Facit

Sommarmatte 2

(Skillnad mellan versioner)

Versionen från 18 juli 2007 kl. 10.37

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

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

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

Övning 1.3:1

a)	Funktionen har en kritisk punkt då $x = 1$. Funktionen saknar terrasspunkt. Då $x = 1$ har funktionen som extrempunkt ett lokalt och globalt minimum. Funktionen är strängt avtagande i intervallet $x\le 0$, funktionen är strängt växande i intervallet $x\ge 0$.	b)	Funktionen har en kritisk punkt då $x = -1$ och då $x=1$. Funktionen saknar terrasspunkt. Funktionen har som lokala extrempunkter ett lokalt minimum då $x = -1$ och ett lokalt maximum då $x=1$. Funktionen har ett lokalt och globalt minimum i den vänstra ändpunkten för funktionens definitionsintervall och ett lokalt och globalt maximum i den högra ändpunkten. Funktionen är strängt växande i intervallen $x\le -1$ och $x\ge 1$, funktionen är strängt avtagande i intervallet $-1\le x\le 1$.

c)	Funktionen har kritiska punkter då $x = -2$, då $x=-1$ och då $x=1/2$. Funktionen har en terrasspunkt då $x=-1$. Funktionen har som extrempunkter ett lokalt och globalt minimum då $x = -2$, ett lokalt maximum då $x=1/2$, ett lokalt och globalt maximum i vänstra ändpunkten för funktionens definitionsintervall och ett lokalt minimum i den högra ändpunkten för definitionsintervallet. Funktionen är strängt avtagande i intervallet $x\le -2$, strängt växande i intervallet $-2\le x\le 1/2$ och strängt avtagande i intervallet $x\ge 1/2$.	d)	Funktionen har kritiska punkter då $x = -5/2$ och då $x=1/2$. Funktionen saknar terrasspunkt. Funktionen har som extrempunkter ett lokalt minimum i vänstra ändpunkten för funktionens definitionsintervall, ett lokalt och globalt minimum då $x = -5/2$, ett lokalt och globalt maximum då $x=-1$, ett lokalt miminum då $x=-1/2$, ett lokalt maximum då $x=1/2$ och ett lokalt maximum i högra ändpunkten för funktionens definitionsintervall. Funktionen är strängt avtagande i intervallet $x\le -5/2$, strängt växande i intervallet $-5/2\le x\le -3/4$, strängt avtagande i intervallet $-3/4\le x\le -1/2$, strängt växande då $-1/2\le x\le 1/2$ och strängt avtagande i intervallet $x\ge 1/2$.

Övning 1.3:2

a)	$x=1\,$ (lokal minimipunkt)	b)	$x=\frac{3}{2}\,$ (lokal maximipunkt)

c)	$x=-2\,$ (lokal maximipunkt) $x=1\,$ (lokal minimipunkt)	d)	lokal extrempunkt saknas

Övning 1.3:3

a)	$x=0\,$ (lokal maximipunkt)	b)	$x=-\frac{1}{3}\ln\frac{5}{3}\,$ (lokal minimipunkt)

c)	$x=1/e\,$ (lokal minimipunkt)	d)	$x=-\sqrt{\sqrt{2}-1}\,$ (lokal maximipunkt) $x=0\,$ (lokal minimipunkt) $x=\sqrt{\sqrt{2}-1}\,$ (lokal maximipunkt)

e)	$x=-3\,$ (lokal minimipunkt) $x=-2\,$ (lokal maximipunkt) $x=1\,$ (lokal minimipunkt) $x=3\,$ (lokal maximipunkt)

Övning 1.3:4

$P = \bigl(1/\sqrt{3},2/3\bigr)$

Övning 1.3:5

$\alpha=\pi/6$

Övning 1.3:6

radie ${}={}$ höjd $\displaystyle {}=\sqrt[3]{\frac{V}{\pi}}$

Övning 1.3:7

Vinkeln $2\pi\bigl(1-\sqrt{\frac{2}{3}}\,\bigr)\,$ radianer ska tas bort.

Övning 2.1:1

a)	$6$	b)	$2$

c)	$2$	d)	$\displaystyle\frac{5}{2}$

Övning 2.1:2

a)	$\displaystyle\frac{44}{3}$	b)	$\displaystyle-\frac{9}{2}$

c)	$\displaystyle\frac{32}{3}$	d)	$1$

Övning 2.1:3

a)	$-\cos x + C$	b)	$\displaystyle-\frac{\cos 2x}{2}+C$

c)	$\displaystyle\frac{e^{3x}}{3}+\frac{e^{2x}}{2}+C$	d)	$\displaystyle\frac{x^2}{2}+\ln x + C$

Övning 2.1:4

a) $3-\displaystyle\frac{1}{\sqrt2}$ a.e.

b) $\displaystyle 4.\sqrt{3}$ a.e.

c) $32$ a.e.

d) $\sqrt{2}-1-\ln(\sqrt{2}-1)\,$ a.e.

e) $\displaystyle\frac{9}{2}$ a.e.

Övning 2.1:5

a) $\displaystyle\frac{2}{27}\left((x+9)\sqrt{x+9}+x\sqrt{x}\right)+C$

b) $-\displaystyle\frac{\sin2x}{4}+\frac{x}{2}+C$

Övning 2.2:1

a)	$\displaystyle\frac{13}{1000}$
b)	$\displaystyle\frac{(x^2+3)^6}{12}+C$
c)	$\displaystyle\frac{1}{3}e^{\scriptstyle x^3}+C$

Övning 2.2:2

a)	$0$	b)	$\displaystyle\frac{1}{2}(e^4-e^3)$
c)	$14$	d)	$\displaystyle\frac{3}{4}$

Övning 2.2:3

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

Övning 2.2:4

a)	$\displaystyle\frac{1}{2}\arctan\left(\frac{x}{2}\right)+C$	b)	$\displaystyle\frac{1}{\sqrt3}\arctan\left(\frac{x-1}{\sqrt3}\right)+C$
c)	$\displaystyle\frac{1}{2}\arctan\left(\frac{x+2}{2}\right)+C$	d)	$x-\arctan x + C$

Övning 2.3:1

a)	$-2(x+1)e^{-x}+C$	b)	$-(x+1)\cos x+\sin x + C$
c)	$2x\cos x + (x^2-2)\sin x + C$	d)	$\displaystyle\frac{x^2}{2}\left(\ln x - \frac{1}{2}\right) + C$

Övning 2.3:2

a)	$2e^{\sqrt{x}}\left(\sqrt{x}-1\right)+C$	b)	$\displaystyle\frac{1}{2}$
c)	$-\ln\|\cos x\|+C$	d)	$x(\ln x-1)+C$

Övning 3.1:1

a)	$8+3i$	b)	$-2+4i$
c)	$-3+2i$	d)	$31+i$
e)	$7-i$	f)	$1-i$

Övning 3.1:2

a)	$\displaystyle \frac{1}{2} - \frac{5}{2}\,i$	b)	$\displaystyle -\frac{19}{26} + \frac{2}{13}\,i$
c)	$\displaystyle -\frac{11}{4}-\frac{5\sqrt3}{4}\,i$	d)	$\displaystyle \frac{7}{130} -\frac{93}{65}\,i$

Övning 3.1:3

$a=-6$

Övning 3.1:4

a)	$z=2+3i$	b)	$z=\displaystyle\frac{4}{5} + \frac{7}{5}i$
c)	$z=2+i$	d)	$z=\displaystyle \frac{3}{5} - \frac{1}{5}i$
e)	$z=\displaystyle \frac{2}{3}-i$	f)	$z=3+i$

Övning 3.2:1

a)	b)
c)	d)

Övning 3.2:2

a)	b)
c)	d)
e)	f)

Övning 3.2:3

$2+4i$

Övning 3.2:4

a)	$5$	b)	$\sqrt{53}$
c)	$5\sqrt{13}$	d)	$\displaystyle\frac{5}{\sqrt{13}}$

Övning 3.2:5

a)	$\pi$	b)	$\displaystyle\frac{3\pi}{4}$
c)	$-\displaystyle\frac{\pi}{12}\,$ eller $\,\displaystyle\frac{23}{12}\pi$	d)	$\displaystyle\frac{\pi}{4}$

Övning 3.2:6

a)	$\displaystyle3(\cos 0 + i\,\sin 0)$	b)	$\displaystyle11\left(\cos \frac{3\pi}{2} + i\,\sin\frac{3\pi}{2}\right)$
c)	$\displaystyle4\sqrt2\left(\cos \frac{5\pi}{4} + i\,\sin\frac{5\pi}{4}\right)$	d)	$\displaystyle2\sqrt{10}\left(\cos \frac{\pi}{3} + i\,\sin\frac{\pi}{3}\right)$
e)	$\displaystyle\sqrt2\left(\cos \frac{\pi}{12} + i\,\sin\frac{\pi}{12}\right)$	f)	$\displaystyle\frac{\sqrt2}{3}\left(\cos \frac{\pi}{4} + i\,\sin\frac{\pi}{4}\right)$

Övning 3.3:1

a)	$-64$	b)	$1$
c)	$2^{65}+2^{65}\sqrt{3}\,i$	d)	$-64$
e)	$\displaystyle\frac{\sqrt{3}}{32} - \frac{i}{32} $

Övning 3.3:2

a)	$z= \left\{\begin{matrix} \phantom{-}1 \\ -1 \\ \phantom{-}i \\ -i\\ \end{matrix}\right.$	b)	$z = \left\{\begin{matrix} \frac{1}{2}+\frac{1}{2}\sqrt{3}\,i \\ -1\phantom{{}-\frac{1}{2}\sqrt{3}\,i} \\ \frac{1}{2}-\frac{1}{2}\sqrt{3}\,i \\ \end{matrix}\right.$	c)	$\displaystyle z=2^{1/10}\exp\Bigl(\frac{\pi i}{4}+\frac{2k\pi i}{5}\Bigr)$ för $\ k=0,1,2,3,4$
d)	$z= \left\{\begin{matrix} 2+i \\ 2-i \\ \phantom{-}i \\ -i\\ \end{matrix}\right.$	e)	$z = \left\{\begin{matrix} \phantom{-}1 \\ -1 \end{matrix}\right.$

Övning 3.3:3

a)	$(z+1)^2+2$	b)	$\left(z+\frac{3}{2}i\,\right)^2+2$
c)	$-(z-2+i)^2+4(1-i)$	d)	$i\bigl(z+\frac{3}{2}-i\bigl)^2-4-\frac{5}{4}\,i$

Övning 3.3:4

a)	$z= \left\{\begin{matrix} \phantom{-}(1+i)/\sqrt{2}\\ -(1+i)/\sqrt{2}\\ \end{matrix}\right.$	b)	$z = \left\{\begin{matrix} 2+i \\ 2-i \\ \end{matrix}\right.$
c)	$z= \left\{\begin{matrix} -1 \\ \phantom{-}3 \\ \end{matrix}\right. $	d)	$z= \left\{\begin{matrix} (1+i\sqrt{15})/4\\ (1-i\sqrt{15})/4 \end{matrix}\right.$

Övning 3.3:5

a)	$z= \left\{\begin{matrix} 2+1 \\ i \\ \end{matrix}\right.$	b)	$z = \left\{\begin{matrix} 1+i\phantom{2} \\ 1-2i \\ \end{matrix}\right.$
c)	$z= \left\{\begin{matrix} \phantom{-}2+i\phantom{2} \\ -1+2i \\ \end{matrix}\right. $	d)	$z= \left\{\begin{matrix} i \\ 1+4i \\ \end{matrix}\right.$

Övning 3.3:6

Lösningar:

$z= \left\{\eqalign{&\textstyle\sqrt[\scriptstyle 4]{2}\bigl(\cos\frac{\pi}{8}+i\,\sin\frac{\pi}{8}\bigr)\cr &\textstyle\sqrt[\scriptstyle 4]{2}\bigl(\cos\frac{9\pi}{8}+i\,\sin\frac{9\pi}{8}\bigr)}\right. = \left\{\eqalign{&\textstyle\phantom{-}{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}+2}\vphantom{2^{2^{\scriptstyle 2}}}}+i\,{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}-2}\vphantom{2^{2^{\scriptstyle 2}}}}\cr &\textstyle -{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}+2}\vphantom{2^{2^{\scriptstyle 2}}}}-i\,{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}-2}\vphantom{2^{2^{\scriptstyle 2}}}}}\right.$

Uttryck:

$\displaystyle\tan \frac{\pi}{8} = \sqrt{2} - 1$

Övning 3.4:1

a)	$x+1$	b)	$\displaystyle x-1+\frac{1}{x+1}$	c)	$x^2-ax+a^2$
d)	$x^2-x+2$	e)	$\displaystyle x-1+\frac{2x+2}{x^2+3x+1}$

Övning 3.4:2

$ z = \Bigl\{\eqalign{&1+i\cr &1-i}$

Övning 3.4:3

$z = \left\{\begin{matrix}-1+i\cr -1-i\cr \phantom{-}2i\cr -2i\end{matrix}\right.$

Övning 3.4:4

Välj $\,a=1\,$ och $\,b=10\,$. Lösningarna är $\ z = \left\{ \begin{matrix} 1-2i \\ 1+2i \\ -2 \end{matrix} \right.$

Övning 3.4:5

Två fall:

Välj $\,a=8\,$ och $\,b=-3\,$. Lösningarna är $\,z=1\,$ (trippelroten) och $\,z=-3\,$.
Välj $\,a=-8\,$ och $\,b=-3\,$. Lösningarna är $\,z=-1\,$ (trippelroten) och $\,z=3\,$.

Övning 3.5:6

$ z = \left\{ \begin{matrix} \phantom{-}i\sqrt{6}\\ -i\sqrt{6} \\ -\frac{3}{2} + \frac{1}{2}\sqrt{29}\,i \\ -\frac{3}{2} - \frac{1}{2}\sqrt{29}\,i \end{matrix} \right.$

Övning 2.5:7

$(z-1)(z-2)(z-4) = z^3 -7z^2 + 14z - 8$

$(z+1-i)(z+1+i) = z^2+2z+2 $

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

 <td class="ntext">d)</td>
 <td class="ntext" width="50%">$x-\arctan x + C$</td>
+</tr>
+</table>
+==Övning 2.3:1==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$-2(x+1)e^{-x}+C$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$-(x+1)\cos x+\sin x + C$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$2x\cos x + (x^2-2)\sin x + C$</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$\displaystyle\frac{x^2}{2}\left(\ln x - \frac{1}{2}\right) + C$</td>
+</tr>
+</table>
+==Övning 2.3:2==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$2e^{\sqrt{x}}\left(\sqrt{x}-1\right)+C$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$\displaystyle\frac{1}{2}$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$-\ln|\cos x|+C$</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$x(\ln x-1)+C$</td>
+</tr>
+</table>
+==Övning 3.1:1==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$8+3i$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$-2+4i$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$-3+2i$</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$31+i$</td>
+</tr>
+<tr align="left">
+<td class="ntext">e)</td>
+<td class="ntext" width="50%">$7-i$</td>
+<td class="ntext">f)</td>
+<td class="ntext" width="50%">$1-i$</td>
+</tr>
+</table>
+==Övning 3.1:2==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$\displaystyle \frac{1}{2} - \frac{5}{2}\,i$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$\displaystyle -\frac{19}{26} + \frac{2}{13}\,i$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$\displaystyle -\frac{11}{4}-\frac{5\sqrt3}{4}\,i$</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$\displaystyle \frac{7}{130} -\frac{93}{65}\,i$</td>
+</tr>
+</table>
+==Övning 3.1:3==
+$a=-6$
+==Övning 3.1:4==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$z=2+3i$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$z=\displaystyle\frac{4}{5} + \frac{7}{5}i$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$z=2+i$</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$z=\displaystyle \frac{3}{5} - \frac{1}{5}i$</td>
+</tr>
+<tr align="left">
+<td class="ntext">e)</td>
+<td class="ntext" width="50%">$z=\displaystyle \frac{2}{3}-i$</td>
+<td class="ntext">f)</td>
+<td class="ntext" width="50%">$z=3+i$</td>
+</tr>
+</table>
+==Övning 3.2:1==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)<br\>[[Bild:f_3_2_1a.gif]]</td>
+<td class="ntext">b)<br\>[[Bild:f_3_2_1b.gif]]</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)<br\>[[Bild:f_3_2_1c.gif]]</td>
+<td class="ntext">d)<br\>[[Bild:f_3_2_1d.gif]]</td>
+</tr>
+</table>
+==Övning 3.2:2==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)<br\>[[Bild:f_3_2_2a.gif]]</td>
+<td class="ntext">b)<br\>[[Bild:f_3_2_2b.gif]]</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)<br\>[[Bild:f_3_2_2c.gif]]</td>
+<td class="ntext">d)<br\>[[Bild:f_3_2_2d.gif]]</td>
+</tr>
+<tr align="left">
+<td class="ntext">e)<br\>[[Bild:f_3_2_2e.gif]]</td>
+<td class="ntext">f)<br\>[[Bild:f_3_2_2f.gif]]</td>
+</tr>
+</table>
+==Övning 3.2:3==
+$2+4i$
+==Övning 3.2:4==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$5$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$\sqrt{53}$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$5\sqrt{13}$</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$\displaystyle\frac{5}{\sqrt{13}}$</td>
+</tr>
+</table>
+==Övning 3.2:5==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$\pi$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$\displaystyle\frac{3\pi}{4}$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$-\displaystyle\frac{\pi}{12}\,$ eller $\,\displaystyle\frac{23}{12}\pi$</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$\displaystyle\frac{\pi}{4}$</td>
+</tr>
+</table>
+==Övning 3.2:6==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$\displaystyle3(\cos 0 + i\,\sin 0)$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$\displaystyle11\left(\cos \frac{3\pi}{2} + i\,\sin\frac{3\pi}{2}\right)$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$\displaystyle4\sqrt2\left(\cos \frac{5\pi}{4} + i\,\sin\frac{5\pi}{4}\right)$</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$\displaystyle2\sqrt{10}\left(\cos \frac{\pi}{3} + i\,\sin\frac{\pi}{3}\right)$</td>
+</tr>
+<tr align="left">
+<td class="ntext">e)</td>
+<td class="ntext" width="50%">$\displaystyle\sqrt2\left(\cos \frac{\pi}{12} + i\,\sin\frac{\pi}{12}\right)$</td>
+<td class="ntext">f)</td>
+<td class="ntext" width="50%">$\displaystyle\frac{\sqrt2}{3}\left(\cos \frac{\pi}{4} + i\,\sin\frac{\pi}{4}\right)$</td>
+</tr>
+</table>
+==Övning 3.3:1==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$-64$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$1$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$2^{65}+2^{65}\sqrt{3}\,i$</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$-64$</td>
+</tr>
+<tr align="left">
+<td class="ntext">e)</td>
+<td class="ntext" width="50%">$\displaystyle\frac{\sqrt{3}}{32} - \frac{i}{32} $</td>
+</tr>
+</table>
+==Övning 3.3:2==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="33%">$z= \left\{\begin{matrix} \phantom{-}1 \\ -1 \\ \phantom{-}i \\ -i\\ \end{matrix}\right.$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="33%">$z = \left\{\begin{matrix} \frac{1}{2}+\frac{1}{2}\sqrt{3}\,i \\ -1\phantom{{}-\frac{1}{2}\sqrt{3}\,i} \\ \frac{1}{2}-\frac{1}{2}\sqrt{3}\,i \\  \end{matrix}\right.$</td>
+<td class="ntext">c)</td>
+<td class="ntext" width="33%">$\displaystyle z=2^{1/10}\exp\Bigl(\frac{\pi i}{4}+\frac{2k\pi i}{5}\Bigr)$ för $\ k=0,1,2,3,4$</td>
+</tr>
+<tr align="left">
+<td class="ntext">d)</td>
+<td class="ntext" width="33%">$z= \left\{\begin{matrix} 2+i \\ 2-i \\ \phantom{-}i \\ -i\\ \end{matrix}\right.$</td>
+<td class="ntext">e)</td>
+<td class="ntext" width="33%">$z = \left\{\begin{matrix} \phantom{-}1 \\ -1 \end{matrix}\right.$</td>
+</tr>
+</table>
+==Övning 3.3:3==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$(z+1)^2+2$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$\left(z+\frac{3}{2}i\,\right)^2+2$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$-(z-2+i)^2+4(1-i)$</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$i\bigl(z+\frac{3}{2}-i\bigl)^2-4-\frac{5}{4}\,i$</td>
+</tr>
+</table>
+==Övning 3.3:4==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$z= \left\{\begin{matrix} \phantom{-}(1+i)/\sqrt{2}\\ -(1+i)/\sqrt{2}\\ \end{matrix}\right.$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$z = \left\{\begin{matrix} 2+i \\ 2-i \\  \end{matrix}\right.$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$z= \left\{\begin{matrix} -1 \\ \phantom{-}3  \\ \end{matrix}\right. $</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$z= \left\{\begin{matrix} (1+i\sqrt{15})/4\\ (1-i\sqrt{15})/4 \end{matrix}\right.$</td>
+</tr>
+</table>
+==Övning 3.3:5==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$z= \left\{\begin{matrix} 2+1 \\ i \\ \end{matrix}\right.$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$z = \left\{\begin{matrix} 1+i\phantom{2} \\ 1-2i \\  \end{matrix}\right.$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$z= \left\{\begin{matrix} \phantom{-}2+i\phantom{2} \\ -1+2i  \\ \end{matrix}\right. $</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$z= \left\{\begin{matrix} i \\ 1+4i \\ \end{matrix}\right.$</td>
+</tr>
+</table>
+==Övning 3.3:6==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">Lösningar:</td>
+<td class="ntext" width="100%">$z= \left\{\eqalign{&\textstyle\sqrt[\scriptstyle 4]{2}\bigl(\cos\frac{\pi}{8}+i\,\sin\frac{\pi}{8}\bigr)\cr &\textstyle\sqrt[\scriptstyle 4]{2}\bigl(\cos\frac{9\pi}{8}+i\,\sin\frac{9\pi}{8}\bigr)}\right. = \left\{\eqalign{&\textstyle\phantom{-}{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}+2}\vphantom{2^{2^{\scriptstyle 2}}}}+i\,{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}-2}\vphantom{2^{2^{\scriptstyle 2}}}}\cr &\textstyle -{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}+2}\vphantom{2^{2^{\scriptstyle 2}}}}-i\,{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}-2}\vphantom{2^{2^{\scriptstyle 2}}}}}\right.$</td>
+</tr>
+<tr align="left">
+<td class="ntext">Uttryck:</td>
+<td class="ntext" width="100%">$\displaystyle\tan \frac{\pi}{8} = \sqrt{2} - 1$</td>
+</tr>
+</table>
+==Övning 3.4:1==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="33%">$x+1$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="33%">$\displaystyle x-1+\frac{1}{x+1}$</td>
+<td class="ntext">c)</td>
+<td class="ntext" width="33%">$x^2-ax+a^2$</td>
+</tr>
+<tr align="left">
+<td class="ntext">d)</td>
+<td class="ntext" width="33%">$x^2-x+2$</td>
+<td class="ntext">e)</td>
+<td class="ntext" width="33%">$\displaystyle x-1+\frac{2x+2}{x^2+3x+1}$</td>
+</tr>
+</table>
+==Övning 3.4:2==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">
+$ z = \Bigl\{\eqalign{&1+i\cr &1-i}$
+</td>
+</tr>
+</table>
+==Övning 3.4:3==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">
+$z =  \left\{\begin{matrix}-1+i\cr -1-i\cr \phantom{-}2i\cr -2i\end{matrix}\right.$
+</td>
+</tr>
+</table>
+==Övning 3.4:4==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">
+Välj $\,a=1\,$ och $\,b=10\,$. Lösningarna är  $\ z = \left\{ \begin{matrix} 1-2i \\ 1+2i \\ -2 \end{matrix} \right.$
+</td>
+</tr>
+</table>
+==Övning 3.4:5==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">
+Två fall:
+* Välj $\,a=8\,$ och $\,b=-3\,$. Lösningarna är $\,z=1\,$ (trippelroten) och $\,z=-3\,$.
+* Välj $\,a=-8\,$ och $\,b=-3\,$. Lösningarna är $\,z=-1\,$ (trippelroten) och $\,z=3\,$.
+</td>
+</tr>
+</table>
+==Övning 3.5:6==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">
+$ z = \left\{ \begin{matrix} \phantom{-}i\sqrt{6}\\ -i\sqrt{6} \\ -\frac{3}{2} + \frac{1}{2}\sqrt{29}\,i \\ -\frac{3}{2} - \frac{1}{2}\sqrt{29}\,i  \end{matrix} \right.$
+</td>
+</tr>
+</table>
+==Övning 2.5:7==
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$(z-1)(z-2)(z-4) = z^3 -7z^2 + 14z - 8$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$(z+1-i)(z+1+i) = z^2+2z+2 $</td>
 </tr>
 </table>