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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

Find equations of the tangent line and normal line to the curve at the given point.

y = (2 + x)ex, (0, 2)

To determine

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

Explanation

Given:

The equation of the curve y=(2+x)ex.

The curve passing through the point (0,2).

Derivative rules:

Product Rule: ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)]

Formula used:

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

The equation of normal line at (x1,y1) is, yy1=1m(xx1) (2)

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((2+x)ex) 

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

dydx=(2+x)ddx(ex)+exddx(2+x)=(2+x)(ex)+ex(0+1)=(2x+1)ex=(1x)ex

Therefore, the derivative of the function y=(2+x)ex is dydx=(1x)ex

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