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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

(a) The curve y = | x | / 2 x 2 is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1).

(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

(a)

To determine

To find: The equation of the tangent line to the curve at the point.

Explanation

Given:

The function is y=|x|2x2 and the point is (1,1).

Result used:

The Power Rule combined with the Chain Rule:

If n is any real number and g(x) is differentiable function, then

ddx[g(x)]n=n[g(x)]n1g(x) (1)

Quotient Rule:

If f(x). and g(x) are both differentiable function, then

ddx[f(x)g(x)]=g(x)ddx[f(x)]f(x)ddx[g(x)][g(x)]2 (2)

Formula used:

The equation of the tangent line at (x1,y1) is, yy1=m(xx1) (3)

where, m is the slope of the tangent line at (x1,y1) and m=dydx|x=x1.

Calculation:

For x>0,|x|=x and f(x)=x2x2.

The derivative of y is dydx, which is obtained as follows,

dydx=ddx(y)=ddx(|x|2x2)=ddx(x2x2)

Apply the quotient rule as shown in equation (2),

dydx=2x2ddx[x]xddx[2x2][2x2]2

Apply the power rule combined with the chain rule as shown in equation (1),

dydx=2x2[1x11]xddx(2x2)12[(2x2)12]2=2x2[1]x12(2x</

(b)

To determine

To sketch: The graph of the curve and the tangent line.

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