https://peda.net/id/83a1d5de499 2020-02-07T12:42:19+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/ed599d1c49a 2020-02-07T14:08:52+02:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5Cleft(5x-1%5Cright)%3D%5Cln%5Cleft(2x%5Cright)" alt="\ln\left(5x-1\right)=\ln\left(2x\right)"/></div> <div>määrittelyehto</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=5x-1%3E0" alt="5x-1&gt;0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=5x%3E1" alt="5x&gt;1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3E%5Cfrac%7B1%7D%7B5%7D" alt="x&gt;\frac{1}{5}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%3E0" alt="2x&gt;0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3E0" alt="x&gt;0"/></div> <div>määrittelyehto on siis <img src="https://math-demo.abitti.fi/math.svg?latex=x%3E%5Cfrac%7B1%7D%7B5%7D" alt="x&gt;\frac{1}{5}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5Cleft(5x-1%5Cright)%3D%5Cln%5Cleft(2x%5Cright)%7B%2C%7D%5C%20jos%5C%205x-1%3D2x" alt="\ln\left(5x-1\right)=\ln\left(2x\right){,}\ jos\ 5x-1=2x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=5x-2x%3D1" alt="5x-2x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D1" alt="3x=1"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1%7D%7B3%7D" alt="x=\frac{1}{3}"/><!--filtered attribute: style="display: inline;"--></div> <div>ratkaisu toteuttaa määrittelyehdon<br/> <br/> </div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Clog_2x%3D%5Clog_2%5Cleft(3x%2B4%5Cright)" alt="2\log_2x=\log_2\left(3x+4\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_2x%5E2%3D%5Clog_2%5Cleft(3x%2B4%5Cright)" alt="\log_2x^2=\log_2\left(3x+4\right)"/></div> <div>määrittelyehto</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3E0" alt="x^2&gt;0"/></div> <div>kaikki luvut toteuttavat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3x%2B4%3E0" alt="3x+4&gt;0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3E-4" alt="3x&gt;-4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3E-%5Cfrac%7B3%7D%7B4%7D" alt="x&gt;-\frac{3}{4}"/></div> <div>määrittelyehto on siis <img src="https://math-demo.abitti.fi/math.svg?latex=x%3E-%5Cfrac%7B3%7D%7B4%7D" alt="x&gt;-\frac{3}{4}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_xx%5E2%3D%5Clog_2%5Cleft(3x%2B4%5Cright)%7B%2C%7D%5C%20jos%5C%20x%5E2%3D3x%2B4" alt="\log_xx^2=\log_2\left(3x+4\right){,}\ jos\ x^2=3x+4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-3x-4%3D0" alt="x^2-3x-4=0"/><br/> ratkaistaan toisen asteen yhtälön ratkaisukaavalla<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-%5Cleft(-3%5Cright)%5Cpm%5Csqrt%7B%5Cleft(-3%5Cright)%5E2%2B4%5Ccdot1%5Ccdot%5Cleft(-4%5Cright)%7D%7D%7B2%5Ccdot1%7D%3D%5Cfrac%7B3%5Cpm%5Csqrt%7B9%2B16%7D%7D%7B2%7D%3D%5Cfrac%7B3%2B5%7D%7B2%7D%3D4%5C%20tai%5C%20%5Cfrac%7B3-5%7D%7B2%7D%3D-1" alt="x=\frac{-\left(-3\right)\pm\sqrt{\left(-3\right)^2+4\cdot1\cdot\left(-4\right)}}{2\cdot1}=\frac{3\pm\sqrt{9+16}}{2}=\frac{3+5}{2}=4\ tai\ \frac{3-5}{2}=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D4%5C%20tai%5C%20x%3D-1" alt="x=4\ tai\ x=-1"/></div> <div>molemmat ratkaisut toteuttavat määrittelyehdon</div> 2020-02-07T14:04:00+02:00 https://peda.net/id/a71fc79649a 2020-02-07T13:54:53+02:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Clg%5Cleft(x-1%5Cright)" alt="f\left(x\right)=\lg\left(x-1\right)"/></div> <div>määrittelyjoukko</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x-1%3E0" alt="x-1&gt;0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3E1" alt="x&gt;1"/></div> <div>nollakohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clg%5Cleft(x-1%5Cright)%3D0" alt="\lg\left(x-1\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-1%3D1" alt="x-1=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D2" alt="x=2"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cln%5Cleft(x%5E2-3%5Cright)" alt="f\left(x\right)=\ln\left(x^2-3\right)"/></div> <div>määrittelyjoukko</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-3%3E0" alt="x^2-3&gt;0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3E3" alt="x^2&gt;3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3E%5Csqrt%7B3%7Dtai%5C%20x%3C-%5Csqrt%7B3%7D" alt="x&gt;\sqrt{3}tai\ x&lt;-\sqrt{3}"/></div> <div>nollakohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5Cleft(x%5E2-3%5Cright)%3D0" alt="\ln\left(x^2-3\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-3%3D1" alt="x^2-3=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D4" alt="x^2=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D2%5C%20tai%5C%20x%3D-2" alt="x=2\ tai\ x=-2"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cln%5Cleft(x%5E2%2B3%5Cright)" alt="f\left(x\right)=\ln\left(x^2+3\right)"/></div> <div>määrittelyjoukko</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2B3%3E0" alt="x^2+3&gt;0"/></div> <div>funktio on määritelty kaikilla x:n arvoilla </div> <div>nollakohdat</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5Cleft(x%5E2%2B3%5Cright)%3D0" alt="\ln\left(x^2+3\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2B3%3D1" alt="x^2+3=1"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D-2" alt="x^2=-2"/></div> <div>yhtälöllä ei ole nollakohtia</div> 2020-02-07T13:54:53+02:00 https://peda.net/id/720e3490499 2020-02-07T13:39:05+02:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cln%20x" alt="f\left(x\right)=\ln x"/></div> <div>funktion f määrittelyjoukko on <img src="https://math-demo.abitti.fi/math.svg?latex=x%3E0" alt="x&gt;0"/></div> <div>b) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D0%5C%20" alt="f\left(x\right)=0\ "/>ratkaisu on <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1" alt="x=1"/>, koska <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln1%3D0" alt="\ln1=0"/> </div> <div> </div> 2020-02-07T13:39:05+02:00 https://peda.net/id/79a5acfc499 2020-02-07T13:32:08+02:00 a) kasvava, positiivinen<br/> b) vähenevä, negatiivinen<br/> c) kasvava, positiivinen 2020-02-07T13:32:08+02:00 https://peda.net/id/b3eac48a499 2020-02-07T13:19:27+02:00 <p>A2<br/> B3<br/> C1<br/> D4</p> 2020-02-07T13:19:27+02:00