   Chapter 3.2, Problem 35E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# (a) The curve y = 1/(1 + x2) is called a witch of Maria Agnesi. Find an equation of the tangent line to this curve at the point ( − 1 , 1 2 ) .(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 equation of the curve y=11+x2.

The curve passing through the point (1,12).

Derivative rules:

(1) Quotient Rule: If f1(x) and f2(x) are both differentiable, then

ddx[f1(x)f2(x)]=f2(x)ddx[f1(x)]f1(x)ddx[f2(x)][f2x]2

(2) Power Rule: ddx(xn)=nxn1

(3) Sum rule: ddx(f+g)=ddx(f)+ddx(g)

Formula used:

The equation of the tangent line at (x1,y1) is, yy1=m(xx1) (1)

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

Calculation:

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

dydx=ddx(y) =ddx(11+x2)

Apply the quotient rule (1) and simplify the terms,

dydx=(1+x2)ddx(1)(1)ddx(1+x2)(1+x2)2

Apply the derivative rules (3) ,(2) and simplify the terms,

dydx=(1+x2)dd

(b)

To determine

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

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