https://peda.net/id/6e9f5f42433 2020-01-30T11:03:50+02:00 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/2d8c52d8442 2020-01-31T14:06:57+02:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cpi%3Ee%3E%5Cfrac%7B1%7D%7B2%7D%3E%5Cfrac%7B1%7D%7Be%7D%3E%5Cfrac%7B1%7D%7B3%7D%3E%5Cfrac%7B1%7D%7B%5Cpi%7D" alt="\pi&gt;e&gt;\frac{1}{2}&gt;\frac{1}{e}&gt;\frac{1}{3}&gt;\frac{1}{\pi}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e-1%3E%5Cfrac%7Be%7D%7B3%7D" alt="e-1&gt;\frac{e}{3}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e-1%5Capprox1%7B%2C%7D72" alt="e-1\approx1{,}72"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Be%7D%7B3%7D%5Capprox0%7B%2C%7D91" alt="\frac{e}{3}\approx0{,}91"/></div> 2020-01-31T14:06:57+02:00 https://peda.net/id/4bbe5108442 2020-01-31T14:00:38+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cfrac%7B1%7D%7B2%7De%5Ex%5Cleft(%5Csin%20x%2B%5Ccos%20x%5Cright)" alt="f\left(x\right)=\frac{1}{2}e^x\left(\sin x+\cos x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D%5Cfrac%7B1%7D%7B2%7De%5Ex%5Cleft(%5Ccos%20x-%5Csin%20x%5Cright)" alt="f'\left(x\right)=\frac{1}{2}e^x\left(\cos x-\sin x\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(0%5Cright)%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot1%5Ccdot%5Cleft(1-0%5Cright)%3D%5Cfrac%7B1%7D%7B2%7D" alt="f'\left(0\right)=\frac{1}{2}\cdot1\cdot\left(1-0\right)=\frac{1}{2}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E3e%5Ex" alt="f\left(x\right)=x^3e^x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3Dx%5E3e%5Ex%2B3x%5E2e%5Ex" alt="f'\left(x\right)=x^3e^x+3x^2e^x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D0" alt="f'\left(x\right)=0"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-3%5C%20tai%5C%20x%3D0" alt="x=-3\ tai\ x=0"/></div> 2020-01-31T14:00:38+02:00 https://peda.net/id/33e3003e442 2020-01-31T13:52:48+02:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(e%5E%7Bx%5E3-1%7D%5Cright)%3D3x%5E2e%5E%7Bx%5E3-1%7D" alt="D\left(e^{x^3-1}\right)=3x^2e^{x^3-1}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7Be%5Ex%7D%7Bx%7D%3D%5Cfrac%7Be%5Ex%5Ccdot%20x-1%5Ccdot%20e%5Ex%7D%7Bx%5E2%7D%3D%5Cfrac%7Be%5Ex%5Cleft(x-1%5Cright)%7D%7Bx%5E2%7D" alt="D\ \frac{e^x}{x}=\frac{e^x\cdot x-1\cdot e^x}{x^2}=\frac{e^x\left(x-1\right)}{x^2}"/></div> <div>c)</div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7Bx%2B1%7D%7Be%5Ex%7D%3D%5Cfrac%7B1%5Ccdot%20e%5E%7B2x%7D-2e%5E%7B2x%7D%5Ccdot%5Cleft(x%2B1%5Cright)%7D%7Be%5E%7B2x%7D%7D%3D%3D-x%2Be%5E%7B-x%7D" alt="D\ \frac{x+1}{e^x}=\frac{1\cdot e^{2x}-2e^{2x}\cdot\left(x+1\right)}{e^{2x}}==-x+e^{-x}"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cfrac%7B2%7D%7Be%5E%7B2x%7D%7D%3D2e%5E%7B-2x%7D%3D-4e%5E%7B-2x%7D" alt="D\frac{2}{e^{2x}}=2e^{-2x}=-4e^{-2x}"/></div> <div>e)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(x%5E2e%5E%7B2x%7D%5Cright)%3D2x%5Ccdot%20e%5E%7B2x%7D%2Bx%5E2%5Ccdot%20e%5E%7B2x%7D%3De%5E%7B2x%7D%5Cleft(x%5E2%2B2x%5Cright)" alt="D\left(x^2e^{2x}\right)=2x\cdot e^{2x}+x^2\cdot e^{2x}=e^{2x}\left(x^2+2x\right)"/></div> <div>f)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Cfrac%7Be%5Ex%2Bx%7D%7Bx%7D%5Cright)%3D%5Cfrac%7Bx%5Cleft(e%5Ex%2B1%5Cright)-1%5Cleft(e%5Ex%2Bx%5Cright)%7D%7Bx%5E2%7D%3D%5Cfrac%7B%5Cleft(x-1%5Cright)%5Cleft(e%5Ex%2Bx%5Cright)%7D%7Bx%5E2%7D" alt="D\left(\frac{e^x+x}{x}\right)=\frac{x\left(e^x+1\right)-1\left(e^x+x\right)}{x^2}=\frac{\left(x-1\right)\left(e^x+x\right)}{x^2}"/></div> 2020-01-31T13:52:48+02:00 https://peda.net/id/af3a0638441 2020-01-31T13:20:28+02:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(2e%5Ex%2B2x%5Cright)%3D2e%5Ex%2B2" alt="D\left(2e^x+2x\right)=2e^x+2"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(e%5E%7B3x%7D%2Be%5Cright)%3D3xe%5E%7B3x%7D" alt="D\left(e^{3x}+e\right)=3xe^{3x}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(%5Cfrac%7Be%5E%7B6x%7D%7D%7B3%7D%5Cright)%3D%5Cfrac%7B6e%5E%7B6x%7D%5Ccdot3-e%5E%7B6x%7D%5Ccdot0%7D%7B3%5E2%7D%3D%5Cfrac%7B18e%5E%7B6x%7D%7D%7B9%7D%3D2e%5E%7B6x%7D" alt="D\left(\frac{e^{6x}}{3}\right)=\frac{6e^{6x}\cdot3-e^{6x}\cdot0}{3^2}=\frac{18e^{6x}}{9}=2e^{6x}"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(xe%5Ex%5Cright)%3D1%5Ccdot%20e%5Ex%2Bx%5Ccdot%20e%5Ex%3D%5Cleft(1%2Bx%5Cright)e%5Ex" alt="D\left(xe^x\right)=1\cdot e^x+x\cdot e^x=\left(1+x\right)e^x"/></div> <div> </div> 2020-01-31T13:20:28+02:00 https://peda.net/id/d3d7a36a434 2020-01-30T11:42:27+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f(0)%3D1" alt="f(0)=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f(1)%3De" alt="f(1)=e"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex%3D1" alt="e^x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0" alt="x=0"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B-1%7D" alt="e^{-1}"/> 2020-01-30T11:42:27+02:00