BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 5.4, Problem 20E

a.

To determine

To find: the value of x at which the local maximum and minimum values of g occur.

Expert Solution

Answer to Problem 20E

The value of x at which the local maximum and minimum values of g occur at 2, 6 and 10 and at 4 and 8 respectively.

Explanation of Solution

Given information:

The function is g(x)=0xf(t)dt and the graph of f(t) is shown below:

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 5.4, Problem 20E , additional homework tip  1

Concept used:

The fundamental theorem of calculus, part 1 is defined by,

If f is continuous on [a,b] , then the function g defined by,

  g(x)=axf(t)dt axb is an antiderivative of f , that is g(x)=f(x) for a<x<b .

Using Part 1 of the Fundamental Theorem of Calculus,

  g(x)=f(x) [since, f(t) is continuous and 0<x<b ]

Since, a function has a local maximum or local minimum at a point x0 if and only if the first derivative of a function is equal to zero at x=x0 .

Hence, the function g has a local maximum and minimum at a point where g(x) that is f(x) is concave up and concave down respectively and touches the x -axis.

From the graph provided in the question, it can be observed that f is concave up between 0x2 , 4x6 and 8x10 and is concave down between 2x4 and 6x8 .

Therefore,

The value of x at which the local maximum and minimum values of g occur at 2, 6 and 10 and at 4 and 8 respectively.

b.

To determine

To find: the value of x at which the absolute maximum values of g occur.

Expert Solution

Answer to Problem 20E

The value of x at which the absolute maximum values of g occur at x=2 .

Explanation of Solution

Given information:

The function is g(x)=0xf(t)dt and the graph of f(t) is shown below:

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 5.4, Problem 20E , additional homework tip  2

Concept used:

The fundamental theorem of calculus, part 1 is defined by,

If f is continuous on [a,b] , then the function g defined by,

  g(x)=axf(t)dt axb is an antiderivative of f , that is g(x)=f(x) for a<x<b .

Using Part 1 of the Fundamental Theorem of Calculus,

  g(x)=f(x) [since, f(t) is continuous and 0<x<b ]

Since, a function has a local maximum or local minimum at a point x0 if and only if the first derivative of a function is equal to zero at x=x0 .

Hence, the function g has a local maximum at a point where g(x) that is f(x) has highest concave up peak and touches the x -axis.

From the graph provided in the question, it can be observed that f has absolute maximum at x=2 .

Therefore,

The value of x at which the absolute maximum values of g occur at x=2 .

c.

To determine

To find: the interval at which g is concave downward.

Expert Solution

Answer to Problem 20E

The interval at which the function g is concave down is between 2x4 and 6x8 .

Explanation of Solution

Given information:

The function is g(x)=0xf(t)dt and the graph of f(t) is shown below:

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 5.4, Problem 20E , additional homework tip  3

Concept used:

The fundamental theorem of calculus, part 1 is defined by,

If f is continuous on [a,b] , then the function g defined by,

  g(x)=axf(t)dt axb is an antiderivative of f , that is g(x)=f(x) for a<x<b .

Using Part 1 of the Fundamental Theorem of Calculus,

  g(x)=f(x) [since, f(t) is continuous and 0<x<b ]

Since, a function has a local maximum or local minimum at a point x0 if and only if the first derivative of a function is equal to zero at x=x0 .

Hence, the function g has a local maximum and minimum at a point where g(x) that is f(x) is concave up and concave down respectively and touches the x -axis.

From the graph provided in the question, it can be observed that f is concave down between 2x4 and 6x8 .

Therefore,

The interval at which the function g is concave down is between 2x4 and 6x8 .

d.

To determine

To sketch the graph of g

Expert Solution

Explanation of Solution

Given information:

The function is g(x)=0xf(t)dt and the graph of f(t) is shown below:

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 5.4, Problem 20E , additional homework tip  4

Concept used:

The fundamental theorem of calculus, part 1 is defined by,

If f is continuous on [a,b] , then the function g defined by,

  g(x)=axf(t)dt axb is an antiderivative of f , that is g(x)=f(x) for a<x<b .

Graph:

From the graph of first derivative of g that is f(x) , the following observation is made about the graph of g :

  • The derivative g(x)=f(x) is zero at x=2,4,6,8,10 , thus these are the turning point of the graph of the function g .
  • The function g is increasing on the interval (0,2),(4,6) and (8,10) because its first derivative is positive at these interval.
  • The function g is decreasing on the interval (2,4) and (6,8) because its first derivative is negative at these interval.

Hence, from the above observation,

The graph of g is obtained as:

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 5.4, Problem 20E , additional homework tip  5

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