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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

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BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

Gross domestic product The table shows U.S. gross domestic product (GDP) in billions of dollars for selected years from 2000 to 2070 (actual and projected).

Year GDP Year GDP
2000 9143 2040 79,680
2005 12,145 2045 103,444
2010 16,174 2050 133,925
2015 21,270 2055 173,175
2020 27,683 2060 224,044
2025 35,919 2065 290,042
2030 46,765 2070 375,219
2035 61,100

Source: Social Security Administration Trustees Report

Assume the GDP can be modeled with the function

G ( t )   = 212.9 ( 0.2 t   +   5 ) 3   5016 ( 0.2 t   +   5 ) 2 +   8810.4 t   +   104 , 072

where G(t) is in billions of dollars and t is the number of years past 2000.

(a) Use the model to find and interpret the instantaneous rates of change of the GDP in 2005 and 2015.

(b) Use the data in the table to find the average rate of change of the GDP from 2005 to 2015.

(c) How well does your answer from part (b) approximate the instantaneous rate of change of GDP in 2010?

(a)

To determine

To calculate: The instantaneous rate of change of the GDP in 2005 and 2015 and interpret it. The table shows U.S. gross domestic product (GDP) in billions of dollars for selected years from 2000 to 2070. Assume that the GDP can be modeled with the function G(t)=212.9(0.2t+5)35016(0.2t+5)2+8810.4t+104,072 where G(t) is in billions of dollars and t is the number of years past 2000.

Explanation

Given Information:

The table shows U.S. gross domestic product (GDP) in billions of dollars,

Year GDP Year GDP
2000 9143 2040 79,680
2005 12,145 2045 103,444
2010 16,174 2050 133,925
2015 21,270 2055 173,175
2020 27,683 2060 224,044
2025 35,919 2065 290,042
2030 46,765 2070 375,219
2035 61,100

Formula used:

According to the power rule, if f(x)=xn, then,

f(x)=nxn1

According to the property of differentiation, if a function is of the form, g(x)=cf(x), then,

g(x)=cf(x)

According to the property of differentiation, if a function is of the form f(x)=u(x)+v(x), then,

f(x)=u(x)+v(x)

According to the product rule, if f(x)=u(x)v(x), then

f(x)=u(x)v(x)+v(x)u(x)

According to the property of differentiation, if a function is of the form y=un, where u=g(x),

dydx=nun1dudx

Calculation:

Consider the provided function,

G(t)=212.9(0.2t+5)35016(0.2t+5)2+8810.4t+104,072

Consider (0.2t+5) to be u,

G(t)=212.9(u)35016(u)2+8810.4t+104,072

Differentiate both sides with respect to t,

dGdt=ddt(212.9(u)35016(u)2+8810.4t+104,072)

Simplify using the power rule,

dGdt=ddt(212.9u3)ddt(5016u2)+ddt(8810

(b)

To determine

To calculate: The average rate of change of the GDP from 2005 to 2015 using the data given. The table shows U.S. gross domestic product (GDP) in billions of dollars for selected years from 2000 to 2070. Assume that the GDP can be modeled with the function G(t)=212.9(0.2t+5)35016(0.2t+5)2+8810.4t+104,072 where G(t) is in billions of dollars and t is the number of years past 2000.

(c)

To determine

The wellness of average rate of change of the GDP from 2005 to 2015 approximates the instantaneous rate of change of GDP in 2020. The table shows U.S. gross domestic product (GDP) in billions of dollars for selected years from 2000 to 2070. Assume that the GDP can be modeled with the function G(t)=212.9(0.2t+5)35016(0.2t+5)2+8810.4t+104,072 where G(t) is in billions of dollars and t is the number of years past 2000.

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