https://peda.net/id/8b877052ea5 2019-10-09T08:20:45+03:00 https://peda.net/:static/537/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/ac7d13c0f3e 2019-10-21T13:15:23+03:00 a) -2 -1<br/> b) -3, 1<br/> c) <br/> <a href="https://peda.net/p/oskari.lahtinen/maa6p-derivaatta/3d/347/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/maa6p-derivaatta/3d/347/sieppaa-png:file/photo/38658d4bb7db5bc286f8ff196cb040b0f24834d8/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a> 2019-10-21T13:15:14+03:00 https://peda.net/id/eb00c66cf3e 2019-10-21T12:55:31+03:00 A g<br/> B f<br/> C h 2019-10-21T12:55:31+03:00 https://peda.net/id/ab34b278f3e 2019-10-21T12:53:44+03:00 A1<br/> B3<br/> C2 2019-10-21T12:53:44+03:00 https://peda.net/id/70e9201cea6 2019-10-09T09:53:03+03:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(kx%5Cright)%3Dk" alt="D\left(kx\right)=k"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%20a%7D%5C%20%5Cfrac%7Bkx-ka%7D%7Bx-a%7D%3D%5Cfrac%7Bk%5Cleft(x-a%5Cright)%7D%7Bx-a%7D%3Dk" alt="\lim_{x\rightarrow a}\ \frac{kx-ka}{x-a}=\frac{k\left(x-a\right)}{x-a}=k"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(k%5Ccdot%20f%5Cleft(x%5Cright)%5Cright)%3Dk%5Ccdot%20Df%5Cleft(x%5Cright)" alt="D\left(k\cdot f\left(x\right)\right)=k\cdot Df\left(x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7Bx%5Crightarrow%20a%7D%3D%5Cfrac%7Bkf%5Cleft(x%5Cright)-kf%5Cleft(a%5Cright)%7D%7Bkx-ka%7D%3Dk%5C%20%5Cfrac%7Bf%5Cleft(x%5Cright)-f%5Cleft(a%5Cright)%7D%7Bx-a%7D" alt="\lim_{x\rightarrow a}=\frac{kf\left(x\right)-kf\left(a\right)}{kx-ka}=k\ \frac{f\left(x\right)-f\left(a\right)}{x-a}"/> 2019-10-09T09:53:03+03:00 https://peda.net/id/7d46cfbeea5 2019-10-09T09:46:40+03:00 <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E3%2B7x" alt="f\left(x\right)=x^3+7x"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D3x%5E2%2B7" alt="f'\left(x\right)=3x^2+7"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E2%2B7%3D0" alt="3x^2+7=0"/></div> <div>laskimen mukaan yhtälö on epätosi, siis yhtälön kuvaajan kasvunopeus ei ole missään kohdassa 0, eli mikään käyrälle asetetuista tangenteista ei ole vaakasuora</div> 2019-10-09T09:39:05+03:00 https://peda.net/id/37fd0608ea5 2019-10-09T09:37:09+03:00 a) joo<br/> b) ei<br/> c) joo 2019-10-09T09:37:09+03:00 https://peda.net/id/19d5306aea5 2019-10-21T13:07:21+03:00 tangentti on vaakasuora, kun derivaattafunktion arvo on 0<br/> derivoidaan paraabelin yhtälö<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D-x%5E2%2B2x%2B3" alt="f\left(x\right)=-x^2+2x+3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D-2x%2B2" alt="f'\left(x\right)=-2x+2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-2x%2B2%3D0" alt="-2x+2=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-2x%3D-2" alt="-2x=-2"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1" alt="x=1"/></div> <br/> pisteeseen (1,0) 2019-10-09T09:36:18+03:00 https://peda.net/id/bd166510ea5 2019-10-09T09:33:43+03:00 <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D4x%5E%7B11%7D-11x%5E4" alt="f\left(x\right)=4x^{11}-11x^4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D44x%5E%7B10%7D-44x%5E3" alt="f'\left(x\right)=44x^{10}-44x^3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(-1%5Cright)%3D44%2B44%3D88" alt="f'\left(-1\right)=44+44=88"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D2x%5E%7B13%7D-13x%5E2" alt="g\left(x\right)=2x^{13}-13x^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g'%5Cleft(x%5Cright)%3D26x%5E%7B12%7D-26x" alt="g'\left(x\right)=26x^{12}-26x"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(-1%5Cright)%3D26%2B26%3D52" alt="g\left(-1\right)=26+26=52"/></div> <div>funktion f muutosnopeus on suurempi</div> 2019-10-09T09:33:43+03:00 https://peda.net/id/0e35a7b8ea5 2019-10-09T09:53:45+03:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(x%5E3-x%5E2-x%5Cright)%3D3x%5E2-2x-1" alt="D\left(x^3-x^2-x\right)=3x^2-2x-1"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft(-8x%2B3%5Cright)%3D-8" alt="\frac{d}{dx}\left(-8x+3\right)=-8"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(-5x%5E3-7x%2B1%5Cright)%3D-15x%5E2-7" alt="D\left(-5x^3-7x+1\right)=-15x^2-7"/><br/> 2019-10-09T09:28:49+03:00 https://peda.net/id/affcbdc6ea5 2019-10-09T09:26:11+03:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E4" alt="f\left(x\right)=x^4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D4x%5E3" alt="f'\left(x\right)=4x^3"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E3%2B1" alt="f\left(x\right)=x^3+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D3x%5E2" alt="f'\left(x\right)=3x^2"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D-3x" alt="f\left(x\right)=-3x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D-3" alt="f'\left(x\right)=-3"/><!--filtered attribute: style="display: inline;"--> 2019-10-09T09:26:11+03:00 https://peda.net/id/7ab35742ea5 2019-10-09T09:24:42+03:00 Voidaanko tiedosta f'(1)=2 päätellä, että funktion f derivaattafunktion kuvaaja on nouseva suora?<br/> ei voida, se voi olla mitä vain<br/> kuitenkin funktion f kuvaaja on nopeudella 2 nouseva suora kohdassa x=1 2019-10-09T09:24:42+03:00 https://peda.net/id/45d34384ea5 2019-10-09T09:23:13+03:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D3x%5E2" alt="f\left(x\right)=3x^2"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(1%5Cright)%3D%5Clim_%7Bx%5Crightarrow1%7D%5C%20%5Cfrac%7B3x%5E2-3%7D%7Bx-1%7D%3D%5Cfrac%7B3%5Cleft(x-1%5Cright)%5Cleft(x%2B1%5Cright)%7D%7Bx-1%7D%3D3x%2B3%3D6" alt="f'\left(1\right)=\lim_{x\rightarrow1}\ \frac{3x^2-3}{x-1}=\frac{3\left(x-1\right)\left(x+1\right)}{x-1}=3x+3=6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(a%5Cright)%3D%5Clim_%7Bx%5Crightarrow%20a%7D%5C%20%5Cfrac%7B3x%5E2-3a%7D%7Bx-a%7D%3D%5Cfrac%7B3%5Cleft(x-a%5Cright)%5Cleft(x%2Ba%5Cright)%7D%7Bx-a%7D%3D3x%2B3a%3D6a" alt="f'\left(a\right)=\lim_{x\rightarrow a}\ \frac{3x^2-3a}{x-a}=\frac{3\left(x-a\right)\left(x+a\right)}{x-a}=3x+3a=6a"/></div> <div>b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D6x" alt="f'\left(x\right)=6x"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(-2%5Cright)%3D6%5Cleft(-2%5Cright)%3D-12" alt="f'\left(-2\right)=6\left(-2\right)=-12"/></div> 2019-10-09T09:23:13+03:00 https://peda.net/id/d7646accea5 2019-10-09T09:20:08+03:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(2x%5E5-x%2B3%5Cright)%3D10x%5E4-1" alt="D\left(2x^5-x+3\right)=10x^4-1"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(4x%5E2%2B5x%5Cright)%3D8x%2B5" alt="D\left(4x^2+5x\right)=8x+5"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(4x%5E7%2Bx-1%5Cright)%3D28x%5E6%2B1" alt="D\left(4x^7+x-1\right)=28x^6+1"/> 2019-10-09T09:20:08+03:00 https://peda.net/id/a2ecfce6ea5 2019-10-09T09:18:40+03:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D-2x%5E3%2B3x%5E2%2Bx" alt="f\left(x\right)=-2x^3+3x^2+x"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D-6x%5E2%2B6x%2B1" alt="f'\left(x\right)=-6x^2+6x+1"/><br/> <br/> b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(2%5Cright)%3D-24%2B12%2B1%3D-11" alt="f'\left(2\right)=-24+12+1=-11"/></div> 2019-10-09T09:18:40+03:00 https://peda.net/id/4d983a12ea5 2019-10-09T09:16:17+03:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E7" alt="f\left(x\right)=x^7"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D7x%5E6" alt="f'\left(x\right)=7x^6"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E5%2B2" alt="f\left(x\right)=x^5+2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D5x%5E4" alt="f'\left(x\right)=5x^4"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D6x" alt="f\left(x\right)=6x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D6" alt="f'\left(x\right)=6"/></div> 2019-10-09T09:16:17+03:00 https://peda.net/id/006acb38ea5 2019-10-09T09:14:07+03:00 a)<br/> f(x)=3x<br/> f'(x)=3<br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow%20a%7D%5C%20%5Cfrac%7B3x-3a%7D%7Bx-a%7D%3D%5Clim_%7Bx%5Crightarrow%20a%7D%5Cfrac%7B3%5Cleft(x-a%5Cright)%7D%7Bx-a%7D%3D3" alt="f'\left(x\right)=\lim_{x\rightarrow a}\ \frac{3x-3a}{x-a}=\lim_{x\rightarrow a}\frac{3\left(x-a\right)}{x-a}=3"/> 2019-10-09T09:14:07+03:00 https://peda.net/id/5f67bc24ea5 2019-10-09T09:02:28+03:00 <div>Funktion f derivaattafunktio on <img src="https://math-demo.abitti.fi/math.svg?latex=f'" alt="f'"/></div> <div>ja <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)" alt="f'\left(x\right)"/> on derivaatan arvo kohdassa x</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3An" alt="f'\left(x\right):n"/> määrittelyjoukko muodostuu niistä pisteistä, joissa f on derivoituva</div> <div> </div> <div> </div> <div>esim</div> <div>määritä f(x) derivaattafunktio f'(x)</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E2" alt="f\left(x\right)=x^2"/></div> <div>määritetään derivaatta kohdassa a</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(a%5Cright)%3D%5Clim_%7Bx%5Crightarrow%20a%7D%5Cfrac%7Bf%5Cleft(x%5Cright)-f%5Cleft(a%5Cright)%7D%7Bx-a%7D%3D%5Clim_%7B%5Crightarrow%20a%7D%5Cfrac%7Bx%5E2-a%5E2%7D%7Bx-a%7D%3D%5Cfrac%7B%5Cleft(x-a%5Cright)%5Cleft(x%2Ba%5Cright)%7D%7Bx-a%7D%3D2a" alt="f'\left(a\right)=\lim_{x\rightarrow a}\frac{f\left(x\right)-f\left(a\right)}{x-a}=\lim_{\rightarrow a}\frac{x^2-a^2}{x-a}=\frac{\left(x-a\right)\left(x+a\right)}{x-a}=2a"/></div> <div>kohdassa a, derivaatta on 2a</div> <div>kohdassa x derivaatta on 2x, eli <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D2x" alt="f'\left(x\right)=2x"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E3" alt="f\left(x\right)=x^3"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(a%5Cright)%3D%5Clim_%7Bx%5Crightarrow%20a%7D%5Cfrac%7Bf%5Cleft(x%5Cright)-f%5Cleft(a%5Cright)%7D%7Bx-a%7D%3D%5Clim_%7B%5Crightarrow%20a%7D%5Cfrac%7Bx%5E3-a%5E3%7D%7Bx-a%7D%3D%5Cfrac%7B%5Cleft(x-a%5Cright)%5Cleft(x%2Ba%5Cright)%7D%7Bx-a%7D%3D3a%5E2" alt="f'\left(a\right)=\lim_{x\rightarrow a}\frac{f\left(x\right)-f\left(a\right)}{x-a}=\lim_{\rightarrow a}\frac{x^3-a^3}{x-a}=\frac{\left(x-a\right)\left(x+a\right)}{x-a}=3a^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D3x%5E2" alt="f'\left(x\right)=3x^2"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E4" alt="f\left(x\right)=x^4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(a%5Cright)%3D%5Clim_%7Bx%5Crightarrow%20a%7D%5Cfrac%7Bf%5Cleft(x%5Cright)-f%5Cleft(a%5Cright)%7D%7Bx-a%7D%3D%5Clim_%7B%5Crightarrow%20a%7D%5Cfrac%7Bx%5E4-a%5E4%7D%7Bx-a%7D%3D%5Cfrac%7B%5Cleft(x-a%5Cright)%5Cleft(x%2Ba%5Cright)%7D%7Bx-a%7D%3D4a%5E3" alt="f'\left(a\right)=\lim_{x\rightarrow a}\frac{f\left(x\right)-f\left(a\right)}{x-a}=\lim_{\rightarrow a}\frac{x^4-a^4}{x-a}=\frac{\left(x-a\right)\left(x+a\right)}{x-a}=4a^3"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D4x%5E3" alt="f'\left(x\right)=4x^3"/></div> <div> </div> <div>Lause <img src="https://math-demo.abitti.fi/math.svg?latex=x%5En" alt="x^n"/>on derivoituva ja </div> <div> </div> <div>Derivoimista voidaan merkitä myös</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bd%7D%7Bdx%7Dx%5En%3D%5Cfrac%7Bdx%5En%7D%7Bdx%7D%3Dnx%5E%7Bn-1%7D" alt="\frac{d}{dx}x^n=\frac{dx^n}{dx}=nx^{n-1}"/></div> <div> </div> <div>Lause</div> <div>kun k on vakio ja f ja g ovat derivoituvia</div> <div>a) <img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20k%3D0" alt="D\ k=0"/></div> <div>b) <img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20kx%3Dk" alt="D\ kx=k"/></div> <div>c) <img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cleft(f%5Cleft(x%5Cright)%2Bg%5Cleft(x%5Cright)%5Cright)%3DD%5C%20f%5Cleft(x%5Cright)%2BD%5C%20g%5Cleft(x%5Cright)" alt="D\ \left(f\left(x\right)+g\left(x\right)\right)=D\ f\left(x\right)+D\ g\left(x\right)"/></div> <div>d) <img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20k%5Ccdot%20f%5Cleft(x%5Cright)%3Dk%5C%20D%5C%20f%5Cleft(x%5Cright)" alt="D\ k\cdot f\left(x\right)=k\ D\ f\left(x\right)"/></div> <div>ESIM</div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(3x%5E2-4x%2B2%5Cright)%3DD%5Cleft(3x%5E2%5Cright)%2BD%5Cleft(-4x%5Cright)%2BD%5Cleft(2%5Cright)%3D3%5Ccdot2x-4%2B0%3D6x-4" alt="D\left(3x^2-4x+2\right)=D\left(3x^2\right)+D\left(-4x\right)+D\left(2\right)=3\cdot2x-4+0=6x-4"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D-3x%5E5%2B2x%5E3-5x" alt="g\left(x\right)=-3x^5+2x^3-5x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g'%5Cleft(x%5Cright)%3D-15x%5E4%2B6x%5E2-5" alt="g'\left(x\right)=-15x^4+6x^2-5"/></div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%5Cleft(x%5Cright)%3D-3x%5E4%2Bx-5" alt="h\left(x\right)=-3x^4+x-5"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h'%5Cleft(x%5Cright)%3D-12x%5E3%2B1" alt="h'\left(x\right)=-12x^3+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=h'%5Cleft(-1%5Cright)%3D13" alt="h'\left(-1\right)=13"/></div> 2019-10-09T09:02:27+03:00