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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Let g ( x ) = 0 x f ( t ) d t , where f is the function whose graph is shown.

(a) At what values of x do the local maximum and minimum values of g occur?

(b) Where does g attain its absolute maximum value?

(c) On what intervals is g concave downward?

(d) Sketch the graph of g.

Chapter 5.3, Problem 74E, Let g(x)=0xf(t)dt, where f is the function whose graph is shown. (a) At what values of x do the

(a)

To determine

The value of x which gives the maximum and minimum value of g.

Explanation

Given information:

The function is g(x)=0xf(t)dt.

Show the graph for the function as in Figure 1.

Show the integral function as follows:

g(x)=0xf(t)dt (1)

Here, g(x) is area under the graph of f from a to x and f(t) is function of t.

The Fundamental Theorem of Calculus, Part 1 is shown below:

g(x)=axf(t)dtaxb

Here, the continuous function on the interval [a,b] is f and the function is g.

Condition for the theorem to be valid:

  • If the function is continuous on the interval [a,b] and differentiable on (a,b)

(b)

To determine

To find: the absolute maximum value of g.

(C)

To determine

the intervals g when it is concave downward.

(d)

To determine

To sketch: the rough graph of g.

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