Chapter 5, Problem 16PS

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Mortgage A $120,000 home mortgage for 35 years at 9 1 2 % has a monthly payment of$985.93. Part of the monthly payment goes for the interest charge on the unpaid balance, and the remainder of the payment is used to reduce the principal. The amount that goes for interest is u = M − ( M − Pr 12 ) ( 1 + r 12 ) 12 t and the amount that goes toward reduction of the principal is v = ( M − Pr 12 ) ( 1 + r 12 ) 12 t In these formulas, P is the amount of the mortgage, r is the interest rate (in decimal form), M is the monthly payment, and t is the time in years.(a) Use a graphing utility to graph each function in the same viewing window. (The viewing window should show all 35 years of mortgage payments.)(b) In the early years of the mortgage, the larger part of the monthly payment goes for what purpose? Approximate the time when the monthly payment is evenly divided between interest and principal reduction.(c) Use the graphs in part (a) to make a conjecture about the relationship between the slopes of the tangent lines to the two curves for a specified value of t. Give an analytical argument to verify your conjecture. Find u ' ( 15 ) and v ' ( 15 ) .(d) Repeat parts (a) and (b) for a repayment period of 20 years ( M = $1118.56 ) . What can you conclude? (a) To determine To graph: The functions u=M(MPr12)(1+r12)12t and v=(MPr12)(1+r12)12t for 35 years of mortgage payments. Explanation Given: A home mortgage of$120,000 for 35 years at 9.5% has a monthly payment of \$985.93.

The function u=M(MPr12)(1+r12)12t gives the interest amount and the function v=(MPr12)(1+r12)12t gives the capital repayment.

Graph:

Now replace 985.93 for M and 0.095 for r to obtain:

u=985.93(985.93(120000)(0.095)12)(1+0.09512)12tv=(985.93(120000)(0.095)12)(1+0

(b)

To determine
The purpose that larger part of the payment fulfils in the earlier years and to approximate the time when both the payment divisions are equal.

(c)

To determine
The relationship between the slopes of u and v both graphically and algebraically. Then calculate the slopes of both the functions at 15.

(d)

To determine

To graph: The functions u=M(MPr12)(1+r12)12t and v=(MPr12)(1+r12)12t for 20 years of mortgage payments and to determine the purpose that larger part of the payment fulfils in the earlier years and to approximate the time when both the payment divisions are equal.

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