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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 5.2, Problem 60E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Marginal Propensity to Consume In

Exercises 59 and 60, (a) use the marginal propensity to consume, dQ/dx, to write Q as a function of x, where x is the income (in dollars) and Q is the income consumed (in dollars). Assume that families who have annual incomes of $25,000 or less consume 100% of their income. (b) Use the result of part (a) and a spreadsheet to complete the table showing the income consumed and the income saved, x - Q, for various incomes. (c) Use a graphing utility to represent graphically the income consumed and saved. See Example 8.

X 25,000 50,000 100,000 150,000
Q
x-Q

d Q d x = 0.93 ( x − 24 , 999 ) 0.007 , x ≥ 25 , 000

(a)

To determine

To calculate: The income consumed Q as a function of x for the provided equation

dQdx=0.93(x−24,999)0.07 when the annual income is $25,000 and the income consumed is 100%

that is $25,000.

Explanation

Given Information:

The marginal propensity to consume income x is modeled by the equation

dQdx=0.93(x−24,999)0.07 where Q is income consumed and x is annual income.

Formula Used:

According to the general power rule for integration,

If u is a differentiable function of x, then

∫undu=un+1n+1+C

where n≠−1

Calculation:

Consider the provided equation dQdx=0.93(x−24,999)0.07.

Integrate dQdx to obtain Q.

Q=∫0.93(x−24,999)0.07dx …… (1)

Let x−24,999=u

Differentiate the above equation with respect to x,

ddx(x−24,999)=dudxddx(x)−ddx(24,999)=dudx1=dudxdx=du

Substitute the value of dx=du and x−24,999=u in equation (1),

Q=0.93∫duu0.07=0.95∫u−0

(b)

To determine

To calculate: The values of income consumed Q and income saved x−Q by using the result of

part (a) for the following table.

x25,00050,000100,000150,000Qx−Q

(c)

To determine

To graph: The income consumed Q and income saved x−Q using graphing utility from the table obtained in part (b).

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Research Methods for the Behavioral Sciences (MindTap Course List)

Oil wells are expensive to drill, and dry wells are a great concern to oil exploration companies. The domestic ...

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The report referenced in the previous exercise also gave data for representative samples of 250 adults in Canad...

Introduction To Statistics And Data Analysis

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