Chapter 5.2, Problem 59E

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

Chapter
Section

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Marginal Propensity to Consume InExercises 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 = 0.95 ( x − 24 , 999 ) 0.05 , x ≥ 25 , 000 (a) To determine To calculate: The income consumed Q as a function of x for the provided equation dQdx=0.95(x24,999)0.05 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=0.95(x24,999)0.05 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 n1

Calculation:

Consider the provided equation dQdx=0.95(x24,999)0.05.

Integrate dQdx to obtain Q.

Q=0.95(x24,999)0.05dx …… (1)

Let x24,999=u

Differentiate the above equation with respect to x,

ddx(x24,999)=dudxddx(x)ddx(24,999)=dudx1=dudxdx=du

Substitute the value of dx=du and x24,999=u in equation (1),

Q=0.95duu0.05=0.95u0

(b)

To determine

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

part (a) for the following table.

x25,00050,000100,000150,000QxQ

(c)

To determine

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

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