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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

Let g(x) = ecx + f(x) and h(x) = ekxf(x), where f(0) = 3, f′(0) = 5, and f″(0) = –2.

(a) Find g′(0) and g″(0) in terms of c.

(b) In terms of k, find an equation of the tangent line to the graph of h at the point where x = 0.

(a)

To determine

To find: The derivative of g(0) and g(0) in terms of c.

Explanation

Given:

The function is g(x)=ecx+f(x),

Derivative Rule: Sum Rule

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

ddt[f(t)+g(t)]=ddt[f(t)]+ddt[g(t)] (1)

Calculation:

Obtain the first derivative of g(x)=ecx+f(x).

g(x)=ddx(g(x))=ddx(ecx+f(x))

Apply the sum rule as shown in equation (1),

g(x)=ddx(ecx)+ddx(f(x))=cecx+f(x)

Substitute x=0,

g(0)=cec(0)+f(0)

Substitute f(0)=5 and obtain the first derivative of g(x) at x=0.

g(0)=ce0+5=c1+5       (Q e0=1 )=c+5

Therefore, the first derivative of g(x)=ecx+f(x) is g(x)=c+5

(b)

To determine

To find: The equation of the tangent line to the graph of h at the point where x=0.

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