   Chapter 3.1, Problem 37E

Chapter
Section
Textbook Problem

Find equations of the tangent line and normal line to the curve at the given point.y = x4 + 2ex, (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=x4+2ex.

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

Derivative rules:

(1) Constant Multiple Rule: ddx[cf(x)]=cddxf(x)

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

(3) Sum Rule: ddx[f(x)+g(x)]=ddx(f(x))+ddx(g(x))

(4) Derivative of natural exponential function: ddx(ex)=ex

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(x4+2ex)

Apply sum rule (3) and constant multiple rule (1),

dydx=ddx(x4)+ddx(2ex)  =ddx(x4)+2ddx(ex)

Apply rule (

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