https://peda.net/id/f11a0e660d1 2019-11-22T12:36:28+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/e8b66b460d2 2019-11-22T20:08:26+02:00 202<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2%5Csin%20x-1" alt="f\left(x\right)=2\sin x-1"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(3%5Cpi%5Cright)%3D2%5Csin3%5Cpi-1" alt="f\left(3\pi\right)=2\sin3\pi-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Csin%5Cpi-1" alt="2\sin\pi-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot0-1" alt="2\cdot0-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(3%5Cpi%5Cright)%3D-1" alt="f\left(3\pi\right)=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)" alt="f\left(-\frac{\pi}{2}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot%5Cleft(-1%5Cright)-1" alt="2\cdot\left(-1\right)-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%3D-3" alt="f\left(-\frac{\pi}{2}\right)=-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Cright)" alt="f\left(-\frac{\pi}{6}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot%5Cleft(-%5Cfrac%7B1%7D%7B2%7D%5Cright)-1" alt="2\cdot\left(-\frac{1}{2}\right)-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Cright)%3D-2" alt="f\left(-\frac{\pi}{6}\right)=-2"/></div> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(0%5Cright)" alt="f\left(0\right)"/><!--filtered attribute: style="display: inline;"--><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=f%5Cleft(0%5Cright)%3D2%5Csin0-1" alt="f\left(0\right)=2\sin0-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(0%5Cright)%3D-1" alt="f\left(0\right)=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D0%3D2%5Csin%20x-1" alt="f\left(x\right)=0=2\sin x-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Csin%20x%3D1" alt="2\sin x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D%5Cfrac%7B1%7D%7B2%7D" alt="\sin x=\frac{1}{2}"/><br/> yksi nollakohta on<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B6%7D" alt="x=\frac{\pi}{6}"/><br/> täydellinen ratkaisu<br/> <div>nollakohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B6%7D%2Bn%5Ccdot2%5Cpi" alt="x=\frac{\pi}{6}+n\cdot2\pi"/></div> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpi-%5Cfrac%7B%5Cpi%7D%7B6%7D%2Bn%5Ccdot2%5Cpi" alt="x=\pi-\frac{\pi}{6}+n\cdot2\pi"/></div> varmistetaan tulokset kuvasta<br/> <a href="https://peda.net/p/oskari.lahtinen/mtf/2sjk/teht%C3%A4vi%C3%A4/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/mtf/2sjk/teht%C3%A4vi%C3%A4/sieppaa-png:file/photo/b94a462e7add1b23e2b4da6302e9c187f7cb1689/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a><br/> 209<br/> a)<br/> <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/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-%5Cfrac%7B5%5Cpi%7D%7B6%7D%5Cright)%3Df%5Cleft(%5Cfrac%7B7%5Cpi%7D%7B6%7D%5Cright)" alt="f\left(-\frac{5\pi}{6}\right)=f\left(\frac{7\pi}{6}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos%5Cfrac%7B7%5Cpi%7D%7B6%7D-1" alt="2\cos\frac{7\pi}{6}-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot%5Cleft(-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Cright)-1" alt="2\cdot\left(-\frac{\sqrt{3}}{2}\right)-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Csqrt%7B3%7D-1" alt="-\sqrt{3}-1"/><br/> b)<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=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/> yksi ratkaisu<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B3%7D" alt="x=\frac{\pi}{3}"/></div> <div>täydellinen ratkaisu</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>tai<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>kun <img src="https://math-demo.abitti.fi/math.svg?latex=n%5Cin%5Cmathbb%7BZ%7D" alt="n\in\mathbb{Z}"/><br/> c)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-3%5Cle-1%2B2%5Csin%20x%5Cle1" alt="-3\le-1+2\sin x\le1"/><!--filtered attribute: style="display: inline;"--></div> [-3,1]</div> d)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D-1" alt="f\left(x\right)=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos%20x-1%3D-1" alt="2\cos x-1=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos%20x%3D0" alt="2\cos x=0"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D0" alt="\cos x=0"/><!--filtered attribute: style="display: inline;"--><br/> yksi ratkaisu<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D" alt="x=\frac{\pi}{2}"/></div> <div>täydellinen ratkaisu</div> <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>tai</div> <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>kun <img src="https://math-demo.abitti.fi/math.svg?latex=n%5Cin%5Cmathbb%7BZ%7D" alt="n\in\mathbb{Z}"/><br/> <br/> 210<br/> a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2%2B3%5Ccos%20x" alt="f\left(x\right)=2+3\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D-1%2B2%5Csin%20x" alt="g\left(x\right)=-1+2\sin x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-1%5Cle%5Ccos%20x%5Cle1" alt="-1\le\cos x\le1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-3%5Cle3%5Ccos%20x%5Cle3" alt="-3\le3\cos x\le3"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-1%5Cle2%2B3%5Ccos%20x%5Cle5" alt="-1\le2+3\cos x\le5"/></div> <div>[-1,5]</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-3%5Cle-1%2B2%5Csin%20x%5Cle1" alt="-3\le-1+2\sin x\le1"/><!--filtered attribute: style="display: inline;"--></div> [-3,1]<br/> kuva<br/> <a href="https://peda.net/p/oskari.lahtinen/mtf/2sjk/teht%C3%A4vi%C3%A4/sieppaa-png2#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/mtf/2sjk/teht%C3%A4vi%C3%A4/sieppaa-png2:file/photo/e88d4e30fe4846936cdf163f10c6635e1e6445f9/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a><br/> b)<br/> f(x)=3 on olemassa, sillä se kuuluu funktion arvojoukkoon<br/> g(x)=3 ei ole olemassa, sillä se ei kuulu arvojoukkoon<br/> 213<br/> <a href="https://peda.net/p/oskari.lahtinen/mtf/2sjk/teht%C3%A4vi%C3%A4/sieppaa-pngf#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/mtf/2sjk/teht%C3%A4vi%C3%A4/sieppaa-pngf:file/photo/5b7cf984edf1e42ed20a716ae60c115d18b6860a/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a><br/> molempien käyrien jakso on 2π<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D1-%5Ccos%20x" alt="\cos x=1-\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos%20x%3D1" alt="2\cos x=1"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D%5Cfrac%7B1%7D%7B2%7D" alt="\cos x=\frac{1}{2}"/></div> <div>yksi ratkaisu</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B3%7D" alt="x=\frac{\pi}{3}"/></div> <div>funktion arvo kohdassa</div> <div><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}"/></div> <div>kaikki ratkaisut</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>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>ratkaisut välillä <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%5B2%5Cpi%7B%2C%7D%5C%204%5Cpi%5Cright%5D" alt="\left[2\pi{,}\ 4\pi\right]"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B7%5Cpi%7D%7B3%7D%7B%2C%7D%5C%20%5Cfrac%7B11%5Cpi%7D%7B3%7D%7B%2C%7D%5Cfrac%7B13%5Cpi%7D%7B3%7D" alt="\frac{7\pi}{3}{,}\ \frac{11\pi}{3}{,}\frac{13\pi}{3}"/></div> <div>leikkauspisteet</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cfrac%7B7%5Cpi%7D%7B3%7D%7B%2C%7D%5C%20%5Cfrac%7B1%7D%7B2%7D%5Cright)%7B%2C%7D%5C%20%5Cleft(%5Cfrac%7B11%5Cpi%7D%7B3%7D%7B%2C%7D%5C%20%5Cfrac%7B1%7D%7B2%7D%5Cright)%7B%2C%7D%5C%20%5Cleft(%5Cfrac%7B13%5Cpi%7D%7B3%7D%7B%2C%7D%5C%20%5Cfrac%7B1%7D%7B2%7D%5Cright)" alt="\left(\frac{7\pi}{3}{,}\ \frac{1}{2}\right){,}\ \left(\frac{11\pi}{3}{,}\ \frac{1}{2}\right){,}\ \left(\frac{13\pi}{3}{,}\ \frac{1}{2}\right)"/></div> <br/> <br/> <br/> 216<br/> a) arvojoukko [-1,5;1,5], kerroin A=1,5<br/> b) funktion f perusjakso 4π, kerroin C=0,5<br/> c)<br/> <a href="https://peda.net/p/oskari.lahtinen/mtf/2sjk/teht%C3%A4vi%C3%A4/sieppaa-png3#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/mtf/2sjk/teht%C3%A4vi%C3%A4/sieppaa-png3:file/photo/beea22f2d1dd2452107a29134b28a1af55d373f8/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a></div> 2019-11-22T14:37:56+02:00