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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

Energy use Energy use per dollar of GDP indexed to 1980 means that energy use for any year is viewed as a percent of the use per dollar of GDP in 1980. The following data show the energy use per dollar of GDP, as a percent, for selected years from 1985 and projected to 2035.

Energy Use per Dollar of GDP

Year Percent Year Percent
1985 83 2015 51
1990 79 2020 45
1995 75 2025 41
2000 67 2030 37
2005 60 2035 34
2010 56

Source: U.S. Department of Energy

These data can be modeled with the function

E ( t ) = 0.0039 ( 0.4 t + 2 ) 3 0.13 ( 0.4 t + 2 ) 2 1.4 ( 0.4 t + 2 ) + 91 where E(t) is the energy use per dollar of GDP (indexed to 1980) and t is the number of years past 1980.

(a) Find E'(t).

(b) Use this model to find and interpret the instantaneous rates of change of energy use per dollar of GDP in 2000 and 2025.

(c) Use the data in the table to find an average rate of change that approximates the 2025 instantaneous rate.

(a)

To determine

To calculate: The partial derivative E'(t) of the function E(t)=0.0039(0.4t+2)30.13(0.4t+2)21.4(0.4t+2)+91, where E(t) is the energy use per dollar and t is the number of years pat 1980.

Explanation

Given Information:

The function E(t)=0.0039(0.4t+2)30.13(0.4t+2)21.4(0.4t+2)+91 where E(t) is the energy use per dollar and t is the number of years pat 1980.

Year Percent Year Percent
1985 83 2015 51
1990 79 2020 45
1995 75 2025 41
2000 67 2030 37
2005 60 2035 34
2010 56

Formula used:

The power rule of differentiation,

ddxxn=nxn1

Calculation:

Consider the provided function,

E(t)=0.0039(0.4t+2)30.13(0.4t+2)21.4(0.4t+2)+91

The instantaneous rate of change of total national health care expenditures is given by A'(t).

Consider (0.4t+2) to be u,

E(t)=0.0039u30.13u21.4u+91

Differentiate both sides with respect to t,

dEdt=ddt(0.0039u30.13u21.4u+91)

Simplify using the power rule,

dEdt=ddt(0.0039u3)ddt(0.13u2)ddt(1.4u)+ddt(91)=0.0039ddtu30.13ddtu21

(b)

To determine

To calculate: The instantaneous rate of change of energy the function E(t)=0.0039(0.4t+2)30.13(0.4t+2)21.4(0.4t+2)+91 use per dollar of GDP in 2000 and 2025 and interpret it.

(c)

To determine

To calculate: The average rate of change that approximates the instantaneous rate of change function E(t)=0.0039(0.4t+2)30.13(0.4t+2)21.4(0.4t+2)+91 where E(t) is the energy use per dollar and t is the number of years for 2025.

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