   Chapter 11.5, Problem 106E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

Subprime Mortgage Debt during the Housing Bubble (Compare Exercise 104.) During the real estate run-up in 2000–2008 the value of subprime (normally classified as risky) mortgage debt outstanding in the United States could be approximated by A ( t ) = 1 , 350 x 1 + 4.2 ( 1.7 ) − t  percent ( 0 ≤ t ≤ 8 ) t years after the start of 2000.58a. How fast, to the nearest 1%.was the percentage increasing at the start of 2005?b. Compute lim t → + ∞ A ( t ) and lim t → + ∞ A ' ( t ) . What do the answers tell you about subprime mortgages?

(a)

To determine

To calculate: The rate of increase in subprime mortgage debt to the about \$1 billion at the start of 2005(t=5) when the value of risky mortgages debt remaining in the United States during the real estate run-up during 20002008 is approximated by,

A(t)=1,3501+4.2(1.7)tbillion dollars

Where, t is the time since 2000 (0t8).

Explanation

Given information:

The value of risky mortgages debt remaining in the United States during the real estate run-up during 20002008 is approximated by

A(t)=1,3501+4.2(1.7)tbillion dollars

Where, t is the time since 2000 (0t8).

Formula used:

Quotient rule of derivative of differentiable functions, f(x) and g(x) is,

ddx[f(x)g(x)]=f'(x)g(x)f(x)g'(x)[g(x)]2 Where, g(x)0.

Constant multiple rule of derivative of function f(x) is f'(cx)=cf'(x) where c is constant.

The derivative of constant b raised to a function is ddxbu=bulnbdudx.

Calculation:

Consider the function, A(t)=1,3501+4.2(1.7)t

The derivative of the function is,

A(t)=ddt(1,3501+4.2(1.7)t)

Apply the quotient rule of derivative,

A(t)=ddt(1350)(1+4.2(1.7)t)1350ddt(1+4.2(1.7)t)(1+4.2(1.7)t)2=0(1+4.2(1.7)t)1350(ddt(1)+ddt[4.2(1.7)t])(1+4.2(1.7)t)2=1350(ddt(1)+ddt[4.2(1.7)t])(1+4.2(1.7)t)2

Apply the constant multiple rule,

A(t)=1350(0+4

(b)

To determine

To calculate: The value of limt+A(t) and limt+A(t) and interpret the answer for the subprime or risky mortgage. when the value of risky mortgages debt remaining in the United States during the real estate run-up during 20002008 is approximated by,

A(t)=1,3501+4.2(1.7)tbillion dollars

Where, t is the time since 2000 (0t8).

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