><br/> <br/> </div> <div>151</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=Koska%5C%202%5Ccdot2-4%3D0%7B%2C%7D%5C%20jakossa%5C%20tapahtuu%5C%20sijoitamisen%5C%20j%C3%A4lkeen%5C%20nollalla%5C%20jakaamista" alt="Koska\ 2\cdot2-4=0{,}\ jakossa\ tapahtuu\ sijoitamisen\ jälkeen\ nollalla\ jakaamista"/></div> <span>Tällöin voidaan hyödyntää hopitalin sääntöä</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow2%7D%5Cleft(%5Cfrac%7Bf%27%5Cleft(x%5Cright)%7D%7Bg%27%5Cleft(x%5Cright)%7D%5Cright)%3D%5Cfrac%7B-6x%5E2-8x%2B18%7D%7B2%7D%3D%5Cfrac%7B-6%5Ccdot2%5E2-8%5Ccdot2%2B18%7D%7B2%7D%3D%5Cfrac%7B-24-16%2B18%7D%7B2%7D%3D-11" alt="\lim_{x\rightarrow2}\left(\frac{f'\left(x\right)}{g'\left(x\right)}\right)=\frac{-6x^2-8x+18}{2}=\frac{-6\cdot2^2-8\cdot2+18}{2}=\frac{-24-16+18}{2}=-11"/><br/> <br/> <div>152</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26x%5E2%2B3%5C%5C%0A%5Chline%0A3x-2%263x%5E3-2x%5E2%2Bkx-6%5C%5C%0A-%26%5Cleft(3x%5E3-2x%5E2%5Cright)%5C%5C%0A%26-------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20kx-6%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(9x-6%5Cright)%5C%5C%0A%26-----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;x^2+3\\&#10;\hline&#10;3x-2&amp;3x^3-2x^2+kx-6\\&#10;-&amp;\left(3x^3-2x^2\right)\\&#10;&amp;-------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ kx-6\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(9x-6\right)\\&#10;&amp;-----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=k%3D9" alt="k=9"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%262x-1%5C%5C%0A%5Chline%0Ax%5E2-3%262x%5E3-x%5E2-6x%2Bk%5C%5C%0A-%26%5Cleft(2x%5E3%2B0x-6x%5Cright)%5C%5C%0A%26---------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-x%5E2%2Bk%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-x%5E2%2B3%5Cright)%5C%5C%0A%26-----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;2x-1\\&#10;\hline&#10;x^2-3&amp;2x^3-x^2-6x+k\\&#10;-&amp;\left(2x^3+0x-6x\right)\\&#10;&amp;---------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -x^2+k\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(-x^2+3\right)\\&#10;&amp;-----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=k%3D3" alt="k=3"/><br/> <br/> <div>153</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-2x%5E3-x%5E2%2B4x-1%3DQ%5Cleft(x%5Cright)%5Ccdot%5Cleft(-2x%5E2-5x-11%5Cright)-34" alt="-2x^3-x^2+4x-1=Q\left(x\right)\cdot\left(-2x^2-5x-11\right)-34"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=-2x%5E3-x%5E2%2B4x%2B33%3DQ%5Cleft(x%5Cright)%5Ccdot%5Cleft(-2x%5E2-5x-11%5Cright)" alt="-2x^3-x^2+4x+33=Q\left(x\right)\cdot\left(-2x^2-5x-11\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26x-3%5C%5C%0A%5Chline%0A-2x%5E2-5x-11%26-2x%5E3%2Bx%5E2%2B4x%2B33%5C%5C%0A-%26%5Cleft(-2x%5E3-5x%5E2-11x%5Cright)%5C%5C%0A%26------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%206x%5E2%2B15x-33%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(6x%5E2%2B15x%2B33%5Cright)%5C%5C%0A%26--------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;x-3\\&#10;\hline&#10;-2x^2-5x-11&amp;-2x^3+x^2+4x+33\\&#10;-&amp;\left(-2x^3-5x^2-11x\right)\\&#10;&amp;------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ 6x^2+15x-33\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \left(6x^2+15x+33\right)\\&#10;&amp;--------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/><br/> <br/> </div> </div> <div>154</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%2612x%5E2-10x%2B2%5C%5C%0A%5Chline%0Ax%2B3%2612x%5E3%2B26x%5E2-28x%2B6%5C%5C%0A-%26%5Cleft(12x%5E3%2B36x%5E2%5Cright)%5C%5C%0A%26--------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-10x%5E2-28x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-10x%5E2-30x%5Cright)%5C%5C%0A%26------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%202x%2B6%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(2x%2B6%5Cright)%5C%5C%0A%26--------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;12x^2-10x+2\\&#10;\hline&#10;x+3&amp;12x^3+26x^2-28x+6\\&#10;-&amp;\left(12x^3+36x^2\right)\\&#10;&amp;--------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ -10x^2-28x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \left(-10x^2-30x\right)\\&#10;&amp;------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x+6\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(2x+6\right)\\&#10;&amp;--------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=12x%5E2-10x%2B2" alt="12x^2-10x+2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D12%7B%2C%7D%5C%20b%3D-10%7B%2C%7D%5C%20c%3D2" alt="a=12{,}\ b=-10{,}\ c=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Ccdot%20v%3D24%5C%20ja%5C%20u%2Bv%3D-10" alt="u\cdot v=24\ ja\ u+v=-10"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%7B%2C%7D2%7B%2C%7D3%7B%2C%7D4%7B%2C%7D6%7B%2C%7D8%7B%2C%7D12%7B%2C%7D24" alt="1{,}2{,}3{,}4{,}6{,}8{,}12{,}24"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%3D-4%5C%20ja%5C%20v%3D-6" alt="u=-4\ ja\ v=-6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(12x%5E2-4x%5Cright)%2B%5Cleft(-6x%2B2%5Cright)%3D4x%5Cleft(3x-1%5Cright)-2%5Cleft(3x-1%5Cright)%3D%5Cleft(4x-2%5Cright)%5Cleft(3x-1%5Cright)" alt="\left(12x^2-4x\right)+\left(-6x+2\right)=4x\left(3x-1\right)-2\left(3x-1\right)=\left(4x-2\right)\left(3x-1\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B4x-2%7D%7B4%7D%3Dx-%5Cfrac%7B1%7D%7B2%7D%3D%5Cfrac%7B1%7D%7B2%7D" alt="\frac{4x-2}{4}=x-\frac{1}{2}=\frac{1}{2}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3x-1%7D%7B3%7D%3Dx-%5Cfrac%7B1%7D%7B3%7D%3D%5Cfrac%7B1%7D%7B3%7D" alt="\frac{3x-1}{3}=x-\frac{1}{3}=\frac{1}{3}"/><br/> <br/> <div>155</div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%262x%5E3-5x%5E2-x%2B6%5C%5C%0A%5Chline%0Ax%2B1%262x%5E4-3x%5E3-6x%5E2%2B5x%2B6%5C%5C%0A-%26%5Cleft(2x%5E4%2B2x%5E3%5Cright)%5C%5C%0A%26-------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-5x%5E3-6x%5E2%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-5x%5E3-5x%5E2%5Cright)%5C%5C%0A%26----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-x%5E2%2B5x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-x%5E2-x%5Cright)%5C%5C%0A%26------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%206x%2B6%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(6x%2B6%5Cright)%5C%5C%0A%26---------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;2x^3-5x^2-x+6\\&#10;\hline&#10;x+1&amp;2x^4-3x^3-6x^2+5x+6\\&#10;-&amp;\left(2x^4+2x^3\right)\\&#10;&amp;-------\\&#10;&amp;\ \ \ \ \ \ \ \ -5x^3-6x^2\\&#10;-&amp;\ \ \ \ \ \ \ \left(-5x^3-5x^2\right)\\&#10;&amp;----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -x^2+5x\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(-x^2-x\right)\\&#10;&amp;------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6x+6\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(6x+6\right)\\&#10;&amp;---------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%262x%5E2-7x%2B6%5C%5C%0A%5Chline%0Ax%2B1%262x%5E3-5x%5E2-x%2B6%5C%5C%0A-%26%5Cleft(2x%5E%7B%5E3%7D%2B2x%5E2%5Cright)%5C%5C%0A%26-------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20-7x%5E2-x%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(-7x%5E2-7x%5Cright)%5C%5C%0A%26----------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%206x%2B6%5C%5C%0A-%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(6x%2B6%5Cright)%5C%5C%0A%26------------%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;2x^2-7x+6\\&#10;\hline&#10;x+1&amp;2x^3-5x^2-x+6\\&#10;-&amp;\left(2x^{^3}+2x^2\right)\\&#10;&amp;-------\\&#10;&amp;\ \ \ \ \ \ \ \ -7x^2-x\\&#10;-&amp;\ \ \ \ \ \ \ \left(-7x^2-7x\right)\\&#10;&amp;----------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6x+6\\&#10;-&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(6x+6\right)\\&#10;&amp;------------\\&#10;&amp;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0&#10;\end{array}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2-7x%2B6%3D0" alt="2x^2-7x+6=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B7%5Cpm%5Csqrt%5B%5D%7B%5Cleft(-7%5Cright)%5E2-4%5Ccdot2%5Ccdot6%7D%7D%7B2%5Ccdot2%7D%3D%5Cfrac%7B7%5Cpm%5Csqrt%5B%5D%7B1%7D%7D%7B4%7D%3D%5Cfrac%7B7%5Cpm1%7D%7B4%7D" alt="x=\frac{7\pm\sqrt[]{\left(-7\right)^2-4\cdot2\cdot6}}{2\cdot2}=\frac{7\pm\sqrt[]{1}}{4}=\frac{7\pm1}{4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B7%2B1%7D%7B4%7D%3D2" alt="x=\frac{7+1}{4}=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=tai" alt="tai"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B7-1%7D%7B4%7D%3D%5Cfrac%7B6%7D%7B4%7D%3D%5Cfrac%7B3%7D%7B2%7D" alt="x=\frac{7-1}{4}=\frac{6}{4}=\frac{3}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%2B1%5Cright)%5E2%5Cleft(x-2%5Cright)%5Cleft(2x-3%5Cright)" alt="\left(x+1\right)^2\left(x-2\right)\left(2x-3\right)"/></div> <br/> <div> </div> <br/> <br/> </div> </div> </div> </div> 2019-03-11T11:36:23+02:00 https://peda.net/id/eba3c27a432 2019-03-16T19:32:45+02:00 <span>121</span> <div>a)</div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2x%5E2%2B8x%2B8%7D%7Bx%2B2%7D%3D%5Cfrac%7B2%5Cleft(x%5E2%2B4x%2B4%5Cright)%7D%7Bx%2B2%7D" alt="\frac{2x^2+8x+8}{x+2}=\frac{2\left(x^2+4x+4\right)}{x+2}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2B4x%2B4" alt="x^2+4x+4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D1%7B%2C%7D%5C%20b%3D4%7B%2C%7D%5C%20c%3D4" alt="a=1{,}\ b=4{,}\ c=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Ccdot%20v%3D4%5C%20ja%5C%20u%2Bv%3D4" alt="u\cdot v=4\ ja\ u+v=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%3D2%5C%20ja%5C%20v%3D2" alt="u=2\ ja\ v=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2B4x%2B4%3D%5Cleft(x%5E2%2B2x%5Cright)%2B%5Cleft(2x%2B4%5Cright)%3Dx%5Cleft(x%2B2%5Cright)%2B2%5Cleft(x%2B2%5Cright)%3D%5Cleft(x%2B2%5Cright)%5Cleft(x%2B2%5Cright)%3D%5Cleft(x%2B2%5Cright)%5E2" alt="x^2+4x+4=\left(x^2+2x\right)+\left(2x+4\right)=x\left(x+2\right)+2\left(x+2\right)=\left(x+2\right)\left(x+2\right)=\left(x+2\right)^2"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2x%5E2%2B8x%2B8%7D%7Bx%2B2%7D%3D%5Cfrac%7B2%5Cleft(x%5E2%2B4x%2B4%5Cright)%7D%7Bx%2B2%7D%3D%5Cfrac%7B2%5Cleft(x%2B2%5Cright)%5E2%7D%7B%5Cleft(x%2B2%5Cright)%7D%3D2%5Cleft(x%2B2%5Cright)%3D2x%2B4" alt="\frac{2x^2+8x+8}{x+2}=\frac{2\left(x^2+4x+4\right)}{x+2}=\frac{2\left(x+2\right)^2}{\left(x+2\right)}=2\left(x+2\right)=2x+4"/></div> <div> </div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%5E2%2B2x%2B2%7D%7Bx%2B1%7D%3D%5Cfrac%7Bx%5E2%2B2x%2B1%2B1%7D%7Bx%2B1%7D%3D%5Cfrac%7B%5Cleft(x%2B1%5Cright)%5E2%2B1%7D%7Bx%2B1%7D%3Dx%2B1%2B%5Cfrac%7B1%7D%7Bx%2B1%7D" alt="\frac{x^2+2x+2}{x+1}=\frac{x^2+2x+1+1}{x+1}=\frac{\left(x+1\right)^2+1}{x+1}=x+1+\frac{1}{x+1}"/></div> </div> <br/> 122<br/> <div>a)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(1%5Cright)%3D3%5Ccdot1%5E2-1-2%3D3-1-2%3D0" alt="P\left(1\right)=3\cdot1^2-1-2=3-1-2=0"/><br/> <div>On</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E2-x-2" alt="3x^2-x-2"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=ax%5E2%2Bbx%2Bc%3D%5Cleft(ax%5E2%2Bux%5Cright)%2B%5Cleft(vx%2Bc%5Cright)" alt="ax^2+bx+c=\left(ax^2+ux\right)+\left(vx+c\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D3%7B%2C%7D%5C%20b%3D-1%7B%2C%7D%5C%20c%3D-2" alt="a=3{,}\ b=-1{,}\ c=-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Ccdot%20v%3Da%5Ccdot%20c%5C%20ja%5C%20u%2Bv%3Db" alt="u\cdot v=a\cdot c\ ja\ u+v=b"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Ccdot%20v%3D-6%5C%20ja%5C%20u%2Bv%3D-1" alt="u\cdot v=-6\ ja\ u+v=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%3D2%5C%20ja%5C%20v%3D-3" alt="u=2\ ja\ v=-3"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3x%5E2%2B2x%5Cright)%2B%5Cleft(-3x-2%5Cright)" alt="\left(3x^2+2x\right)+\left(-3x-2\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cleft(3x%2B2%5Cright)-%5Cleft(3x%2B2%5Cright)%3D%5Cleft(x-1%5Cright)%5Cleft(3x%2B2%5Cright)" alt="x\left(3x+2\right)-\left(3x+2\right)=\left(x-1\right)\left(3x+2\right)"/><br/> <br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(2%5Cright)%3D3%5Ccdot2%5E2-2-2%3D12-4%3D8" alt="P\left(2\right)=3\cdot2^2-2-2=12-4=8"/></div> <span>ei ole <br/> <br/> </span> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(1%5Cright)%3D3%5Ccdot1%5E2-1-2%3D0" alt="P\left(1\right)=3\cdot1^2-1-2=0"/></div> <span>on</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E2-x-2" alt="3x^2-x-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=ax%5E2%2Bbx%2Bc%3D%5Cleft(ax%5E2%2Bux%5Cright)%2B%5Cleft(vx%2Bc%5Cright)" alt="ax^2+bx+c=\left(ax^2+ux\right)+\left(vx+c\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D3%7B%2C%7D%5C%20b%3D-1%7B%2C%7D%5C%20c%3D-2" alt="a=3{,}\ b=-1{,}\ c=-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Ccdot%20v%3Da%5Ccdot%20c%5C%20ja%5C%20u%2Bv%3Db" alt="u\cdot v=a\cdot c\ ja\ u+v=b"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Ccdot%20v%3D-6%5C%20ja%5C%20u%2Bv%3D-1" alt="u\cdot v=-6\ ja\ u+v=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%3D2%5C%20ja%5C%20v%3D-3" alt="u=2\ ja\ v=-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3x%5E2%2B2x%5Cright)%2B%5Cleft(-3x-2%5Cright)" alt="\left(3x^2+2x\right)+\left(-3x-2\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cleft(3x%2B2%5Cright)-%5Cleft(3x%2B2%5Cright)%3D%5Cleft(x-1%5Cright)%5Cleft(3x%2B2%5Cright)" alt="x\left(3x+2\right)-\left(3x+2\right)=\left(x-1\right)\left(3x+2\right)"/> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft(x-1%5Cright)%5Cleft(3x%2B2%5Cright)%7D%7B2x-2%7D%3D%5Cfrac%7B%5Cleft(x-1%5Cright)%5Cleft(3x%2B2%5Cright)%7D%7B2%5Cleft(x-1%5Cright)%7D%3D%5Cfrac%7B3x%2B2%7D%7B2%7D" alt="\frac{\left(x-1\right)\left(3x+2\right)}{2x-2}=\frac{\left(x-1\right)\left(3x+2\right)}{2\left(x-1\right)}=\frac{3x+2}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D%5Cleft(2x-2%5Cright)%5Cleft(%5Cfrac%7B3%7D%7B2%7Dx%2B1%5Cright)" alt="P\left(x\right)=\left(2x-2\right)\left(\frac{3}{2}x+1\right)"/><br/> <br/> 123 <div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E2-6x%2B3" alt="3x^2-6x+3"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D3%7B%2C%7D%5C%20b%3D-6%7B%2C%7D%5C%20c%3D3" alt="a=3{,}\ b=-6{,}\ c=3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Ccdot%20v%3D9%5C%20ja%5C%20u%2Bv%3D-6" alt="u\cdot v=9\ ja\ u+v=-6"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=u%3D-3%5C%20ja%5C%20v%3D-3" alt="u=-3\ ja\ v=-3"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3x%5E2-3x%5Cright)%2B%5Cleft(-3x%2B3%5Cright)%3D3x%5Cleft(x-1%5Cright)-3%5Cleft(x-1%5Cright)%3D%5Cleft(3x-3%5Cright)%5Cleft(x-1%5Cright)" alt="\left(3x^2-3x\right)+\left(-3x+3\right)=3x\left(x-1\right)-3\left(x-1\right)=\left(3x-3\right)\left(x-1\right)"/></div> <span>b)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2-7x-4" alt="2x^2-7x-4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D2%7B%2C%7D%5C%20b%3D-7%7B%2C%7D%5C%20c%3D-4" alt="a=2{,}\ b=-7{,}\ c=-4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Ccdot%20v%3D-8%5C%20ja%5C%20u%2Bv%3D-7" alt="u\cdot v=-8\ ja\ u+v=-7"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%3D-8%5C%20ja%5C%20v%3D1" alt="u=-8\ ja\ v=1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2x%5E2-8x%5Cright)%2B%5Cleft(x-4%5Cright)%3D2x%5Cleft(x-4%5Cright)%2B1%5Cleft(x-4%5Cright)%3D%5Cleft(2x%2B1%5Cright)%5Cleft(x-4%5Cright)" alt="\left(2x^2-8x\right)+\left(x-4\right)=2x\left(x-4\right)+1\left(x-4\right)=\left(2x+1\right)\left(x-4\right)"/><br/> <div>c)</div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3-2x%5E2%2B3x%3Dx%5Cleft(x%5E2-2x%2B3%5Cright)" alt="x^3-2x^2+3x=x\left(x^2-2x+3\right)"/></div> <br/> 124<br/> <div>a)</div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B4x%5E2-4%7D%7Bx%2B1%7D%3D%5Cfrac%7B4%5Cleft(x-1%5Cright)%5Cleft(x%2B1%5Cright)%7D%7B%5Cleft(x%2B1%5Cright)%7D%3D4%5Cleft(x-1%5Cright)%3D4x-4" alt="\frac{4x^2-4}{x+1}=\frac{4\left(x-1\right)\left(x+1\right)}{\left(x+1\right)}=4\left(x-1\right)=4x-4"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4x%5E2-4%3D%5Cleft(4x-4%5Cright)%5Cleft(x%2B1%5Cright)" alt="4x^2-4=\left(4x-4\right)\left(x+1\right)"/></div> b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%5E3%2Bx%5E2-12x%7D%7Bx%2B4%7D%3D%5Cfrac%7Bx%5Cleft(x%5E2%2Bx-12%5Cright)%7D%7Bx%2B4%7D" alt="\frac{x^3+x^2-12x}{x+4}=\frac{x\left(x^2+x-12\right)}{x+4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-1%5Cpm%5Csqrt%5B%5D%7B1%5E2-4%5Ccdot1%5Ccdot%5Cleft(-12%5Cright)%7D%7D%7B2%5Ccdot1%7D%3D%5Cfrac%7B-1%5Cpm%5Csqrt%5B%5D%7B49%7D%7D%7B2%7D%3D%5Cfrac%7B-1%5Cpm7%7D%7B2%7D" alt="x=\frac{-1\pm\sqrt[]{1^2-4\cdot1\cdot\left(-12\right)}}{2\cdot1}=\frac{-1\pm\sqrt[]{49}}{2}=\frac{-1\pm7}{2}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-1%2B7%7D%7B2%7D%3D3" alt="x=\frac{-1+7}{2}=3"/> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-1-7%7D%7B2%7D%3D%5Cfrac%7B-8%7D%7B2%7D%3D-4" alt="x=\frac{-1-7}{2}=\frac{-8}{2}=-4"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%5Cleft(x%5E2%2Bx-12%5Cright)%7D%7Bx%2B4%7D%3D%5Cfrac%7Bx%5Cleft(x-3%5Cright)%5Cleft(x%2B4%5Cright)%7D%7B%5Cleft(x%2B4%5Cright)%7D%3Dx%5Cleft(x-3%5Cright)" alt="\frac{x\left(x^2+x-12\right)}{x+4}=\frac{x\left(x-3\right)\left(x+4\right)}{\left(x+4\right)}=x\left(x-3\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3%2Bx%5E2-12x%3Dx%5Cleft(x-3%5Cright)%5Cleft(x%2B4%5Cright)" alt="x^3+x^2-12x=x\left(x-3\right)\left(x+4\right)"/></div> <br/> 125<br/> a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%3D6%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E2-%5Cfrac%7B1%7D%7B2%7D-2%3D1%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7B2%7D-2%3D-1" alt="P\left(\frac{1}{2}\right)=6\cdot\left(\frac{1}{2}\right)^2-\frac{1}{2}-2=1\frac{1}{2}-\frac{1}{2}-2=-1"/></div> <span>ei ole jaollinen</span> <div> </div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%3D4%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E4-%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E3-%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E2%3D%5Cfrac%7B4%7D%7B16%7D-%5Cfrac%7B1%7D%7B8%7D-%5Cfrac%7B1%7D%7B8%7D%3D%5Cfrac%7B4%7D%7B16%7D-%5Cfrac%7B1%7D%7B4%7D%3D%5Cfrac%7B4%7D%7B16%7D-%5Cfrac%7B4%7D%7B16%7D%3D0" alt="P\left(\frac{1}{2}\right)=4\cdot\left(\frac{1}{2}\right)^4-\left(\frac{1}{2}\right)^3-\frac{1}{2}\cdot\left(\frac{1}{2}\right)^2=\frac{4}{16}-\frac{1}{8}-\frac{1}{8}=\frac{4}{16}-\frac{1}{4}=\frac{4}{16}-\frac{4}{16}=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B4x%5E4-x%5E3-%5Cfrac%7B1%7D%7B2%7Dx%5E2%7D%7B2x-1%7D%3D%5Cfrac%7B4x%5E4-x%5E3-%5Cfrac%7Bx%5E2%7D%7B2%7D%7D%7B2x-1%7D%3D%5Cfrac%7B%5Cfrac%7B4x%5E42%7D%7B2%7D-%5Cfrac%7Bx%5E32%7D%7B2%7D-%5Cfrac%7Bx%5E2%7D%7B2%7D%7D%7B2x-1%7D%3D%5Cfrac%7B%5Cfrac%7B8x%5E4-2x%5E3-x%5E2%7D%7B2%7D%7D%7B2x-1%7D%3D%5Cfrac%7B8x%5E4-2x%5E3-x%5E2%7D%7B2%5Cleft(2x-1%5Cright)%7D%3D%5Cfrac%7Bx%5E2%5Cleft(8x%5E2-2x-1%5Cright)%7D%7B2%5Cleft(2x-1%5Cright)%7D" alt="\frac{4x^4-x^3-\frac{1}{2}x^2}{2x-1}=\frac{4x^4-x^3-\frac{x^2}{2}}{2x-1}=\frac{\frac{4x^42}{2}-\frac{x^32}{2}-\frac{x^2}{2}}{2x-1}=\frac{\frac{8x^4-2x^3-x^2}{2}}{2x-1}=\frac{8x^4-2x^3-x^2}{2\left(2x-1\right)}=\frac{x^2\left(8x^2-2x-1\right)}{2\left(2x-1\right)}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D8x%5E2-2x-1" alt="f\left(x\right)=8x^2-2x-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D8%7B%2C%7D%5C%20b%3D-2%7B%2C%7D%5C%20c%3D-1" alt="a=8{,}\ b=-2{,}\ c=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Ccdot%20v%3D-8%5C%20ja%5C%20u%2Bv%3D-2" alt="u\cdot v=-8\ ja\ u+v=-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%3D2%5C%20ja%5C%20v%3D-4" alt="u=2\ ja\ v=-4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(8x%5E2%2B2x%5Cright)%2B%5Cleft(-4x-1%5Cright)%3D2x%5Cleft(4x%2B1%5Cright)-1%5Cleft(4x%2B1%5Cright)%3D%5Cleft(2x-1%5Cright)%5Cleft(4x%2B1%5Cright)" alt="\left(8x^2+2x\right)+\left(-4x-1\right)=2x\left(4x+1\right)-1\left(4x+1\right)=\left(2x-1\right)\left(4x+1\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%5E2%5Cleft(8x%5E2-2x-1%5Cright)%7D%7B2%5Cleft(2x-1%5Cright)%7D%3D%5Cfrac%7Bx%5E2%5Cleft(2x-1%5Cright)%5Cleft(4x%2B1%5Cright)%7D%7B2%5Cleft(2x-1%5Cright)%7D%3D%5Cfrac%7Bx%5E2%5Cleft(4x%2B1%5Cright)%7D%7B2%7D" alt="\frac{x^2\left(8x^2-2x-1\right)}{2\left(2x-1\right)}=\frac{x^2\left(2x-1\right)\left(4x+1\right)}{2\left(2x-1\right)}=\frac{x^2\left(4x+1\right)}{2}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D%5Cleft(%5Cfrac%7Bx%5E2%5Cleft(4x%2B1%5Cright)%7D%7B2%7D%5Cright)%5Ccdot%5Cleft(2x-1%5Cright)" alt="P\left(x\right)=\left(\frac{x^2\left(4x+1\right)}{2}\right)\cdot\left(2x-1\right)"/><br/> <br/> <span>126</span> <div>A I </div> <div>B II, IV</div> <div>C II, IV</div> <div>D III</div> <div> </div> <div>127</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%5E2%2B1%2B%5Cfrac%7B3%7D%7Bx%2B3%7D%5Cright)%5Ccdot%5Cleft(x%2B3%5Cright)" alt="\left(x^2+1+\frac{3}{x+3}\right)\cdot\left(x+3\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2B1%2B%5Cfrac%7B3%7D%7Bx%2B3%7D%3D%5Cfrac%7Bx%5E2%5Cleft(x%2B3%5Cright)%7D%7Bx%2B3%7D%2B%5Cfrac%7Bx%2B3%7D%7Bx%2B3%7D%2B%5Cfrac%7B3%7D%7Bx%2B3%7D%3D%5Cfrac%7Bx%5E3%2B3x%5E2%2Bx%2B6%7D%7Bx%2B3%7D" alt="x^2+1+\frac{3}{x+3}=\frac{x^2\left(x+3\right)}{x+3}+\frac{x+3}{x+3}+\frac{3}{x+3}=\frac{x^3+3x^2+x+6}{x+3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D%5Cfrac%7Bx%5E3%2B3x%5E2%2Bx%2B6%7D%7Bx%2B3%7D%5Ccdot%5Cleft(x%2B3%5Cright)%3Dx%5E3%2B3x%5E2%2Bx%2B6" alt="P\left(x\right)=\frac{x^3+3x^2+x+6}{x+3}\cdot\left(x+3\right)=x^3+3x^2+x+6"/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2x%5E3-5x%5E2%2B3x%7D%7B2x-3%7D%3D%5Cfrac%7Bx%5Cleft(2x%5E2-5x%2B3%5Cright)%7D%7B2x-3%7D" alt="\frac{2x^3-5x^2+3x}{2x-3}=\frac{x\left(2x^2-5x+3\right)}{2x-3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2-5x%2B3" alt="2x^2-5x+3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D2%7B%2C%7D%5C%20b%3D-5%7B%2C%7D%5C%20c%3D3" alt="a=2{,}\ b=-5{,}\ c=3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Ccdot%20v%3D6%5C%20ja%5C%20u%2Bv%3D-5" alt="u\cdot v=6\ ja\ u+v=-5"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%3D-2%5C%20ja%5C%20v%3D-3" alt="u=-2\ ja\ v=-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2x%5E2-2x%5Cright)%2B%5Cleft(-3x%2B3%5Cright)%3D2x%5Cleft(x-1%5Cright)%2B-3%5Cleft(x-1%5Cright)%3D%5Cleft(2x-3%5Cright)%5Cleft(x-1%5Cright)" alt="\left(2x^2-2x\right)+\left(-3x+3\right)=2x\left(x-1\right)+-3\left(x-1\right)=\left(2x-3\right)\left(x-1\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(x%5Cright)%3D%5Cfrac%7Bx%5Cleft(2x-3%5Cright)%5Cleft(x-1%5Cright)%7D%7B2x-3%7D%3Dx%5Cleft(x-1%5Cright)%3Dx%5E2-x" alt="P\left(x\right)=\frac{x\left(2x-3\right)\left(x-1\right)}{2x-3}=x\left(x-1\right)=x^2-x"/><br/> <br/> <br/> <div>128</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-12x%2Ba_1%3Da_2%5E2%2B2a_2b%2Bb%5E2" alt="x^2-12x+a_1=a_2^2+2a_2b+b^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a_2%3Dx%7B%2C%7D%5C%20b%3D-6%7B%2C%7D%5C%20b%5E2%3Da_1%3D36" alt="a_2=x{,}\ b=-6{,}\ b^2=a_1=36"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=9x%5E2%2Bax%2B4%3D%5Cleft(3x%5Cright)%5E2%2B%5Cleft(2%5Ccdot%20b%5Ccdot3x%5Cright)%2B4%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C4%3D2%5E2%7B%2C%7D%5C%20joten%5C%20b%3D%5Cpm2" alt="9x^2+ax+4=\left(3x\right)^2+\left(2\cdot b\cdot3x\right)+4\ \ \ \ \ \left|\right|4=2^2{,}\ joten\ b=\pm2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3x%5Cright)%5E2%2B%5Cleft(2%5Ccdot2%5Ccdot3x%5Cright)%2B2%5E2%3D9x%5E2%2B12x%2B4" alt="\left(3x\right)^2+\left(2\cdot2\cdot3x\right)+2^2=9x^2+12x+4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D%5Cpm12" alt="a=\pm12"/> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=ax%5E2-8x%2B4%3Dax%5E2%2B%5Cleft(2%5Ccdot-2%5Ccdot%20ax%5Cright)%2B%5Cleft(-2%5Cright)%5E2%3D%5Cleft(2x-2%5Cright)%5E2" alt="ax^2-8x+4=ax^2+\left(2\cdot-2\cdot ax\right)+\left(-2\right)^2=\left(2x-2\right)^2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2x-2%5Cright)%5E2%3D4x%5E2%2B2%5Ccdot%5Cleft(-2%5Cright)%5Ccdot2x%2B%5Cleft(-2%5Cright)%5E2%3D4x%5E2-8x%2B4" alt="\left(2x-2\right)^2=4x^2+2\cdot\left(-2\right)\cdot2x+\left(-2\right)^2=4x^2-8x+4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D4" alt="a=4"/><br/> <br/> <div>129</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%5E3-4x%7D%7Bx%5E2%2B2x%7D%3D%5Cfrac%7Bx%5Cleft(x%5E2-4%5Cright)%7D%7Bx%5Cleft(x%2B2%5Cright)%7D%3D%5Cfrac%7Bx%5Cleft(x%2B2%5Cright)%5Cleft(x-2%5Cright)%7D%7Bx%5Cleft(x%2B2%5Cright)%7D%3Dx-2" alt="\frac{x^3-4x}{x^2+2x}=\frac{x\left(x^2-4\right)}{x\left(x+2\right)}=\frac{x\left(x+2\right)\left(x-2\right)}{x\left(x+2\right)}=x-2"/><br/> <br/> <div>130</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%5Cleft(x-1%5Cright)-4%5Cleft(x-1%5Cright)%3D%5Cleft(x%5E2-4%5Cright)%5Cleft(x-1%5Cright)%3D%5Cleft(x-2%5Cright)%5Cleft(x%2B2%5Cright)%5Cleft(x-1%5Cright)" alt="x^2\left(x-1\right)-4\left(x-1\right)=\left(x^2-4\right)\left(x-1\right)=\left(x-2\right)\left(x+2\right)\left(x-1\right)"/></div> <span>b)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3%2B8x%5E2%2Bx%2B8%3Dx%5E2%5Cleft(x%2B8%5Cright)%2B%5Cleft(x%2B8%5Cright)%3D%5Cleft(x%5E2%2B1%5Cright)%5Cleft(x%2B8%5Cright)" alt="x^3+8x^2+x+8=x^2\left(x+8\right)+\left(x+8\right)=\left(x^2+1\right)\left(x+8\right)"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3-3x%5E2%2B2x-6%3Dx%5E2%5Cleft(x-3%5Cright)%2B2%5Cleft(x-3%5Cright)%3D%5Cleft(x%5E2%2B2%5Cright)%5Cleft(x-3%5Cright)" alt="x^3-3x^2+2x-6=x^2\left(x-3\right)+2\left(x-3\right)=\left(x^2+2\right)\left(x-3\right)"/><br/> <br/> <div>131</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%2B2x-24%3D0" alt="2x^2+2x-24=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-2%5Cpm%5Csqrt%5B%5D%7B2%5E2-4%5Ccdot2%5Ccdot%5Cleft(-24%5Cright)%7D%7D%7B2%5Ccdot2%7D%3D%5Cfrac%7B-2%5Cpm14%7D%7B4%7D" alt="x=\frac{-2\pm\sqrt[]{2^2-4\cdot2\cdot\left(-24\right)}}{2\cdot2}=\frac{-2\pm14}{4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-2%2B14%7D%7B4%7D%3D3" alt="x=\frac{-2+14}{4}=3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=tai" alt="tai"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-2-14%7D%7B4%7D%3D-4" alt="x=\frac{-2-14}{4}=-4"/><br/> <div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(2%5Cright)%3D2%5E2-2k-k%5E2-1%3D4-2k-k%5E2-1%3D-k%5E2-2k%2B3" alt="P\left(2\right)=2^2-2k-k^2-1=4-2k-k^2-1=-k^2-2k+3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-k%5E2-2k%2B3%3D0" alt="-k^2-2k+3=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7B2%5Cpm%5Csqrt%5B%5D%7B%5Cleft(-2%5Cright)%5E2-4%5Ccdot%5Cleft(-1%5Cright)%5Ccdot3%7D%7D%7B2%5Ccdot%5Cleft(-1%5Cright)%7D%3D%5Cfrac%7B2%5Cpm4%7D%7B-2%7D" alt="k=\frac{2\pm\sqrt[]{\left(-2\right)^2-4\cdot\left(-1\right)\cdot3}}{2\cdot\left(-1\right)}=\frac{2\pm4}{-2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7B2%2B4%7D%7B-2%7D%3D-3" alt="k=\frac{2+4}{-2}=-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=tai" alt="tai"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7B2-4%7D%7B-2%7D%3D1" alt="k=\frac{2-4}{-2}=1"/><br/> <br/> <div>132</div> <div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%2Bax%2B2a" alt="2x^2+ax+2a"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%2Ba%5Cleft(x%2B2%5Cright)" alt="2x^2+a\left(x+2\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D0" alt="a=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%2B0%5Ccdot%5Cleft(x%2B2%5Cright)%3D2x%5E2%3D2x%5Ccdot%20x" alt="2x^2+0\cdot\left(x+2\right)=2x^2=2x\cdot x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D16" alt="a=16"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%2B16%5Cleft(x%2B2%5Cright)%3D2x%5E2%2B16x%2B32%3D2%5Cleft(x%5E2%2B8x%2B16%5Cright)%3D2%5Cleft(x%2B4%5Cright)%5E2" alt="2x^2+16\left(x+2\right)=2x^2+16x+32=2\left(x^2+8x+16\right)=2\left(x+4\right)^2"/><br/> <br/> <div>133</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3x%5E2%2B5x-2%7D%7B2x%2B4%7D%3D%5Cfrac%7Bx%5Cleft(3x%2B5%5Cright)-2%7D%7B2%5Cleft(x%2B2%5Cright)%7D" alt="\frac{3x^2+5x-2}{2x+4}=\frac{x\left(3x+5\right)-2}{2\left(x+2\right)}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-2%7D%5Cleft(%5Cfrac%7Bx%5Cleft(3x%2B5%5Cright)-2%7D%7B2%5Cleft(x%2B2%5Cright)%7D%5Cright)%3D%5Cfrac%7B-2%5Cleft(3%5Ccdot%5Cleft(-2%5Cright)%2B5%5Cright)-2%7D%7B2%5Cleft(-2%2B2%5Cright)%7D%3D%5Cfrac%7B-2%5Cleft(-6%2B5%5Cright)-2%7D%7B2%5Ccdot0%7D%3DNollalla%5C%20jako" alt="\lim_{x\rightarrow-2}\left(\frac{x\left(3x+5\right)-2}{2\left(x+2\right)}\right)=\frac{-2\left(3\cdot\left(-2\right)+5\right)-2}{2\left(-2+2\right)}=\frac{-2\left(-6+5\right)-2}{2\cdot0}=Nollalla\ jako"/></div> <span>Käytetään Hopitalin sääntöä</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow-2%7D%3D%5Cfrac%7Bf%27%5Cleft(x%5Cright)%7D%7Bg%27%5Cleft(x%5Cright)%7D%3D%5Cfrac%7B6x%2B5%7D%7B2%7D%3D%5Cfrac%7B6%5Ccdot%5Cleft(-2%5Cright)%2B5%7D%7B2%7D%3D-%5Cfrac%7B7%7D%7B2%7D" alt="\lim_{x\rightarrow-2}=\frac{f'\left(x\right)}{g'\left(x\right)}=\frac{6x+5}{2}=\frac{6\cdot\left(-2\right)+5}{2}=-\frac{7}{2}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2x%5E3-2x%5E2-3x%2B3%7D%7Bx-1%7D%3D%5Cfrac%7Bx%5E2%5Cleft(2x-2%5Cright)-3%5Cleft(x-1%5Cright)%7D%7Bx-1%7D%3D%5Cfrac%7Bx%5E2%5Cleft(2%5Cleft(x-1%5Cright)%5Cright)-3%5Cleft(x-1%5Cright)%7D%7Bx-1%7D%3D%5Cfrac%7B2x%5E2%5Cleft(x-1%5Cright)-3%5Cleft(x-1%5Cright)%7D%7Bx-1%7D%3D%5Cfrac%7B%5Cleft(2x%5E2-3%5Cright)%5Cleft(x-1%5Cright)%7D%7B%5Cleft(x-1%5Cright)%7D%3D2x%5E3-3" alt="\frac{2x^3-2x^2-3x+3}{x-1}=\frac{x^2\left(2x-2\right)-3\left(x-1\right)}{x-1}=\frac{x^2\left(2\left(x-1\right)\right)-3\left(x-1\right)}{x-1}=\frac{2x^2\left(x-1\right)-3\left(x-1\right)}{x-1}=\frac{\left(2x^2-3\right)\left(x-1\right)}{\left(x-1\right)}=2x^3-3"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow1%7D%3D2x%5E3-3%3D2%5Ccdot1%5E3-3%3D-1" alt="\lim_{x\rightarrow1}=2x^3-3=2\cdot1^3-3=-1"/><br/> <br/> <div>134</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2x%5E2%2B5x-3%7D%7B3x%2Ba%7D" alt="\frac{2x^2+5x-3}{3x+a}"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%2B5x-3" alt="2x^2+5x-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D2%7B%2C%7D%5C%20b%3D5%7B%2C%7D%5C%20c%3D-3" alt="a=2{,}\ b=5{,}\ c=-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Ccdot%20v%3D-6%5C%20ja%5C%20u%2Bv%3D5" alt="u\cdot v=-6\ ja\ u+v=5"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%7B%2C%7D2%7B%2C%7D3%7B%2C%7D6" alt="1{,}2{,}3{,}6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%3D-1%5C%20ja%5C%20v%3D6" alt="u=-1\ ja\ v=6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2x%5E2-x%5Cright)%2B%5Cleft(6x-3%5Cright)%3Dx%5Cleft(2x-1%5Cright)%2B3%5Cleft(2x-1%5Cright)%3D%5Cleft(x%2B3%5Cright)%5Cleft(2x-1%5Cright)" alt="\left(2x^2-x\right)+\left(6x-3\right)=x\left(2x-1\right)+3\left(2x-1\right)=\left(x+3\right)\left(2x-1\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft(x%2B3%5Cright)%5Cleft(2x-1%5Cright)%7D%7B3x%2Ba%7D" alt="\frac{\left(x+3\right)\left(2x-1\right)}{3x+a}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft(x%2B3%5Cright)%7D%7B3x%2Ba%7D%3D%5Cfrac%7Bx%2B3%7D%7B3%5Cleft(x%2B%5Cfrac%7Ba%7D%7B3%7D%5Cright)%7D" alt="\frac{\left(x+3\right)}{3x+a}=\frac{x+3}{3\left(x+\frac{a}{3}\right)}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba%7D%7B3%7D%3D3" alt="\frac{a}{3}=3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D9" alt="a=9"/> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2x-1%7D%7B3x%2Ba%7D%3D%5Cfrac%7B2%5Cleft(x-%5Cfrac%7B1%7D%7B2%7D%5Cright)%7D%7B3%5Cleft(x%2B%5Cfrac%7Ba%7D%7B3%7D%5Cright)%7D" alt="\frac{2x-1}{3x+a}=\frac{2\left(x-\frac{1}{2}\right)}{3\left(x+\frac{a}{3}\right)}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-%5Cfrac%7B1%7D%7B2%7D%3Dx%2B%5Cfrac%7Ba%7D%7B3%7D" alt="x-\frac{1}{2}=x+\frac{a}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-%5Cfrac%7B1%7D%7B2%7D%3D%5Cfrac%7B3%7D%7B2%7D%5Cleft(x%2B%5Cfrac%7Ba%7D%7B3%7D%5Cright)" alt="x-\frac{1}{2}=\frac{3}{2}\left(x+\frac{a}{3}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-%5Cfrac%7B1%7D%7B2%7D%3D%5Cfrac%7B3%7D%7B2%7Dx%2B%5Cfrac%7B3a%7D%7B6%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3A%5C%20%5Cfrac%7B3%7D%7B2%7Dx%3D%5Cfrac%7B3x%7D%7B2%7D%7B%2C%7D%5Cfrac%7B3a%7D%7B6%7D%3D%5Cfrac%7Ba%7D%7B2%7D" alt="x-\frac{1}{2}=\frac{3}{2}x+\frac{3a}{6}\ \ \ \ \ \left|\right|:\ \frac{3}{2}x=\frac{3x}{2}{,}\frac{3a}{6}=\frac{a}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-%5Cfrac%7B1%7D%7B2%7D%3D%5Cfrac%7B3x%7D%7B2%7D%2B%5Cfrac%7Ba%7D%7B2%7D" alt="x-\frac{1}{2}=\frac{3x}{2}+\frac{a}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-%5Cfrac%7B1%7D%7B2%7D%3D%5Cfrac%7B3x%2Ba%7D%7B2%7D" alt="x-\frac{1}{2}=\frac{3x+a}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B3x%2Ba%2B1%7D%7B2%7D" alt="x=\frac{3x+a+1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3x%2Ba%2B1%7D%7B2%7D-x%3D0" alt="\frac{3x+a+1}{2}-x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3x%2Ba%2B1-2x%7D%7B2%7D%3D0" alt="\frac{3x+a+1-2x}{2}=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%2Ba%2B1%7D%7B2%7D%3D0" alt="\frac{x+a+1}{2}=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2Ba%2B1%3D0" alt="x+a+1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2Ba%3D-1" alt="x+a=-1"/></div> <br/> <div>135</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E5-x%5E4-8x%5E3%2B4x%5E2%3Dx%5E4%5Cleft(2x-1%5Cright)-4x%5E2%5Cleft(2x-1%5Cright)%3D%5Cleft(x%5E4-4x%5E2%5Cright)%5Cleft(2x-1%5Cright)%3Dx%5E2%5Cleft(x%5E2-4%5Cright)%5Cleft(2x-1%5Cright)%3Dx%5Ccdot%20x%5Cleft(x-2%5Cright)%5Cleft(x%2B2%5Cright)%5Cleft(2x-1%5Cright)" alt="2x^5-x^4-8x^3+4x^2=x^4\left(2x-1\right)-4x^2\left(2x-1\right)=\left(x^4-4x^2\right)\left(2x-1\right)=x^2\left(x^2-4\right)\left(2x-1\right)=x\cdot x\left(x-2\right)\left(x+2\right)\left(2x-1\right)"/></div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> 2019-03-10T14:13:31+02:00 https://peda.net/id/03c23d083f7 2019-03-10T20:16:13+02:00 <span>101</span> <div>a)</div> <div>1)<img src="https://math-demo.abitti.fi/math.svg?latex=826%3A2%3D413" alt="826:2=413"/><br/> 2)<img src="https://math-demo.abitti.fi/math.svg?latex=413%5Crightarrow4130" alt="413\rightarrow4130"/></div> <div>b)</div> <div>1)<img src="https://math-demo.abitti.fi/math.svg?latex=405%3A2%3D202%7B%2C%7D5" alt="405:2=202{,}5"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26202%7B%2C%7D5%5C%5C%0A%5Chline%0A2%26405%5C%5C%0A%2640%5C%5C%0A%26---%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%205%5C%20%5C%20%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%204%5C%5C%0A%26---%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%2010%5C%5C%0A%5C%20%26%5C%20%5C%20%5C%20%5C%2010%5C%5C%0A%26---%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;202{,}5\\&#10;\hline&#10;2&amp;405\\&#10;&amp;40\\&#10;&amp;---\\&#10;&amp;\ \ \ \ 5\ \ \\&#10;&amp;\ \ \ \ 4\\&#10;&amp;---\\&#10;&amp;\ \ \ \ 10\\&#10;\ &amp;\ \ \ \ 10\\&#10;&amp;---\\&#10;&amp;\ \ \ \ \ \ 0&#10;\end{array}"/> <div>2)<img src="https://math-demo.abitti.fi/math.svg?latex=202%7B%2C%7D5%5Crightarrow2025" alt="202{,}5\rightarrow2025"/></div> <br/> 102 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%2B8x%3D0" alt="2x^2+8x=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5Cleft(x%2B4%5Cright)%3D0" alt="2x\left(x+4\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%2B4%3D0" alt="x+4=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-4" alt="x=-4"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=ax%5E2%2Bbx%3D0" alt="ax^2+bx=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cleft(ax%2Bb%5Cright)%3D0" alt="x\left(ax+b\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0%5C%20" alt="x=0\ "/> </div> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=ax%2Bb%3D0" alt="ax+b=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7Bb%7D%7Ba%7D" alt="x=-\frac{b}{a}"/><br/> <div> <br/> <div> <div>103</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26%5C%20%5C%2046%5C%5C%0A%5Chline%0A7%26322%5C%5C%0A-%2628%5C%5C%0A%26---%5C%5C%0A%26%5C%20%5C%2042%5C%5C%0A-%26%5C%20%5C%2042%5C%5C%0A%26---%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;\ \ 46\\&#10;\hline&#10;7&amp;322\\&#10;-&amp;28\\&#10;&amp;---\\&#10;&amp;\ \ 42\\&#10;-&amp;\ \ 42\\&#10;&amp;---\\&#10;&amp;\ \ \ \ 0&#10;\end{array}"/></div> <div>b)</div> <div>Jakoalgoritmissa toistetaan vaiheita</div> <div>- Jaa</div> <div>- Kerro</div> <div>- Vähennä</div> <div>- Pudota</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%261433%5C%5C%0A%5Chline%0A4%265734%5C%5C%0A-%264%5C%5C%0A%26---%5C%5C%0A%2617%5C%5C%0A-%2616%5C%5C%0A%26---%5C%5C%0A%2613%5C%5C%0A-%2612%5C%5C%0A%26---%5C%5C%0A%2614%5C%5C%0A-%2612%5C%5C%0A%26---%5C%5C%0A%26%5C%20%5C%202%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;1433\\&#10;\hline&#10;4&amp;5734\\&#10;-&amp;4\\&#10;&amp;---\\&#10;&amp;17\\&#10;-&amp;16\\&#10;&amp;---\\&#10;&amp;13\\&#10;-&amp;12\\&#10;&amp;---\\&#10;&amp;14\\&#10;-&amp;12\\&#10;&amp;---\\&#10;&amp;\ \ 2&#10;\end{array}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0A%26%5C%20%5C%20106%5C%5C%0A%5Chline%0A59%266254%5C%5C%0A-%2659%5C%5C%0A%26---%5C%5C%0A%26%5C%20%5C%20354%5C%5C%0A-%26%5C%20%5C%20354%5C%5C%0A%26---%5C%5C%0A%26%5C%20%5C%20%5C%20%5C%20%5C%20%5C%200%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;&amp;\ \ 106\\&#10;\hline&#10;59&amp;6254\\&#10;-&amp;59\\&#10;&amp;---\\&#10;&amp;\ \ 354\\&#10;-&amp;\ \ 354\\&#10;&amp;---\\&#10;&amp;\ \ \ \ \ \ 0&#10;\end{array}"/></div> </div> <br/> <div> </div> <div>104</div> <div>a)</div> <div>1) Olkoon p=41. Nyt<img src="https://math-demo.abitti.fi/math.svg?latex=134-p%3D134-41%3D93" alt="134-p=134-41=93"/></div> <div> <div>2) 93&gt;p, joten<img src="https://math-demo.abitti.fi/math.svg?latex=93-p%3D93-41%3D52" alt="93-p=93-41=52"/></div> </div> <div>3)<img src="https://math-demo.abitti.fi/math.svg?latex=52-p%3D52-41%3D11" alt="52-p=52-41=11"/>, 11&lt;p</div> <div>4) Viimeisen vähennyslaskun tulos on 11</div> <div> </div> <div>b) Algoritmilla saadaan selville jakojäännös, kun isompi luvuista jaetaan pienemmällä.</div> <div>c) Algoritmi on ihmisen toteuttamana todella työläs, jos toinen luku on paljon suurempi kuin toinen.</div> <div>Algoritmin ei myöskään kerro, mitä pitää tehdä, jos annetut luvut ovat yhtä suuret tai pienempi luku on nolla. </div> <div> </div> <div>106</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=Jakoj%C3%A4%C3%A4nn%C3%B6s%3D0" alt="Jakojäännös=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=8048%3A4%3D2012" alt="8048:4=2012"/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=Jakoj%C3%A4%C3%A4nn%C3%B6s%3D1" alt="Jakojäännös=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=1026%3D1025%2B1" alt="1026=1025+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1025%3A5%3D205" alt="1025:5=205"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=joten" alt="joten"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1026%3A5%3D1025%3A5%2B1%3D205...1" alt="1026:5=1025:5+1=205...1"/><br/> <br/> <span>107</span> <div>a) Valitaan esimerkiksi luvut a= 17 ja b= 82</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=7%5Ccdot2%3D14" alt="7\cdot2=14"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=7%5Ccdot8%3D56" alt="7\cdot8=56"/><br/> <div> </div> <div>110</div> <div>a) Lasketaan lukujen erotus a-b. Jos erotus positiivinen, luku a on suurempi. Jos erotus ei ole positiivinen, luku b on suurempi.</div> <div> </div> <div>b) Lasketaan lukujen erotus a-b. Jos erotus positiivinen, luku a on suurempi. Jos erotus ei ole positiivinen, lasketaan erotus b-a. Mikäli erotus on positiivinen, on luku b suurempi. Jos kumpikaan erotuksista ei ole positiivinen, niin luvut ovat keskenään yhtä suuria.</div> <div> </div> </div> <div>111</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba%7D%7Bb%7D%2B%5Cfrac%7Bc%7D%7Bd%7D%3D%5Cfrac%7Bad%7D%7Bbd%7D%2B%5Cfrac%7Bbc%7D%7Bbd%7D%3D%5Cfrac%7Bad%2Bbc%7D%7Bbd%7D" alt="\frac{a}{b}+\frac{c}{d}=\frac{ad}{bd}+\frac{bc}{bd}=\frac{ad+bc}{bd}"/></div> <span>b)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba%7D%7Bb%7D%5Ccdot%5Cfrac%7Bc%7D%7Bd%7D%3D%5Cfrac%7Bac%7D%7Bbd%7D" alt="\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba%7D%7Bb%7D%3A%5Cfrac%7Bc%7D%7Bd%7D%3D%5Cfrac%7Bad%7D%7Bbc%7D" alt="\frac{a}{b}:\frac{c}{d}=\frac{ad}{bc}"/></div> <div>  <div> <div>112</div> <div>1) laske arvosanojen summa</div> <div>2) Jakaa summa oppilaiden määrällä</div> <div> </div> <div>113</div> <div>Tulos, kun annettu luku jaetaan viidellä</div> <div> </div> <div>114</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3641-1%3D3640" alt="3641-1=3640"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=3640%3A3%3D1213.333...%5Capprox1213" alt="3640:3=1213.333...\approx1213"/> <div>b)</div> <div>Osamäärän kokonaisosa</div> <div>c)<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba-1%7D%7Bb%7D" alt="\frac{a-1}{b}"/><br/> <br/> </div> </div> <div>115</div> <div>a)<br/> Olkoon funktio y=f(x) kaikkialla määritelty. Seuraavalla algoritmilla, voidaan selvittää pisteen (a,b) sijainti funktioon f nähden: </div> <div>1) Laske funktion arvo f(a).</div> <div>2) Jost saatu arvo on suurempi kuin piseen y-koordinaatti eli f(a)&gt;b, niin piste on kuvaajan alapuolella. Jos f(a)&lt;b, niin piste on kuvaajan yläpuolella. Jos f(a)=b, niin piste on kuvaajalla.</div> <br/> b)<br/> r-säteisen ja <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x_0%7B%2C%7Dy_0%5Cright)" alt="\left(x_0{,}y_0\right)"/>-keskisen ympyrän yhtälö on keskipistemuodossa <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x-x_0%5Cright)%5E2%2B%5Cleft(y-y_0%5Cright)%5E2%3Dr%5E2" alt="\left(x-x_0\right)^2+\left(y-y_0\right)^2=r^2"/>. Seuraavalla algorimilla voidaan selvittää pisteen (a,b) sijaini ympyrään nähden:<br/> 1) Lasketaan luku c kaavalla <img src="https://math-demo.abitti.fi/math.svg?latex=c%3D%5Cleft(a-x_0%5Cright)%5E2%2B%5Cleft(b-y_0%5Cright)%5E2" alt="c=\left(a-x_0\right)^2+\left(b-y_0\right)^2"/>.<br/> 2) Jos <img src="https://math-demo.abitti.fi/math.svg?latex=c%3Er%5E2" alt="c&gt;r^2"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff; color: #333333; font-family: 'Times New Roman'; font-size: 17px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"-->, niin piste on ympyrän ulkopuolella. Jos <img src="https://math-demo.abitti.fi/math.svg?latex=c%3Cr%5E2" alt="c&lt;r^2"/>, niin piste on ympyrän sisäpuolella. Jos <img src="https://math-demo.abitti.fi/math.svg?latex=c%3Dr%5E2" alt="c=r^2"/>, niin piste on ympyrän kehällä.<br/> <br/> </div> <div>116</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%3D%5Cfrac%7B8%7D%7B2%7D%3D4" alt="a=\frac{8}{2}=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=b%3D%5Cfrac%7B9%7D%7B3%7D%3D3" alt="b=\frac{9}{3}=3"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba%7D%7Bb%7D%3D%5Cfrac%7B4%7D%7B3%7D" alt="\frac{a}{b}=\frac{4}{3}"/><br/> <div>b)</div> <div>Murtolukujen osamäärä</div> <div>c)</div> <div>Helppo käyttää</div> <div> </div> <div>117</div> Itse ohjelma:<br/> &lt;p&gt;<br/> <br/> &lt;script&gt;<br/> <br/> var a = 2;<br/> var b = -1;<br/> var c = 4;<br/> var d = -3;<br/> <br/> <span>var a=2;</span><span><br/> </span><span>var b=-7;</span><span><br/> </span><span>var c=4;</span><span><br/> </span><span>var d =5; <br/> </span><span>if a*b*c*d&gt;0:<span> </span></span><span><br/> </span><span class="im"><span> <br/> print('numbers a b c d have the same sign')</span><span><br/> </span><span>else: {print('numbers a b c d have not the same sign')</span></span><br/> }<br/> <br/> &lt;/script&gt; 2019-03-05T20:32:21+02:00