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
Slide 4.2- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Slide 4.2- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Slide 4.2- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Slide 4.2- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Slide 4.2- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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:
Slide 4.2- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Slide 4.2- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Slide 4.2- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3Using Different Compounding Periods to Compare Future Values
(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.
Slide 4.2- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3Using Different Compounding Periods to Compare Future Values
(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
Slide 4.2- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3Using Different Compounding Periods to Compare Future Values
(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
Slide 4.2- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Slide 4.2- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
CONTINUOUS COMPOUND-INTEREST FORMULA
A = amount after t yearsP = principalr = annual interest rate (expressed as a
decimal)t = number of years
APert
Slide 4.2- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Slide 4.2- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Slide 4.2- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5 Calculating the Amount of Repaying a Loan
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.
Slide 4.2- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5 Calculating the Amount of Repaying a Loan
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.
Slide 4.2- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Slide 4.2- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
THE NATURAL EXPONENTIAL FUNCTION
Slide 4.2- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Slide 4.2- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Slide 4.2- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Slide 4.2- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Slide 4.2- 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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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