https://peda.net/id/3b9ad8ae102 2019-11-26T10:10:39+02:00 https://peda.net/:static/535/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/621125c010f 2019-11-27T11:44:41+02:00 funktio on kasvava kun sen derivaatta on positiivinen<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%2B%5Ccos%20x-1" alt="f\left(x\right)=x+\cos x-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D-%5Csin%20x%2B1" alt="f'\left(x\right)=-\sin x+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D0" alt="f'\left(x\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Csin%20x%3D-1" alt="-\sin x=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D1" alt="\sin x=1"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot2%5Cpi" alt="x=\frac{\pi}{2}+n\cdot2\pi"/></div> <div>funktion derivaatta on aina positiivinen tai nolla, funktio on siis kasvava</div> <div>funktion muutosnopeus on nolla funktion nollakohdissa <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot2%5Cpi" alt="x=\frac{\pi}{2}+n\cdot2\pi"/><br/> <img src="https://i.imgur.com/4nG8b6x.png" alt=""/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E3%2B3x%2B2%5Csin%20x" alt="f\left(x\right)=x^3+3x+2\sin x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D3x%5E2%2B3%2B2%5Ccos%20x" alt="f'\left(x\right)=3x^2+3+2\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E2%2B3%2B2%5Ccos%20x%3D0" alt="3x^2+3+2\cos x=0"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos%20x%2B3x%5E2%3D-3" alt="2\cos x+3x^2=-3"/></div> <div>derivaatta ei saa arvoa nolla</div> <img src="https://i.imgur.com/CTFdbcv.png" alt=""/></div> 2019-11-27T11:43:36+02:00 https://peda.net/id/0ac0031e10f 2019-11-27T11:34:00+02:00 <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D3x%5Csin%20x" alt="f\left(x\right)=3x\sin x"/></div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%3D%5Cfrac%7B3%5Cpi%7D%7B2%7D%5Csin%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cfrac%7B3%5Cpi%7D%7B2%7D" alt="f\left(\frac{\pi}{2}\right)=\frac{3\pi}{2}\sin\frac{\pi}{2}=\frac{3\pi}{2}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D3x%5Ccos%2B3%5Csin%20x" alt="f'\left(x\right)=3x\cos+3\sin x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%3D3" alt="f'\left(\frac{\pi}{2}\right)=3"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D3x" alt="g\left(x\right)=3x"/> </div> 2019-11-27T11:34:00+02:00 https://peda.net/id/ebb1b19e10f 2019-11-27T11:25:58+02:00 <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D3-2x%2B6%5Csin%20x" alt="f\left(x\right)=3-2x+6\sin x"/></div> <div>tangentin halutaan olevan suoran <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D-5x%2B6%5C%20" alt="y=-5x+6\ "/>suuntainen</div> <div>tangentin kulmakertoimen, eli funktion derivaatan on siis oltava silloin -5</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D6%5Ccos%20x-2" alt="f'\left(x\right)=6\cos x-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=6%5Ccos%20x%3D-3" alt="6\cos x=-3"/></div> <div>yksi ratkaisu</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B2%5Cpi%7D%7B3%7D" alt="x=\frac{2\pi}{3}"/></div> <div>kaikki ratkaisut</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B2%5Cpi%7D%7B3%7D%2Bn%5Ccdot2%5Cpi%5C%20tai%5C%20-%5Cfrac%7B2%5Cpi%7D%7B3%7D%2Bn%5Ccdot2%5Cpi" alt="x=\frac{2\pi}{3}+n\cdot2\pi\ tai\ -\frac{2\pi}{3}+n\cdot2\pi"/></div> 2019-11-27T11:25:58+02:00 https://peda.net/id/524d4b8a10f 2019-11-27T11:21:41+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7B2%7D%7B%5Ccos%20x%7D%3D%5Cfrac%7B-%5Csin%20x-%5Ccos%20x%7D%7B%5Ccos%5E2x%7D" alt="D\ \frac{2}{\cos x}=\frac{-\sin x-\cos x}{\cos^2x}"/><br/> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7B2x%7D%7B%5Ccos%20x%7D%3D%5Cfrac%7B2%5Ccos%20x%2B2x%5Csin%20x%7D%7B%5Ccos%5E2x%7D%3D%5Cfrac%7B2%5Cleft(%5Ccos%20x%2Bx%5Csin%20x%5Cright)%7D%7B%5Ccos%5E2x%7D" alt="D\ \frac{2x}{\cos x}=\frac{2\cos x+2x\sin x}{\cos^2x}=\frac{2\left(\cos x+x\sin x\right)}{\cos^2x}"/> <br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cfrac%7B%5Ccos%20x%7D%7B2%7D%3D%5Cfrac%7B-2%5Csin%20x%7D%7B4%7D%3D%5Cfrac%7B-%5Csin%20x%7D%7B2%7D" alt="D\frac{\cos x}{2}=\frac{-2\sin x}{4}=\frac{-\sin x}{2}"/></div> 2019-11-27T11:21:41+02:00 https://peda.net/id/334bb5f610f 2019-11-27T11:13:39+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(x%5E%7B10%7D%5Csin%20x%5Cright)" alt="D\left(x^{10}\sin x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E%7B10%7D%7B%2C%7D%5C%20f'%5Cleft(x%5Cright)%3D10x%5E9" alt="f\left(x\right)=x^{10}{,}\ f'\left(x\right)=10x^9"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Csin%20x%7B%2C%7D%5C%20f'%5Cleft(x%5Cright)%3D%5Ccos%20x" alt="f\left(x\right)=\sin x{,}\ f'\left(x\right)=\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(f%5Cleft(x%5Cright)g%5Cleft(x%5Cright)%5Cright)%3Df'%5Cleft(x%5Cright)g%5Cleft(x%5Cright)%2Bf%5Cleft(x%5Cright)g'%5Cleft(x%5Cright)" alt="D\left(f\left(x\right)g\left(x\right)\right)=f'\left(x\right)g\left(x\right)+f\left(x\right)g'\left(x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(x%5E%7B10%7D%5Csin%20x%5Cright)%3D10x%5E9%5Csin%20x%2Bx%5E%7B10%7D%5Ccos%20x" alt="D\left(x^{10}\sin x\right)=10x^9\sin x+x^{10}\cos x"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(x%5E%7B10%7D%2B%5Ccos%20x%5Cright)%3D10x%5E9-%5Csin%20x" alt="D\left(x^{10}+\cos x\right)=10x^9-\sin x"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Csin%20x%5Ccos%20x%5Cright)" alt="D\left(\sin x\cos x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Csin%20x%7B%2C%7D%5C%20f'%5Cleft(x%5Cright)%3D%5Ccos%20x" alt="f\left(x\right)=\sin x{,}\ f'\left(x\right)=\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D%5Ccos%20x%7B%2C%7D%5C%20g'%5Cleft(x%5Cright)%3D-%5Csin%20x" alt="g\left(x\right)=\cos x{,}\ g'\left(x\right)=-\sin x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Csin%20x%5Ccos%20x%5Cright)%3D%5Ccos%5E2x-%5Csin%5E2x" alt="D\left(\sin x\cos x\right)=\cos^2x-\sin^2x"/> 2019-11-27T11:13:39+02:00 https://peda.net/id/e3a512d410f 2019-11-27T10:49:57+02:00 funktion kuvaajalle piirretty tangentti on kohtisuora, kun derivaatta kohdassa on nolla<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2x%2B%5Ccos%20x" alt="f\left(x\right)=2x+\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D-%5Csin%20x%2B2" alt="f'\left(x\right)=-\sin x+2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D0" alt="f'\left(x\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Csin%20x%3D-2" alt="-\sin x=-2"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D2" alt="\sin x=2"/></div> <div>sini saa arvoja vain väliltä [0,1]</div> funktion derivaatalla ei ole nollakohtia, jolloin funktiolle piirretyt tangentit eivät ole koskaan vaakasuoria 2019-11-27T10:49:57+02:00 https://peda.net/id/5e9c80f410f 2019-11-27T10:46:14+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=kuvaaja%5C%201%5C%20f'%5Cleft(x%5Cright)" alt="kuvaaja\ 1\ f'\left(x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=kuvaaja%5C%202%5C%20f%5Cleft(x%5Cright)" alt="kuvaaja\ 2\ f\left(x\right)"/><br/> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D0%7B%2C%7D%5C%20%5Cleft%5B0%7B%2C%7D%5Cpi%5Cright%5D" alt="f'\left(x\right)=0{,}\ \left[0{,}\pi\right]"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0%7B%2C%7D%5C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%7B%2C%7D%5C%20%5Cpi" alt="x=0{,}\ \frac{\pi}{2}{,}\ \pi"/><br/> c)<br/> <div>funktion f suurin muutosnopeus on 2</div> <div>se saavutetaan välillä <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%5B-%5Cfrac%7B%5Cpi%7D%7B2%7D%7B%2C%7D%5C%20%5Cfrac%7B3%5Cpi%7D%7B2%7D%5Cright%5D" alt="\left[-\frac{\pi}{2}{,}\ \frac{3\pi}{2}\right]"/>kohdissa <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20%5Cfrac%7B5%5Cpi%7D%7B4%7D" alt="x=\frac{\pi}{4}{,}\ \frac{5\pi}{4}"/></div> </div> 2019-11-27T10:46:14+02:00 https://peda.net/id/c18f6e0c10f 2019-11-27T10:41:50+02:00 a) <br/> <div>funktion arvo kohdassa nolla</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(0%5Cright)%3D1" alt="f\left(0\right)=1"/></div> <div>derivaattafunktion nollakohdat välillä [0,4π]</div> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D0%7B%2C%7D%5C%20%5Cleft%5B0%7B%2C%7D4%5Cpi%5Cright%5D" alt="f'\left(x\right)=0{,}\ \left[0{,}4\pi\right]"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%7B%2C%7D%5C%20%5Cfrac%7B3%5Cpi%7D%7B2%7D%7B%2C%7D%5C%20%5Cfrac%7B5%5Cpi%7D%7B2%7D%7B%2C%7D%5C%20%5Cfrac%7B7%5Cpi%7D%7B2%7D" alt="x=\frac{\pi}{2}{,}\ \frac{3\pi}{2}{,}\ \frac{5\pi}{2}{,}\ \frac{7\pi}{2}"/> <br/> b)<br/> funktion arvo kohdassa nolla<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2%5Csin%20x%2B1" alt="f\left(x\right)=2\sin x+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(0%5Cright)%3D2%5Ccdot0%2B1%3D1" alt="f\left(0\right)=2\cdot0+1=1"/><br/> derivaatan nollakohdat<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2%5Csin%20x%2B1" alt="f\left(x\right)=2\sin x+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D2%5Ccos%20x" alt="f'\left(x\right)=2\cos x"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos%20x%3D0" alt="2\cos x=0"/></div> yksi ratkaisu<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7Dtai%5C%20-%5Cfrac%7B%5Cpi%7D%7B2%7D" alt="x=\frac{\pi}{2}tai\ -\frac{\pi}{2}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot2%5Cpi%5C%20tai%5C%20x%3D-%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot2%5Cpi" alt="x=\frac{\pi}{2}+n\cdot2\pi\ tai\ x=-\frac{\pi}{2}+n\cdot2\pi"/></div> <div>voidaan yhdistää</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot%5Cpi" alt="x=\frac{\pi}{2}+n\cdot\pi"/></div> <div>lasketaan seuraavaksi kaikki nollakohdat annetulla välillä sijoittamalla arvoja n</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D-%5Cpi%3D-%5Cfrac%7B%5Cpi%7D%7B2%7D%5C%20hyl." alt="x=\frac{\pi}{2}-\pi=-\frac{\pi}{2}\ hyl."/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D" alt="x=\frac{\pi}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2B%5Cpi%3D%5Cfrac%7B3%5Cpi%7D%7B2%7D" alt="x=\frac{\pi}{2}+\pi=\frac{3\pi}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2B2%5Cpi%3D%5Cfrac%7B5%5Cpi%7D%7B2%7D" alt="x=\frac{\pi}{2}+2\pi=\frac{5\pi}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2B3%5Cpi%3D%5Cfrac%7B7%5Cpi%7D%7B2%7D" alt="x=\frac{\pi}{2}+3\pi=\frac{7\pi}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2B4%5Cpi%3D%5Cfrac%7B9%5Cpi%7D%7B2%7D%5C%20hyl." alt="x=\frac{\pi}{2}+4\pi=\frac{9\pi}{2}\ hyl."/></div> <div>kaikki derivaattafunktion nollakohdat välillä [0,4π] ovat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%7B%2C%7D%5C%20%5Cfrac%7B3%5Cpi%7D%7B2%7D%7B%2C%7D%5C%20%5Cfrac%7B5%5Cpi%7D%7B2%7D%7B%2C%7D%5C%20%5Cfrac%7B7%5Cpi%7D%7B2%7D" alt="x=\frac{\pi}{2}{,}\ \frac{3\pi}{2}{,}\ \frac{5\pi}{2}{,}\ \frac{7\pi}{2}"/></div> 2019-11-27T10:41:50+02:00 https://peda.net/id/c31d811610e 2019-11-27T10:27:34+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Ccos%20x" alt="f\left(x\right)=\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D-%5Csin%20x" alt="f'\left(x\right)=-\sin x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f''%5Cleft(x%5Cright)%3D-%5Ccos%20x" alt="f''\left(x\right)=-\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f''%5Cleft(%5Cfrac%7B%5Cpi%7D%7B3%7D%5Cright)%3D-%5Ccos%5Cfrac%7B%5Cpi%7D%7B3%7D%3D-%5Cfrac%7B1%7D%7B2%7D" alt="f''\left(\frac{\pi}{3}\right)=-\cos\frac{\pi}{3}=-\frac{1}{2}"/> 2019-11-27T10:27:34+02:00 https://peda.net/id/7503631a10e 2019-11-27T10:25:23+02:00 <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2%5Csin%20x-x" alt="f\left(x\right)=2\sin x-x"/></div> <div>a)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D2%5Ccos%20x-1" alt="f'\left(x\right)=2\cos x-1"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(0%5Cright)%3D2-1%3D1" alt="f'\left(0\right)=2-1=1"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D0" alt="f'\left(x\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos%20x-1%3D0" alt="2\cos x-1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D%5Cfrac%7B1%7D%7B2%7D" alt="\cos x=\frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B3%7D%2Bn%5Ccdot2%5Cpi" alt="x=\frac{\pi}{3}+n\cdot2\pi"/></div> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B%5Cpi%7D%7B3%7D%2Bn%5Ccdot2%5Cpi" alt="x=-\frac{\pi}{3}+n\cdot2\pi"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(3%5Cpi%5Cright)%3D2%5Ccos3%5Cpi-1%3D-3" alt="f'\left(3\pi\right)=2\cos3\pi-1=-3"/></div> 2019-11-27T10:25:23+02:00 https://peda.net/id/0442b658103 2019-11-26T11:42:09+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D4x-2%5Csin%20x" alt="f\left(x\right)=4x-2\sin x"/><br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D-2%5Ccos%20x%2B4" alt="f'\left(x\right)=-2\cos x+4"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(%5Cfrac%7B2%5Cpi%7D%7B3%7D%5Cright)%3D-2%5Cleft(-%5Cfrac%7B1%7D%7B2%7D%5Cright)%2B4%3D5" alt="f'\left(\frac{2\pi}{3}\right)=-2\left(-\frac{1}{2}\right)+4=5"/> 2019-11-26T11:42:09+02:00 https://peda.net/id/4eea48b6103 2019-11-26T11:38:44+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Csin%20x%3D%5Ccos%20x" alt="D\sin x=\cos x"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%205%5Csin%20x%2B%5Cpi%3D5%5Ccos%20x" alt="D\ 5\sin x+\pi=5\cos x"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%202x%2B7%5Ccos%20x%3D-7%5Csin%20x%2B2" alt="D\ 2x+7\cos x=-7\sin x+2"/><br/> d)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20x%5E4-5x%5E3-4%5Ccos%20x%3D4x%5E3-15x%5E2%2B4%5Csin%20x" alt="D\ x^4-5x^3-4\cos x=4x^3-15x^2+4\sin x"/><br/> e)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7B%5Csin%20x%7D%7B3%7D%3D%5Cfrac%7B3%5Ccos%20x-9%5Csin%20x%7D%7B9%7D" alt="D\ \frac{\sin x}{3}=\frac{3\cos x-9\sin x}{9}"/><br/> f)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7B%5Csin%20x-2%5Ccos%20x%7D%7B2%7D" alt="D\ \frac{\sin x-2\cos x}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Csin%20x-2%5Ccos%20x" alt="f\left(x\right)=\sin x-2\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D%5Ccos%20x%2B2%5Csin%20x" alt="f'\left(x\right)=\cos x+2\sin x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D2" alt="g\left(x\right)=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g'%5Cleft(x%5Cright)%3D0" alt="g'\left(x\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7Bf%5Cleft(x%5Cright)%7D%7Bg%5Cleft(x%5Cright)%7D%3D%5Cfrac%7Bf'%5Cleft(x%5Cright)g%5Cleft(x%5Cright)-f%5Cleft(x%5Cright)g'%5Cleft(x%5Cright)%7D%7B%5Cleft(g%5Cleft(x%5Cright)%5Cright)%5E2%7D%3D%5Cfrac%7B2%5Cleft(%5Ccos%20x%2B2%5Csin%20x%5Cright)%5Ccdot0%5Cleft(%5Ccos%20x%2B2%5Csin%20x%5Cright)%7D%7B4%7D%3D%5Cfrac%7B2%5Ccos%20x%2B4%5Csin%20x%7D%7B4%7D" alt="D\ \frac{f\left(x\right)}{g\left(x\right)}=\frac{f'\left(x\right)g\left(x\right)-f\left(x\right)g'\left(x\right)}{\left(g\left(x\right)\right)^2}=\frac{2\left(\cos x+2\sin x\right)\cdot0\left(\cos x+2\sin x\right)}{4}=\frac{2\cos x+4\sin x}{4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B%5Ccos%20x%2B2%5Csin%20x%7D%7B2%7D" alt="=\frac{\cos x+2\sin x}{2}"/> 2019-11-26T11:37:05+02:00 https://peda.net/id/146b4f9c102 2019-11-26T11:28:18+02:00 <div>Lause</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Csin%20x%3D%5Ccos%20x" alt="D\sin x=\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Ccos%20x%3D-%5Csin%20x" alt="D\cos x=-\sin x"/></div> <div> </div> <div>esimerkki</div> <div>määritä</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)" alt="f'\left(x\right)"/>, kun <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2%5Csin%20x%2B%5Ccos%20x" alt="f\left(x\right)=2\sin x+\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D2%5Ccos%20x-%5Csin%20x" alt="f'\left(x\right)=2\cos x-\sin x"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%7B%2C%7D%5C%20kun%5C%20f%5Cleft(x%5Cright)%3D%5Csin%20x-3%5Ccos%20x" alt="f'\left(\frac{\pi}{2}\right){,}\ kun\ f\left(x\right)=\sin x-3\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D%5Ccos%20x%2B3%5Csin%20x" alt="f'\left(x\right)=\cos x+3\sin x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%3D0%2B3%5Ccdot1%3D3" alt="f'\left(\frac{\pi}{2}\right)=0+3\cdot1=3"/></div> <div> </div> <div>esimerkki</div> <div>derivoi</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%5Cleft(x%5Cright)%3Dx%5E3%5Ccos%20x" alt="h\left(x\right)=x^3\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=h'%5Cleft(x%5Cright)%3D3x%5E2%5Ccos%20x-x%5E3%5Csin%20x" alt="h'\left(x\right)=3x^2\cos x-x^3\sin x"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%5Cleft(x%5Cright)%3D%5Cfrac%7B%5Csin%20x%7D%7B%5Ccos%20x%7D" alt="h\left(x\right)=\frac{\sin x}{\cos x}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=h'%5Cleft(x%5Cright)%3D%5Cfrac%7Bf'%5Cleft(x%5Cright)g%5Cleft(x%5Cright)-f%5Cleft(x%5Cright)g'%5Cleft(x%5Cright)%7D%7B%5Cleft(g%5Cleft(x%5Cright)%5Cright)%5E2%7D%3D%5Cfrac%7B%5Ccos%5E2x%2B%5Csin%5E2x%7D%7B%5Ccos%5E2x%7D%3D%5Cfrac%7B1%7D%7B%5Ccos%5E2x%7D" alt="h'\left(x\right)=\frac{f'\left(x\right)g\left(x\right)-f\left(x\right)g'\left(x\right)}{\left(g\left(x\right)\right)^2}=\frac{\cos^2x+\sin^2x}{\cos^2x}=\frac{1}{\cos^2x}"/></div> 2019-11-26T11:28:18+02:00