https://peda.net/id/71be99ca311 2020-01-07T08:32:15+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/3bd26392311 2020-01-07T09:53:57+02:00 1.<br/> a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot%5Cleft(-3%5Cright)%5E2%3D2%5Ccdot9%3D18" alt="2\cdot\left(-3\right)^2=2\cdot9=18"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-2%5E4%3D-16" alt="-2^4=-16"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4%5Ccdot2%5E%7B-3%7D%3D4%5Ccdot%5Cfrac%7B1%7D%7B8%7D%3D%5Cfrac%7B1%7D%7B2%7D" alt="4\cdot2^{-3}=4\cdot\frac{1}{8}=\frac{1}{2}"/></div> 2.<br/> a) <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E5%2B5x%3Dx%5Cleft(x%5E4%2B5%5Cright)" alt="x^5+5x=x\left(x^4+5\right)"/><br/> b) <img src="https://math-demo.abitti.fi/math.svg?latex=4x%5E3%2B2x%5E2%3D2x%5E2%5Cleft(2x%2B1%5Cright)" alt="4x^3+2x^2=2x^2\left(2x+1\right)"/><br/> c) <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cleft(x%2B1%5Cright)%2B4x%5Cleft(x%2B1%5Cright)%3D%5Cleft(x%2B1%5Cright)%5Cleft(2%2B4x%5Cright)" alt="2\left(x+1\right)+4x\left(x+1\right)=\left(x+1\right)\left(2+4x\right)"/><br/> <br/> 3.<br/> a) f(3)=2<br/> b) f'(3)=1<br/> c) f'(x)=0, kun x=1 tai x=4<br/> d) f'(x)&gt;0, kun <img src="https://math-demo.abitti.fi/math.svg?latex=1%3Cx%3C4" alt="1&lt;x&lt;4"/><br/> <br/> 4.<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E3%2Bx" alt="f\left(x\right)=x^3+x"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D3x%5E2%2B1" alt="f'\left(x\right)=3x^2+1"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cfrac%7B1%7D%7Bx%5E4%7D%7B%2C%7D%5C%20x%5Cne0" alt="f\left(x\right)=\frac{1}{x^4}{,}\ x\ne0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E%7B-4%7D" alt="f\left(x\right)=x^{-4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D-4x%5E%7B-5%7D%3D-%5Cfrac%7B4%7D%7Bx%5E5%7D" alt="f'\left(x\right)=-4x^{-5}=-\frac{4}{x^5}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E2%5Csin%20x" alt="f\left(x\right)=x^2\sin x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f'%5Cleft(x%5Cright)%3D2x%5Csin%20x%2Bx%5E2%5Ccos%20x" alt="f'\left(x\right)=2x\sin x+x^2\cos x"/><br/> <br/> 5.<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%5Cright)%3Du'%5Cleft(s%5Cleft(x%5Cright)%5Cright)s'%5Cleft(x%5Cright)" alt="D\left(u\left(s\left(x\right)\right)\right)=u'\left(s\left(x\right)\right)s'\left(x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cleft(-3x%2B2%5Cright)%5E%7B10%7D" alt="f\left(x\right)=\left(-3x+2\right)^{10}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3Dx%5E%7B10%7D" alt="u\left(x\right)=x^{10}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D-3x%2B2" alt="s\left(x\right)=-3x+2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u'%5Cleft(x%5Cright)%3D10x%5E9" alt="u'\left(x\right)=10x^9"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s'%5Cleft(x%5Cright)%3D-3" alt="s'\left(x\right)=-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%5Cright)%3D-30%5Cleft(-3x%2B2%5Cright)%5E9" alt="D\left(u\left(s\left(x\right)\right)\right)=-30\left(-3x+2\right)^9"/></div> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%5Cright)%3Du'%5Cleft(s%5Cleft(x%5Cright)%5Cright)s'%5Cleft(x%5Cright)" alt="D\left(u\left(s\left(x\right)\right)\right)=u'\left(s\left(x\right)\right)s'\left(x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cleft(2x%5E2-3%5Cright)%5E4" alt="f\left(x\right)=\left(2x^2-3\right)^4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3Dx%5E4" alt="u\left(x\right)=x^4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D2x%5E2-3" alt="s\left(x\right)=2x^2-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u'%5Cleft(x%5Cright)%3D4x%5E3" alt="u'\left(x\right)=4x^3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s'%5Cleft(x%5Cright)%3D4x" alt="s'\left(x\right)=4x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%5Cright)%3D4%5Cleft(2x%5E2-3%5Cright)%5E3%5Ccdot4x" alt="D\left(u\left(s\left(x\right)\right)\right)=4\left(2x^2-3\right)^3\cdot4x"/><br/> <br/> 6. <br/> a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7B49%7D%3D%5Cpm7" alt="\sqrt{49}=\pm7"/><br/> b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B3%5D%7B-8%7D%3D-2" alt="\sqrt[3]{-8}=-2"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B4%5D%7B16%7D%3D%5Cpm2" alt="\sqrt[4]{16}=\pm2"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7B8%7D%3D8%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D" alt="\sqrt{8}=8^{\frac{1}{2}}"/></div> <div>e)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7Ba%5E2%7D%3D%5Cleft%7Ca%5Cright%7C" alt="\sqrt{a^2}=\left|a\right|"/></div> <br/> 7.<br/> a)<br/> 2<br/> b) <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7B0%7B%2C%7D1%7D" alt="\sqrt{0{,}1}"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7Bx%2B1%7D" alt="\sqrt{x+1}"/><!--filtered attribute: style="display: inline;"--><br/> <br/> 8.<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ex%3D32" alt="2^x=32"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=32%3D2%5Ccdot2%5Ccdot2%5Ccdot2%5Ccdot2%3D2%5E5" alt="32=2\cdot2\cdot2\cdot2\cdot2=2^5"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D5" alt="x=5"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=8%5Ex%3D4%5E6" alt="8^x=4^6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4%5E6%3D4096" alt="4^6=4096"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4096%3D8%5Ccdot8%5Ccdot8%5Ccdot8%3D8%5E4" alt="4096=8\cdot8\cdot8\cdot8=8^4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D4" alt="x=4"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ex%5Ccdot5%5Ex%3D1000" alt="2^x\cdot5^x=1000"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ex%5Ccdot5%5Ex%3D10%5Ex" alt="2^x\cdot5^x=10^x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1000%3D10%5Ex" alt="1000=10^x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1000%3D10%5Ccdot10%5Ccdot10%3D10%5E3" alt="1000=10\cdot10\cdot10=10^3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D3" alt="x=3"/> 2020-01-07T08:59:22+02:00