https://peda.net/id/b11897483c1 2020-01-21T09:29:00+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/c49635143dc 2020-01-23T11:36:01+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D1%2B%5Csqrt%7Bx%7D%3D1%2Bx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D" alt="f\left(x\right)=1+\sqrt{x}=1+x^{\frac{1}{2}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D%5Cfrac%7B1%7D%7B2%7Dx%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D" alt="f'\left(x\right)=\frac{1}{2}x^{-\frac{1}{2}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(4%5Cright)%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cfrac%7B1%7D%7B%5Csqrt%7B4%7D%7D%3D%5Cfrac%7B1%7D%7B4%7D%5C%20" alt="f'\left(4\right)=\frac{1}{2}\cdot\frac{1}{\sqrt{4}}=\frac{1}{4}\ "/><br/> <div>tangentti</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B4%7Dx%2B2" alt="\frac{1}{4}x+2"/></div> <div>b)<br/> normaali</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B4%7D%5Ccdot%20z%3D-1" alt="\frac{1}{4}\cdot z=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=z%3D-4" alt="z=-4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-4x%2B19" alt="-4x+19"/></div> 2020-01-23T11:36:01+02:00 https://peda.net/id/5da481f43dc 2020-01-23T11:25:59+02:00 muutosnopeus nähdään derivaattafunktion arvosta<br/> derivoidaan funktio<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D3x%2B%5Cfrac%7B3%7D%7B%5Csqrt%5B3%5D%7Bx%7D%7D%7B%2C%7D%5C%20x%3E0" alt="f\left(x\right)=3x+\frac{3}{\sqrt[3]{x}}{,}\ x&gt;0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D3x%2B3x%5E%7B-%5Cfrac%7B1%7D%7B3%7D%7D" alt="f\left(x\right)=3x+3x^{-\frac{1}{3}}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D3-x%5E%7B-%5Cfrac%7B4%7D%7B3%7D%7D" alt="f'\left(x\right)=3-x^{-\frac{4}{3}}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(%5Cfrac%7B1%7D%7B8%7D%5Cright)%3D3-%5Cfrac%7B1%7D%7B%5Csqrt%5B3%5D%7B%5Cleft(%5Cfrac%7B1%7D%7B8%7D%5Cright)%5E4%7D%7D%3D3-%5Cfrac%7B1%7D%7B%5Csqrt%5B3%5D%7B%5Cfrac%7B1%7D%7B4096%7D%7D%7D%3D3-16%3D-13" alt="f'\left(\frac{1}{8}\right)=3-\frac{1}{\sqrt[3]{\left(\frac{1}{8}\right)^4}}=3-\frac{1}{\sqrt[3]{\frac{1}{4096}}}=3-16=-13"/><br/> <br/> </div> 2020-01-23T11:25:59+02:00 https://peda.net/id/6173ab943dc 2020-01-23T11:18:56+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B2%7D%7B%5Csqrt%7Bx%7D%7D%7B%2C%7D%5C%20x%3E0" alt="-\frac{2}{\sqrt{x}}{,}\ x&gt;0"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(-2x%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D%5Cright)%3D-2%5Ccdot%5Cleft(-%5Cfrac%7B1%7D%7B2%7D%5Cright)x%5E%7B-%5Cfrac%7B3%7D%7B2%7D%7D%3Dx%5E%7B-%5Cfrac%7B3%7D%7B2%7D%7D" alt="D\left(-2x^{-\frac{1}{2}}\right)=-2\cdot\left(-\frac{1}{2}\right)x^{-\frac{3}{2}}=x^{-\frac{3}{2}}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B3%5D%7Bx%5E2%7D%3Dx%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D" alt="\sqrt[3]{x^2}=x^{\frac{2}{3}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(x%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%5Cright)%3D%5Cfrac%7B2%7D%7B3%7Dx%5E%7B-%5Cfrac%7B1%7D%7B3%7D%7D%3D%5Cfrac%7B2%7D%7B3%5Csqrt%5B3%5D%7Bx%7D%7D" alt="D\left(x^{\frac{2}{3}}\right)=\frac{2}{3}x^{-\frac{1}{3}}=\frac{2}{3\sqrt[3]{x}}"/></div> <div>c)</div> <div> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%5E2%7D%7B3%5Csqrt%7Bx%7D%7D%3D%5Cfrac%7B1%7D%7B3%7Dx%5E2%5Ccdot%20x%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D" alt="\frac{x^2}{3\sqrt{x}}=\frac{1}{3}x^2\cdot x^{-\frac{1}{2}}"/><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=f%5Cleft(x%5Cright)%3Dx%5E2" alt="f\left(x\right)=x^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D2x" alt="f'\left(x\right)=2x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3Dx%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D" alt="g\left(x\right)=x^{-\frac{1}{2}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g'%5Cleft(x%5Cright)%3D-%5Cfrac%7B1%7D%7B2%7Dx%5E%7B-%5Cfrac%7B3%7D%7B2%7D%7D" alt="g'\left(x\right)=-\frac{1}{2}x^{-\frac{3}{2}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Cfrac%7B1%7D%7B3%7Dx%5E2x%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D%5Cright)%3D2x%5Ccdot%20x%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D%2Bx%5E2%5Ccdot%5Cleft(-%5Cfrac%7B1%7D%7B2%7Dx%5E%7B-%5Cfrac%7B3%7D%7B2%7D%7D%5Cright)%3D2x%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D-%5Cfrac%7B1%7D%7B2%7Dx%5E%7B-3%7D%3D%5Cfrac%7B2%7D%7B%5Csqrt%7Bx%7D%7D-%5Cfrac%7B1%7D%7B2x%5E3%7D" alt="D\left(\frac{1}{3}x^2x^{-\frac{1}{2}}\right)=2x\cdot x^{-\frac{1}{2}}+x^2\cdot\left(-\frac{1}{2}x^{-\frac{3}{2}}\right)=2x^{-\frac{1}{2}}-\frac{1}{2}x^{-3}=\frac{2}{\sqrt{x}}-\frac{1}{2x^3}"/> 2020-01-23T11:18:56+02:00 https://peda.net/id/b94d844a3db 2020-01-23T11:07:04+02:00 a) <br/> sininen käyrä kuvaa funktiota f<br/> punainen käyrä kuvaa derivaattafunktiota f'<br/> b) <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)%3D0" alt="f\left(1\right)=0"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(1%5Cright)%3D0%7B%2C%7D5" alt="f'\left(1\right)=0{,}5"/></div> <div>c)</div> <div>funktion f ääriarvot välillä [0,2]</div> <div>pakallinen minimi <img src="https://math-demo.abitti.fi/math.svg?latex=%5Capprox-0%7B%2C%7D2" alt="\approx-0{,}2"/> kohdassa <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox0%7B%2C%7D3" alt="x\approx0{,}3"/></div> <div> </div> 2020-01-23T11:07:04+02:00 https://peda.net/id/4b49e1703db 2020-01-23T10:49:40+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Csqrt%7Bx%5E2%2B1%7D%7B%2C%7D%5C%20x%5Cin%5Cmathbb%7BR%7D" alt="f\left(x\right)=\sqrt{x^2+1}{,}\ x\in\mathbb{R}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7Bx%5E2%2B1%7D%3D%5Csqrt%7Bx%5E2%7D%2B%5Csqrt%7B1%7D%3Dx%2B1" alt="\sqrt{x^2+1}=\sqrt{x^2}+\sqrt{1}=x+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D%5Cfrac%7Bx%7D%7B%5Csqrt%7Bx%5E2%2B1%7D%7D" alt="f'\left(x\right)=\frac{x}{\sqrt{x^2+1}}"/><br/> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D%5Cfrac%7Bx%7D%7B%5Csqrt%7Bx%5E2%2B1%7D%7D" alt="f'\left(x\right)=\frac{x}{\sqrt{x^2+1}}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(0%5Cright)%3D%5Cfrac%7B0%7D%7B%5Csqrt%7B0%5E2%2B1%7D%7D%3D%5Cfrac%7B0%7D%7B%5Cleft%7C1%5Cright%7C%7D%3D0" alt="f'\left(0\right)=\frac{0}{\sqrt{0^2+1}}=\frac{0}{\left|1\right|}=0"/></div> <div>c)</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=x%3D0" alt="x=0"/> 2020-01-23T10:49:40+02:00 https://peda.net/id/5a775bf63db 2020-01-23T10:42:56+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D4%5Csqrt%7Bx%7D%7B%2C%7D%5C%20x%3E0" alt="f\left(x\right)=4\sqrt{x}{,}\ x&gt;0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D4x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D" alt="f\left(x\right)=4x^{\frac{1}{2}}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D4%5Ccdot%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20x%5E%7B%5Cfrac%7B1%7D%7B2%7D-1%7D%3D2x%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D%3D%5Cfrac%7B2%7D%7B%5Csqrt%7Bx%7D%7D" alt="f'\left(x\right)=4\cdot\frac{1}{2}\cdot x^{\frac{1}{2}-1}=2x^{-\frac{1}{2}}=\frac{2}{\sqrt{x}}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E2%5Csqrt%7Bx%7D%3Dx%5E2x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D" alt="f\left(x\right)=x^2\sqrt{x}=x^2x^{\frac{1}{2}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D%5Cfrac%7B5%7D%7B2%7Dx%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D" alt="f'\left(x\right)=\frac{5}{2}x^{\frac{3}{2}}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cfrac%7B6%7D%7B%5Csqrt%7Bx%7D%7D%3D6x%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D" alt="f\left(x\right)=\frac{6}{\sqrt{x}}=6x^{-\frac{1}{2}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D6%5Ccdot%5Cleft(-%5Cfrac%7B1%7D%7B2%7D%5Cright)%5Ccdot%20x%5E%7B-%5Cfrac%7B1%7D%7B2%7D-1%7D%3D-3x%5E%7B-%5Cfrac%7B3%7D%7B2%7D%7D" alt="f'\left(x\right)=6\cdot\left(-\frac{1}{2}\right)\cdot x^{-\frac{1}{2}-1}=-3x^{-\frac{3}{2}}"/></div> 2020-01-23T10:42:56+02:00