BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 1.3, Problem 60E
To determine

To find: The composition of AAAA and interpret it; also find an explicit formula for the composition of n copies of A.

Expert Solution

Answer to Problem 60E

The composition function (AAAA)(x)=(1.04)4(x) , which represents the amount after four years.

The formula for the composition of n copies of A, (AAAAup to n times)(x)=(1.04)n(x) .

Explanation of Solution

Given:

It is given that x dollars invested at 4% interest compounded annually. Then the invested amount after one year is A(x)=1.04x .

Calculation:

The composite function (fg)(x) is defined as, (fg)(x)=f(g(x)) .

First, find the composition of (AA)(x)=A(A(x)) .

Substitute A(x)=1.04x for x in A(x) ,

(AA)(1.04x)=1.04(1.04x)=(1.04)2(x)

Therefore, the composition of (AA)(x)=(1.04)2(x) . It represents the amount after two years.

Similarly to find (AAA)(x)=A((AA)(x)) substitute (AA)(x)=(1.04)2x for x in A(x) ,

(AAA)(x)=A((AA)(x))=1.04(1.04)2(x)=(1.04)3(x)

Therefore, the composition of (AAA)(x)=(1.04)3(x) . It represents the amount after three years.

Substitute (AAA)(x)=(1.04)3x for x in A(x) and obtain (AAAA)(x)=A((AAA)(x)) .

(AAAA)(x)=A((AAA)(x))=1.04(1.04)3(x)=(1.04)4(x)

Thus, the composition of (AAAA)(x)=(1.04)4(x) .  It represents the amount after four years.

Observe the pattern of the amount increasing as year increases. It can be concluded that the formula for the composition of n copies of A is, (AAAAup to n times)(x)=(1.04)n(x) .

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