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 3.8, Problem 26E

(a)

To determine

To find: To model the given data with a fourth-degree polynomial.

Expert Solution

Answer to Problem 26E

The fourth-degree polynomial is y=1.2×106t4+9545.853t328.48t2+37757.11t1.88×107.

Explanation of Solution

Given:

The average age of first marriage of Japanese women has varied since 1950 is shown in table (1).

tA(t)
195023.00
195523.80
196024.40
196524.50
197024.20
197524.70
198025.20
198525.50
199025.90
199526.30
200027.00
200528.00

Calculation:

Model the given data with a fourth-degree polynomial.

A(t)=at4+bt3+ct2+dt+e (1)

Plot the graph between t and A(t) using table (1) as shown in figure (1).

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 3.8, Problem 26E , additional homework tip  1

Refer the figure (1).

The fourth-degree polynomial for the curve is as below.

A(t)=1.2×106t4+0.0095t328.48t2+37757.11t1.88×107 (2)

Equate the equation (1) with equation (2).

at4+bt3+ct2+dt+e=1.2×106t4+0.0095t328.48t2+37757.11t1.88×107

Compare the above equation.

a=1.2×106, b=0.0095, c=28.48, d=37757.11 and e=1.88×107.

(b)

To determine

To find: The model for A'(t) by using part (a).

Expert Solution

Answer to Problem 26E

The value of A'(t) is 0.000005t3+28637.55t256.96t+37757.11.

Explanation of Solution

Calculate the value of A'(t).

Differentiate equation (1) with respect to t.

A'(t)=4at3+3bt2+2ct+d (3)

Substitute 1.2×106 for the expression a, 9545.853 for the expression b, 28.48 for the expression c and 37757.11 for the expression d.

A'(t)=4(1.2×106)t3+3(0.0095)t2+2(28.48)t+37757.11

A'(t)=0.000005t3+0.0285t256.96t+37757.11 (4)

Thus, the value of A'(t) is 0.000005t3+0.0285t256.96t+37757.11.

(c)

To determine

To estimate: The rate of change of marriage age for women in 1990.

Expert Solution

Answer to Problem 26E

The rate of change of marriage age for women in 1990 is 2,133.44.

Explanation of Solution

Compute the rate of change of marriage age for women in 1990 using the formula below.

A'(t)=0.000005t3+0.0285t256.96t+37757.11

Substitute 1990 for t.

A'(t)=0.000005×1,9903+0.0285×1,990256.96×1,990+37,757.11=2,133.44

(d)

To determine

To sketch: The graph of data points and models for A and A'.

Expert Solution

Explanation of Solution

Plot the graph between t and A(t) as shown in Figure (2).

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 3.8, Problem 26E , additional homework tip  2

Calculate the difference between the year 1950 and 1955.

Difference=19501945=5

Substitute the value 5 for the expression t in the equation (2).

A'(t)=0.000005(5)3+28637.55(5)256.96(5)+37757.11=0.114614

Similarly, calculate the value of A'(t) for the following values of t in table (1).

Tabulate the value of A'(t) as shown in table (2).

DifferenceA'(t)
00.221834
5.00.114614
100.052734
15.00.027194
200.028994
25.00.049134
300.078614
35.00.108434
400.129594
45.00.133094
500.109934
55.00.051114
60-0.052366

Plot the graph between t and A'(t) using the table (2) as shown in Figure (3).

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 3.8, Problem 26E , additional homework tip  3

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