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Lösung 1.1:4

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If we write the equation of the tangent as


y=kx+m


we know that the tangent's gradient k is equal to the derivative of y=x2 at the point x=1, and since

y=2x, so


\displaystyle k={y}'\left( 1 \right)=2\centerdot 1=2

We can determine the constant \displaystyle m\text{ } with the condition that the tangent should go through the grazing point \displaystyle \left( 1 \right.,\left. 1 \right), i.e. the point \displaystyle \left( 1 \right.,\left. 1 \right) should satisfy the equation of the tangent


\displaystyle 1=2\centerdot 1+m


which gives that \displaystyle m=-\text{1}



The normal to the curve \displaystyle y=x^{\text{2}}\text{ } at the point \displaystyle \left( 1 \right.,\left. 1 \right) 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 \displaystyle k_{1}\centerdot k_{2}=-1, the normal must have a gradient which is equal to


\displaystyle -\frac{1}{k}=-\frac{1}{2}


The equation of the normal can therefore be written as


\displaystyle y=-\frac{1}{2}x+n


where \displaystyle n is some constant.

Since the normal must pass through the line \displaystyle \left( 1 \right.,\left. 1 \right), we can determine the constant \displaystyle n if we substitute the point into the equation of the normal,


\displaystyle 1=-\frac{1}{2}\centerdot +n


and this gives \displaystyle n=\frac{3}{2}.