Lösung 1.1:4
Aus Online Mathematik Brückenkurs 2
K (Lösning 1.1:4 moved to Solution 1.1:4: Robot: moved page) |
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- | {{ | + | If we write the equation of the tangent as |
- | < | + | |
- | < | + | |
- | < | + | <math>y=kx+m</math> |
- | { | + | |
+ | |||
+ | we know that the tangent's gradient | ||
+ | <math>k</math> | ||
+ | is equal to the derivative of | ||
+ | <math>y=x^{\text{2}}\text{ }</math> | ||
+ | at the point | ||
+ | <math>x=\text{1}</math>, and since | ||
+ | |||
+ | <math>{y}'=\text{2}x</math>, so | ||
+ | |||
+ | |||
+ | <math>k={y}'\left( 1 \right)=2\centerdot 1=2</math> | ||
+ | |||
+ | We can determine the constant | ||
+ | <math>m\text{ }</math> | ||
+ | with the condition that the tangent should go through the grazing point | ||
+ | <math>\left( 1 \right.,\left. 1 \right)</math>, i.e. the point | ||
+ | <math>\left( 1 \right.,\left. 1 \right)</math> | ||
+ | should satisfy the equation of the tangent | ||
+ | |||
+ | |||
+ | <math>1=2\centerdot 1+m</math> | ||
+ | |||
+ | |||
+ | which gives that | ||
+ | <math>m=-\text{1}</math> | ||
[[Image:1_1_4_1.gif|center]] | [[Image:1_1_4_1.gif|center]] | ||
+ | |||
+ | |||
+ | The normal to the curve | ||
+ | <math>y=x^{\text{2}}\text{ }</math> | ||
+ | at the point | ||
+ | <math>\left( 1 \right.,\left. 1 \right)</math> | ||
+ | is the straight line which is perpendicular to the tangent at the same point. | ||
+ | |||
+ | Because two straight lines which are perpendicular to each other have gradients which satisfy | ||
+ | <math>k_{1}\centerdot k_{2}=-1</math>, the normal must have a gradient which is equal to | ||
+ | |||
+ | |||
+ | <math>-\frac{1}{k}=-\frac{1}{2}</math> | ||
+ | |||
+ | |||
+ | The equation of the normal can therefore be written as | ||
+ | |||
+ | |||
+ | <math>y=-\frac{1}{2}x+n</math> | ||
+ | |||
+ | |||
+ | where | ||
+ | <math>n</math> | ||
+ | is some constant. | ||
+ | |||
+ | Since the normal must pass through the line | ||
+ | <math>\left( 1 \right.,\left. 1 \right)</math>, we can determine the constant | ||
+ | <math>n</math> | ||
+ | if we substitute the point into the equation of the normal, | ||
+ | |||
+ | |||
+ | <math>1=-\frac{1}{2}\centerdot +n</math> | ||
+ | |||
+ | |||
+ | and this gives | ||
+ | <math>n=\frac{3}{2}</math>. | ||
+ | |||
+ | |||
+ | |||
[[Image:1_1_4-3(3).gif|center]] | [[Image:1_1_4-3(3).gif|center]] |
Version vom 12:12, 10. Okt. 2008
If we write the equation of the tangent as
we know that the tangent's gradient
=2x
1
=2
1=2
We can determine the constant
1
1
1
1
1+m
which gives that
The normal to the curve
1
1
Because two straight lines which are perpendicular to each other have gradients which satisfy
k2=−1
The equation of the normal can therefore be written as
where
Since the normal must pass through the line
1
1
+n
and this gives