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Övningar 2.3

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Versionen från 16 juli 2007 kl. 07.58 (redigera)
KTH.SE:u1zpa8nw (Diskussion | bidrag)
(Ny sida: __NOTOC__ ==&Ouml;vning 2.3:1== <div class="ovning"> Kvadratkomplettera f&ouml;ljande uttryck <table width="100%" cellspacing="10px"> <tr align="left"> <td class="ntext">a)</td> <td class="...)
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KTH.SE:u1zpa8nw (Diskussion | bidrag)

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Rad 1: Rad 1:
__NOTOC__ __NOTOC__
-==&Ouml;vning 2.3:1==+'''&Ouml;vning 2.3:1'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Kvadratkomplettera f&ouml;ljande uttryck Kvadratkomplettera f&ouml;ljande uttryck
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">x^2-2x</td> <td class="ntext" width="25%">x^2-2x</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">x^2+2x-1</td> <td class="ntext" width="25%">x^2+2x-1</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">5+2x-x^2</td> <td class="ntext" width="25%">5+2x-x^2</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">x^2+5x+3</td> <td class="ntext" width="25%">x^2+5x+3</td>
</tr> </tr>
Rad 18: Rad 18:
</div> </div>
-==&Ouml;vning 2.3:2==+'''&Ouml;vning 2.3:2'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
L&ouml;s f&ouml;ljande andragradsekvationer med kvadratkomplettering L&ouml;s f&ouml;ljande andragradsekvationer med kvadratkomplettering
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">x^2-4x+3=0</td> <td class="ntext" width="33%">x^2-4x+3=0</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">y^2+2y-15=0</td> <td class="ntext" width="33%">y^2+2y-15=0</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">y^2+3y+4=0</td> <td class="ntext" width="33%">y^2+3y+4=0</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">4x^2-28x+13=0</td> <td class="ntext" width="33%">4x^2-28x+13=0</td>
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">5x^2+2x-3=0</td> <td class="ntext" width="33%">5x^2+2x-3=0</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">3x^2-10x+8=0</td> <td class="ntext" width="33%">3x^2-10x+8=0</td>
</tr> </tr>
Rad 42: Rad 42:
</div> </div>
-==&Ouml;vning 2.3:3==+'''&Ouml;vning 2.3:3'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
L&ouml;s f&ouml;ljande ekvationer direkt L&ouml;s f&ouml;ljande ekvationer direkt
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">x(x+3)=0</td> <td class="ntext" width="50%">x(x+3)=0</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">(x-3)(x+5)=0</td> <td class="ntext" width="50%">(x-3)(x+5)=0</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">5(3x-2)(x+8)=0</td> <td class="ntext" width="50%">5(3x-2)(x+8)=0</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">x(x+3)-x(2x-9)=0</td> <td class="ntext" width="50%">x(x+3)-x(2x-9)=0</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">(x+3)(x-1)-(x+3)(2x-9)=0</td> <td class="ntext" width="50%">(x+3)(x-1)-(x+3)(2x-9)=0</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">x(x^2-2x)+x(2-x)=0</td> <td class="ntext" width="50%">x(x^2-2x)+x(2-x)=0</td>
</tr> </tr>
Rad 68: Rad 68:
</div> </div>
-==&Ouml;vning 2.3:4==+'''&Ouml;vning 2.3:4'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Best&auml;m en andragradsekvation som har r&ouml;tterna Best&auml;m en andragradsekvation som har r&ouml;tterna
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">-1\ och \ 2</td> <td class="ntext" width="100%">-1\ och \ 2</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">1+\sqrt{3}\ och \ 1-\sqrt{3}</td> <td class="ntext" width="100%">1+\sqrt{3}\ och \ 1-\sqrt{3}</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">3\ och \ \sqrt{3}</td> <td class="ntext" width="100%">3\ och \ \sqrt{3}</td>
</tr> </tr>
Rad 88: Rad 88:
</div> </div>
-==&Ouml;vning 2.3:5==+'''&Ouml;vning 2.3:5'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">Best&auml;m en andragradsekvation som bara har \,-7\, som rot.</td> <td class="ntext" width="100%">Best&auml;m en andragradsekvation som bara har \,-7\, som rot.</td>
</tr> </tr>
<tr> <tr>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">Best&auml;m ett v&auml;rde p&aring; \,x\, som g&ouml;r att uttrycket \,4x^2-28x+48\, &auml;r negativt.</td> <td class="ntext" width="100%">Best&auml;m ett v&auml;rde p&aring; \,x\, som g&ouml;r att uttrycket \,4x^2-28x+48\, &auml;r negativt.</td>
</tr> </tr>
<tr> <tr>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">Ekvationen \,x^2+4x+b=0\, har en rot \,x=1\,. Best&auml;m v&auml;rdet p&aring; konstanten \,b\,.</td> <td class="ntext" width="100%">Ekvationen \,x^2+4x+b=0\, har en rot \,x=1\,. Best&auml;m v&auml;rdet p&aring; konstanten \,b\,.</td>
</tr> </tr>
Rad 107: Rad 107:
</div> </div>
-==&Ouml;vning 2.3:6==+'''&Ouml;vning 2.3:6'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Best&auml;m det minsta v&auml;rde som f&ouml;ljande polynom antar Best&auml;m det minsta v&auml;rde som f&ouml;ljande polynom antar
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">x^2-2x+1</td> <td class="ntext" width="33%">x^2-2x+1</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">x^2-4x+2</td> <td class="ntext" width="33%">x^2-4x+2</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">x^2-5x+7</td> <td class="ntext" width="33%">x^2-5x+7</td>
</tr> </tr>
Rad 123: Rad 123:
</div> </div>
-==&Ouml;vning 2.3:7==+'''&Ouml;vning 2.3:7'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Best&auml;m det st&ouml;rsta v&auml;rde som f&ouml;ljande polynom antar Best&auml;m det st&ouml;rsta v&auml;rde som f&ouml;ljande polynom antar
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">1-x^2</td> <td class="ntext" width="33%">1-x^2</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">-x^2+3x-4</td> <td class="ntext" width="33%">-x^2+3x-4</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">x^2+x+1</td> <td class="ntext" width="33%">x^2+x+1</td>
</tr> </tr>
Rad 139: Rad 139:
</div> </div>
-==&Ouml;vning 2.3:8==+'''&Ouml;vning 2.3:8'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Skissera grafen till f&ouml;ljande funktioner Skissera grafen till f&ouml;ljande funktioner
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">f(x)=x^2+1</td> <td class="ntext" width="33%">f(x)=x^2+1</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">f(x)=(x-1)^2+2</td> <td class="ntext" width="33%">f(x)=(x-1)^2+2</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">f(x)=x^2-6x+11</td> <td class="ntext" width="33%">f(x)=x^2-6x+11</td>
</tr> </tr>
Rad 155: Rad 155:
</div> </div>
-==&Ouml;vning 2.3:9==+'''&Ouml;vning 2.3:9'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Hitta alla sk&auml;rningspunkter mellan x-axeln och kurvan Hitta alla sk&auml;rningspunkter mellan x-axeln och kurvan
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">y=x^2-1</td> <td class="ntext" width="33%">y=x^2-1</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">y=x^2-5x+6</td> <td class="ntext" width="33%">y=x^2-5x+6</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">y=3x^2-12x+9</td> <td class="ntext" width="33%">y=3x^2-12x+9</td>
</tr> </tr>
Rad 171: Rad 171:
</div> </div>
-==&Ouml;vning 2.3:10==+'''&Ouml;vning 2.3:10'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Rita in i ett ''xy''-plan alla punkter vars koordinater \,(x,y)\, uppfyller Rita in i ett ''xy''-plan alla punkter vars koordinater \,(x,y)\, uppfyller
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">y \geq x^2\ och \ y \leq 1 </td> <td class="ntext" width="50%">y \geq x^2\ och \ y \leq 1 </td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">y \leq 1-x^2\ och \ x \geq 2y-3 </td> <td class="ntext" width="50%">y \leq 1-x^2\ och \ x \geq 2y-3 </td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">1 \geq x \geq y^2 </td> <td class="ntext" width="50%">1 \geq x \geq y^2 </td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">x^2 \leq y \leq x </td> <td class="ntext" width="50%">x^2 \leq y \leq x </td>
</tr> </tr>

Versionen från 16 juli 2007 kl. 10.50

Övning 2.3:1

Kvadratkomplettera följande uttryck

a)    x^2-2x b)    x^2+2x-1 c)    5+2x-x^2 d)    x^2+5x+3

Övning 2.3:2

Lös följande andragradsekvationer med kvadratkomplettering

a)    x^2-4x+3=0 b)    y^2+2y-15=0 c)    y^2+3y+4=0
d)    4x^2-28x+13=0 e)    5x^2+2x-3=0 f)    3x^2-10x+8=0

Övning 2.3:3

Lös följande ekvationer direkt

a)    x(x+3)=0 b)    (x-3)(x+5)=0
c)    5(3x-2)(x+8)=0 d)    x(x+3)-x(2x-9)=0
e)    (x+3)(x-1)-(x+3)(2x-9)=0 f)    x(x^2-2x)+x(2-x)=0

Övning 2.3:4

Bestäm en andragradsekvation som har rötterna

a)    -1\ och \ 2
b)    1+\sqrt{3}\ och \ 1-\sqrt{3}
c)    3\ och \ \sqrt{3}

Övning 2.3:5

a)    Bestäm en andragradsekvation som bara har \,-7\, som rot.
b)    Bestäm ett värde på \,x\, som gör att uttrycket \,4x^2-28x+48\, är negativt.
c)    Ekvationen \,x^2+4x+b=0\, har en rot \,x=1\,. Bestäm värdet på konstanten \,b\,.

Övning 2.3:6

Bestäm det minsta värde som följande polynom antar

a)    x^2-2x+1 b)    x^2-4x+2 c)    x^2-5x+7

Övning 2.3:7

Bestäm det största värde som följande polynom antar

a)    1-x^2 b)    -x^2+3x-4 c)    x^2+x+1

Övning 2.3:8

Skissera grafen till följande funktioner

a)    f(x)=x^2+1 b)    f(x)=(x-1)^2+2 c)    f(x)=x^2-6x+11

Övning 2.3:9

Hitta alla skärningspunkter mellan x-axeln och kurvan

a)    y=x^2-1 b)    y=x^2-5x+6 c)    y=3x^2-12x+9

Övning 2.3:10

Rita in i ett xy-plan alla punkter vars koordinater \,(x,y)\, uppfyller

a)    y \geq x^2\ och \ y \leq 1 b)    y \leq 1-x^2\ och \ x \geq 2y-3
c)    1 \geq x \geq y^2 d)    x^2 \leq y \leq x
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