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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 2.7, Problem 34E

a.

To determine

The meaning of P'(t) and also the unit of P'(t) .

Expert Solution

Explanation of Solution

Given information:

  P(t) is the percentage of Americans under the age 18 of at time t

The table gives values of this function in census years from 1950 to 2000

    tP(t)tP(t)
    195031.1198028.0
    196035.7199025.7
    197034.0200025.7

Calculations:

Here P'(t) shows the slope of the tangent of the curve made by the function or rate of change of Americans under the age 18 with respect to time (in years). The unit of P(t) will be Unemployed peer year.

The slope of the tangent is given the formula,

  f'(x)=limh0f(x+h)f(x)h

b.

To determine

The estimated values of P'(t) in a tabulated manner.

Expert Solution

Explanation of Solution

Given information:

  P(t) is the percentage of Americans under the age 18 of at time t

The table gives values of this function in census years from 1950 to 2000

    tP(t)tP(t)
    195031.1198028.0
    196035.7199025.7
    197034.0200025.7

Calculations:

Here P'(t) shows the slope of the tangent of the curve made by the function or rate of change of Americans under the age 18 with respect to time (in years) . The unit of P(t) will be Unemployed peer year.

The slope of the tangent is given the formula,

  f'(x)=limh0f(x+h)f(x)h

    tP(t)P'(t)tP(t)P'(t)
    195031.1198028.0-0.6
    196035.70.46199025.7-0.23
    197034.0-0.17200025.70

c.

To determine

The graph of P(t) and P'(t) .

Expert Solution

Explanation of Solution

Given information:

  P(t) is the percentage of Americans under the age 18 of at time t

The table gives values of this function in census years from 1950 to 2000

    tP(t)tP(t)
    195031.1198028.0
    196035.7199025.7
    197034.0200025.7

Calculations:

Here P'(t) shows the slope of the tangent of the curve made by the function or rate of change of Americans under the age 18 with respect to time (in years) . The unit of P(t) will be Unemployed peer year.

The slope of the tangent is given the formula,

  f'(x)=limh0f(x+h)f(x)h

    tP(t)P'(t)tP(t)P'(t)
    195031.1198028.0-0.6
    196035.70.46199025.7-0.23
    197034.0-0.17200025.70

The graph of P(t) is shown below.

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.7, Problem 34E , additional homework tip  1

The graph of P'(t) is shown below.

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.7, Problem 34E , additional homework tip  2

d.

To determine

The ways to get more accurate values of P'(t) .

Expert Solution

Explanation of Solution

Given information:

  P(t) is the percentage of Americans under the age 18 of at time t

The table gives values of this function in census years from 1950 to 2000

    tP(t)tP(t)
    195031.1198028.0
    196035.7199025.7
    197034.0200025.7

Calculations:

Here P'(t) shows the slope of the tangent of the curve made by the function or rate of change of Americans under the age 18 with respect to time (in years) . The unit of P(t) will be Unemployed peer year.

The slope of the tangent is given the formula,

  f'(x)=limh0f(x+h)f(x)h

The value of P'(t) can be more accurate if the time interval is less and more values of P(t) is obtained.

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