Chapter 11.2, Problem 43E

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Future value If $P is invested for n years at 10% compounded continuously, the future value that results after n years is given by the function S = P e 0.1 n (a) At what rate is the future value growing at any time (for any nonnegativen) 7 (b) At what rate is the future value growing after 1 year (n = 1)?(c) Is the rate of growth of the future value after 1 year greater than 10%? Explain. (a) To determine To calculate: The rate at which the future value of the$P grows at any time n if the $P is invested for n years and is compounded at 10% interest rate, continuously. Explanation Given Information: The future value of the$P is represented by the function, S=Pe0.1n.

Formula used:

If f(x)=cu(x), where, c is a constant and u(x) is a differentiable function of x, then,

f(x)=cu(x)

According to the property of derivatives, if y=ex, where, u is a differentiable function of x,

dydx=ex

According to the power rule of differentiation,

dydx=nxn1

Calculation:

Consider the sales function provided,

S=Pe0.1n …… (1)

Differentiate both sides with respect to n,

S=ddx(Pe0

(b)

To determine

To calculate: The rate at which the future value of the $P grows after 1 year of investment if the$P is invested for n years and is compounded at 10% interest rate, continuously.

(c)

To determine

Whether the rate of growth of the future value after 1 year is greater than 10% if the \$P is invested for n years and is compounded at 10% interest rate, continuously.

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