Explanation 1) Concept: Use The Fundamental Theorem of Calculus to find the critical point. The Fundamental Theorem of Calculus: Suppose f is continuous on [a, b], then ∫ a b f x d x = F b - F ( a ) , where F is antiderivative of f , that is F ' = f . 2) Calculations: Consider, C t = 1 t ∫ 0 t f s + g s d s To find the critical point, differentiate the above equation with respect to t using the power rule C ' t = - 1 t 2 ∫ 0 t f s + g s d s + 1 t { d d t ( ∫ 0 t f s + g s d s } . Using Fundamental Theorem of Calculus, C ' t = - 1 t 2 ∫ 0 t f s + g s d s + 1 t f t + g t In order to find the critical point, equate C ' ( t ) to zero Thus, C ' t To determine (b) The length of time T for D t to equal the initial value of V . To determine (c) The absolute minimum of C on ( 0 , T ] . To determine (d) To sketch: The graph for C and f + g And verify the result in part(a)

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Author: Stewart

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Chapter 4.3, Problem 78E

To determine

(a)

To show,

The critical numbers of C occur at the numbers t where Ct=ft+gt.

To determine

(b)

The length of time T for Dt to equal the initial value of V.

To determine

(c)

The absolute minimum of C on (0, T].

To determine

(d)

To sketch:

The graph for C and f+g

And verify the result in part(a)

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Chapter 4.3, Problem 77E

Chapter 4.3, Problem 79E

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(a) To show, The critical numbers of C occur at the numbers t where C t = f t + g t .

Solution Summary: The author explains that the critical numbers of C occur at the numbers t, where the fundamental theorem of calculus is applied.

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(a)

To show,

The critical numbers of C occur at the numbers t where Ct=ft+gt.

(b)

The length of time T for Dt to equal the initial value of V.

(c)

The absolute minimum of C on (0, T].

(d)

To sketch:

The graph for C and f+g

And verify the result in part(a)

Want to see the full answer?

Chapter 4 Solutions

(a) To show, The critical numbers of C occur at the numbers t where C t = f t + g t .

Solution Summary: The author explains that the critical numbers of C occur at the numbers t, where the fundamental theorem of calculus is applied.

Concept explainers

Videos

(a) To show, The critical numbers of C occur at the numbers t where Ct=ft+gt.

(b) The length of time T for Dt to equal the initial value of V.

(c) The absolute minimum of C on (0, T].

(d) To sketch: The graph for C and f+g And verify the result in part(a)

Want to see the full answer?

Chapter 4 Solutions

(a)

To show,

The critical numbers of C occur at the numbers t where Ct=ft+gt.

(b)

The length of time T for Dt to equal the initial value of V.

(c)

The absolute minimum of C on (0, T].

(d)

To sketch:

The graph for C and f+g

And verify the result in part(a)