slide 4.2- 1 copyright © 2007 pearson education, inc. publishing as pearson addison-wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

The Natural Exponential Function

Learn to develop a compound-interest formula.

Learn to understand the number e.

Learn to graph exponential functions.

Learn to evaluate exponential functions.

SECTION 4.2

1

2

3

4


TerminologyInterest A fee charged for borrowing a lender’s money is called the interest, denoted by I.

Principal The original amount of money borrowed is called the principal, or initial amount, denoted by P.

Time Suppose P dollars is borrowed. The borrower agrees to pay back the initial P dollars, plus the interest, within a specified period. This period is called the time of the loan and is denoted by t.


TerminologyInterest Rate The interest rate is the percent charged for the use of the principal for the given period. The interest rate is expressed as a decimal and denoted by r. Unless stated otherwise, it is assumed to be for one year; that is, r is an annual interest rate.

Simple Interest The amount of interest computed only on the principal is called simple interest.


SIMPLE INTEREST FORMULA

The simple interest I on a principal P at a rate r (expressed as a decimal) per year for t years is

I Prt.


EXAMPLE 1 Calculating Simple Interest

I Prt

I $8000 0.06 5 I $2400

Juanita has deposited 8000 dollars in a bank for five years at a simple interest rate of 6%.a. How much interest will she receive?b. How much money will she receive at the

end of five years?Solution

a. Use the simple interest formula with P = 8000, r = 0.06, and t = 5.


EXAMPLE 1 Calculating Simple Interest

Solution continued

A P I

A $8000 $2400

A $10, 400

b. The total amount due her in five years is the sum of the original principal and the interest earned:


COMPOUND-INTEREST FORMULA

A = amount after t yearsP = principalr = annual interest rate (expressed as a

decimal)n = number of times interest is

compounded each yeart = number of years

AP 1 rn

nt


EXAMPLE 3Using Different Compounding Periods to Compare Future Values

One hundred dollars is deposited in a bank that pays 5% annual interest. Find the future value A after one year if the interest is compounded

(i) Annually.(ii) Semiannually.(iii) Quarterly.(iv) Monthly.(v) Daily.



(i) Annually A P 1r

n

n

A 100 1 0.05 $105.00

Solution

In each of the computations that follow, P = 100 and r = 0.05 and t = 1. Only n, the number of times interest is compounded each year, is changing. Since t = 1, nt = n(1) = n.



(iii) Quarterly

A P 1r

4

4

A 100 10.05

4

4

$105.09

(ii) Semiannually

A P 1r

n

n

A 100 10.05

2

2

$105.06



(iv) MonthlyA P 1

r

12

12

A 100 10.05

12

12

$105.12

(v) Daily

A P 1r

365

365

A 100 10.05

365

365

$105.13


THE VALUE OF eThe number e, an irrational number, is sometimes called the Euler constant.

The value of e to 15 places ise = 2.718281828459045.

Mathematically speaking, e is the fixed number that the expression

approaches as h gets larger and larger.

11

h

h


CONTINUOUS COMPOUND-INTEREST FORMULA

A = amount after t yearsP = principalr = annual interest rate (expressed as a

decimal)t = number of years

APert


EXAMPLE 4 Calculating Continuous Compound Interest

Find the amount when a principal of 8300 dollars is invested at a 7.5% annual rate of interest compounded continuously for eight years and three months.Solution

Convert eight years and three months to 8.25 years. P = $8300 and r = 0.075.

A Pert

A $8300e 0.075 8.25

A $15, 409.83


EXAMPLE 5 Calculating the Amount of Repaying a Loan

How much money did the government owe DeHaven’s descendants for 213 years on a 450,000-dollar loan at the interest rate of 6%?

Solution

a. With simple interest.

A P Prt P 1 rt A $450,000 1 0.06 213 A $6.201 million.



Solution continued

b. With interest compounded yearly.A P 1 r t $450,000 1 0.06 213

A $1.105 1011

A $110.5 million.c. With interest compounded quarterly.

A P 1r

4

4 t

$450,000 10.06

4

4 213

A $1.45305 1011

A $145.305 billion.



Solution continued

d. With interest compounded continuously.

Notice the dramatic difference of more than $14 billion between quarterly and continuously compounding. Notice also the dramatic difference between simple interest and interest compounded yearly.

A Pert $450,000e0.06 213

A $1.5977 1011

A $159.77 billion.


THE NATURAL EXPONENTIAL FUNCTION

with base e is so prevalent in the sciences that it is often referred to as the exponential function or the natural exponential function.

f x ex

The exponential function


THE NATURAL EXPONENTIAL FUNCTION


EXAMPLE 6 Sketching a Graph

Use transformations to sketch the graph ofg x 3x 1 2.

Solution

Shift the graph of f (x) = ex, 1 unit right and 2 units up.


MODEL FOR EXPONENTIALGROWTH OR DECAY

A t A0ekt

A(t) = amount at time t A0 = A(0), the initial amount k = relative rate of growth (k > 0) or

decay (k < 0) t = time


EXAMPLE 7 Modeling Exponential Growth and Decay

In the year 2000, the human population of the world was approximately 6 billion and the annual rate of growth was about 2.1 percent. Using the model on the previous slide, estimate the population of the world in the following years.

a. 2030b. 1990


EXAMPLE 7 Bacterial Growth

A0 6

k 0.021

t 30

A t 6e 0.021 30

A t 11.265663

a. Use year 2000 as t = 0

Solution

The model predicts there will be 11.26 billion people in the world in the year 2030.


EXAMPLE 7 Bacterial Growth

A0 6

k 0.021

t 10

A t 6e 0.021 10

A t 4.8635055

b. Use year 2000 as t = 0

Solution

The model predicts that the world had 4.86 billion people in 1990 (actual was 5.28 billion).

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