Facit

Sommarmatte 2

(Skillnad mellan versioner)

Versionen från 18 juli 2007 kl. 09.20

Övning 1.1:1

a)	$f'(-4)>0, \,\,\,\, f'(1)<0$

b)	$x=-3$ och $x=2$

c)	$-3\le x \le 2$

Övning 1.1:2

a) $f'(x)=2x-3$

b) $f'(x)=-\sin x -\cos x$

c) $f'(x)=e^x-\displaystyle\frac{1}{x}$

d) $f'(x)=\displaystyle\frac{1}{2}x^{-1/2}=\frac{1}{2\sqrt x}$

e) $f'(x)=4x(x^2-1)$

f) $f'(x)=-\sin \left(x+\frac{\pi}{3}\right)$

Övning 1.1:3

$14{,}0\,$ m/s

Övning 1.1:4

Tangentens ekvation: $\ y=2x-1$

Normalens ekvation: $\ y=-\displaystyle\frac{1}{2}x+\frac{3}{2}$

Övning 1.1:5

$\bigl(1-\sqrt2, -3+2\sqrt2\bigr)\,$ och $\,\bigl(1+\sqrt2, -3-2\sqrt2\bigr)$

Övning 1.2:1

a)	$\cos^2x-\sin^2x=\cos2x$	b)	$2x\ln x+ x$	c)	$\displaystyle\frac{x^2+2x-1}{(x+1)^2}=1-\frac{2}{(x+1)^2}$
d)	$\displaystyle\frac{\cos x}{x}-\frac{\sin x}{x^2}$	e)	$\displaystyle\frac{1}{\ln x}-\frac{1}{(\ln x)^2}$	f)	$\displaystyle \frac{\ln x + 1}{\sin x}-\frac{x\ln x \cos x}{\sin^2x}$

Övning 1.2:2

a)	$\cos x^2 \cdot 2x$	b)	$e^{x^2+x}(2x+1)$	c)	$\displaystyle - \frac{\sin x}{2\sqrt{\cos x}}$
d)	$\displaystyle\frac{1}{x\ln x}$	e)	$(2x+1)^3(10x+1)$	f)	$\displaystyle\frac{\sin\sqrt{1-x}}{2\sqrt{1-x}}$

Övning 1.2:3

a)	$\displaystyle\frac{1}{2\sqrt{x}\sqrt{x+1}}$	b)	$\displaystyle - \frac{1}{(x-1)^{3/2}\sqrt{x+1}}$	c)	$\displaystyle - \frac{1-2x^2}{x^2(1-x^2)^{3/2}}$
d)	$-\cos\cos\sin x \cdot \sin\sin x \cdot \cos x$	e)	$e^{\sin x^2}\cdot \cos x^2 \cdot 2x$	f)	$\displaystyle x^{\tan x}\Bigl(\frac{\ln x}{\cos^2x}+\frac{\tan x}{x}\Bigr)$

Övning 1.2:4

$\displaystyle\frac{3x}{(1-x^2)^{5/2}}$

$\displaystyle - \frac{2\sin \ln x}{x}$

Den här artikeln är hämtad från http://wiki.math.se/wikis/sf0601_0701/index.php/Facit

 </tr>
 <tr><td height="5px"/></tr>
+</table>
+==Övning 1.2:1==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="33%">$\cos^2x-\sin^2x=\cos2x$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="33%">$2x\ln x+ x$</td>
+<td class="ntext">c)</td>
+<td class="ntext" width="33%">$\displaystyle\frac{x^2+2x-1}{(x+1)^2}=1-\frac{2}{(x+1)^2}$</td>
+</tr>
+<tr align="left">
+<td class="ntext">d)</td>
+<td class="ntext" width="33%">$\displaystyle\frac{\cos x}{x}-\frac{\sin x}{x^2}$</td>
+<td class="ntext">e)</td>
+<td class="ntext" width="33%">$\displaystyle\frac{1}{\ln x}-\frac{1}{(\ln x)^2}$</td>
+<td class="ntext">f)</td>
+<td class="ntext" width="33%">$\displaystyle \frac{\ln x + 1}{\sin x}-\frac{x\ln x \cos x}{\sin^2x}$</td>
+</tr>
+</table>
+==Övning 1.2:2==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="33%">$\cos x^2 \cdot 2x$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="33%">$e^{x^2+x}(2x+1)$</td>
+<td class="ntext">c)</td>
+<td class="ntext" width="33%">$\displaystyle - \frac{\sin x}{2\sqrt{\cos x}}$</td>
+</tr>
+<tr align="left">
+<td class="ntext">d)</td>
+<td class="ntext" width="33%">$\displaystyle\frac{1}{x\ln x}$</td>
+<td class="ntext">e)</td>
+<td class="ntext" width="33%">$(2x+1)^3(10x+1)$</td>
+<td class="ntext">f)</td>
+<td class="ntext" width="33%">$\displaystyle\frac{\sin\sqrt{1-x}}{2\sqrt{1-x}}$</td>
+</tr>
+</table>
+==Övning 1.2:3==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="33%">$\displaystyle\frac{1}{2\sqrt{x}\sqrt{x+1}}$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="33%">$\displaystyle - \frac{1}{(x-1)^{3/2}\sqrt{x+1}}$</td>
+<td class="ntext">c)</td>
+<td class="ntext" width="33%">$\displaystyle - \frac{1-2x^2}{x^2(1-x^2)^{3/2}}$</td>
+</tr>
+<tr align="left">
+<td class="ntext">d)</td>
+<td class="ntext" width="33%">$-\cos\cos\sin x \cdot \sin\sin x \cdot \cos x$</td>
+<td class="ntext">e)</td>
+<td class="ntext" width="33%">$e^{\sin x^2}\cdot \cos x^2 \cdot 2x$</td>
+<td class="ntext">f)</td>
+<td class="ntext" width="33%">$\displaystyle x^{\tan x}\Bigl(\frac{\ln x}{\cos^2x}+\frac{\tan x}{x}\Bigr)$</td>
+</tr>
+</table>
+==Övning 1.2:4==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="33%">$\displaystyle\frac{3x}{(1-x^2)^{5/2}}$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="33%">$\displaystyle - \frac{2\sin \ln x}{x}$</td>
+</tr>
 </table>