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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

If c > 1 2 , how many lines through the point (0, c) are normal lines to the parabola y = x2? What if c 1 2 ?

To determine

To find: The number of lines that are normal to the parabola y=x2 such that it passes through the point (0,c) when c>12 and c12.

Explanation

Derivative rule:

Power rule: ddx(xn)=nxn1

Calculation:

The derivative of the curve y=x2 is computed as follows.

dydx=ddx(x2)

Apply the power rule (1) and simplify the expression,

dydx=2x21=2x

Thus, the derivative of the curve is dydx=2x.

Therefore, the slope of the tangent line to the curve is 2x.

Obtain the slope of the normal line to the curve by using the slope of the tangent line.

Since every point of the parabola is of the form (a,a2), the slope of the tangent to the curve at (a,a2) is 2a.

Here, the tangent line is perpendicular to the normal line. That is, if m1 and m2 are the slopes of tangent line and normal line, then m1m2=1.

This implies that, the slope of the normal line to the curve is 12a.

Note that, the slope of the line passing through the points (x1,x2) and (y1,y2) is m=y2y1x2x1

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