Copyright © 2011 Pearson Education, Inc. Slide 5.2-1 5.2 Exponential Functions Additional...

Copyright © 2011 Pearson Education, Inc. Slide 5.2-1

5.2 Exponential Functions

Additional Properties of ExponentsFor any real number a > 0, a 0, the following statements are true:

(a) ax is a unique real number for each real number x.(b) ab = ac if and only if b = c.(c) If a > 1 and m < n, then am < an.– If 0 < a < 1 and m < n, then am > an.

32 44 e.g.

3

212

21 e.g.


5.2 Exponential Functions

If a > 0, a 1, then

f (x) = ax

defines the exponential function with base a.


5.2 Graphs of Exponential Functions

Example Graph Determine the domain and range of f.

Solution

There is no x-intercept. Any number to the zero power is 1, so the y-intercept is (0,1). The domain is (– ,), and the range is (0,).

.2)( xxf


5.2 Comparing Graphs

Example Explain why the graph ofis a reflection across the y-axis of the graph of

Analytic SolutionShow that g(x) = f (–x).

xxg 21)(

.2)( xxf

)(2

2

21

)(

1

xf

xg

x

x

x


5.2 Comparing Graphs

Graphical Solution

The graph below indicates that g(x) is a reflection

across the y-axis of f (x).


5.2 Graph of f (x) = ax, a > 1


5.2 Graph of f (x) = ax, 0 < a < 1


5.2 Using Translations to Graph an Exponential Function

Example Explain how the graph of is obtained from the graph of

Solution

2 3 xy

.2xy

units 3 up d translateis 2 32

axis- theacross 2reflect 2xx

xx

y

xy


5.2 Example using Graphs to Evaluate Exponential Expressions

Example Use a graph to evaluate

Solution With we find

that y 2.6651441 from the graph of y = 0.5x.

20.5 .

,414214.12 x


5.2 Exponential Equations (Type I)

ExampleSolve

Solution

.12525 x

12525 x

Write with the same base. 32 55 x

mnnm aa 32 55 x

Set exponents equal and solve.

2332

x

x

. isset solution The.12525 :Verify 232

3


5.2 Using a Graph to Solve Exponential Inequalities

Example Solve the inequality

Solution Using the graph below, the graph lies above the x-axis for values of x less than 0.5.

.05.1 8271 xx

The solution set for y > 0 is (–, 0.5).


5.2 Compound Interest

• Recall simple earned interest where– P is the principal (or initial investment), – r is the annual interest rate, and – t is the number of years.

• If A is the final balance at the end of each year, then

,tPrI

nth rPAn

rPrrPrrPrPArPPrPA

)1(:year

)1()1)(1()1()1(:year 2)1(:year 1

2nd

st


5.2 Compound Interest Formula

Example Suppose that $1000 is invested at an annual rate of 4%, compounded quarterly. Find the total amount in the account after 10 years if no withdrawals are made.

Solution

The final balance is $1488.86.

Suppose that a principal of P dollars is invested at an annual interest rate r (in decimal form), compounded n times per year. Then, the amount A accumulated after t years is given by the formula

.1nt

nr

PA

4 10

0.041000 1 1488.8637

4A


5.2 The Natural Number e

• Named after Swiss mathematician Leonhard Euler

• e involves the expression

• e is an irrational number• Since e is an important

base, calculators are programmed to find powers of e.

:1 1 xx

718281828.2 1, as 1 ex xx


5.2 Continuous Compounding Formula

Example Suppose $5000 is deposited in an account paying 3% compounded continuously for 5 years. Find the total amount on deposit at the end of 5 years.

Solution

The final balance is $5809.17.

If an amount of P dollars is deposited at a rate of interest r (in decimal form) compounded continuously for t years, then the final amount in dollars is

.rtPeA

0.03(5) 0.155000 5000 5809.17A e e


5.2 Modeling the Risk of Alzheimer’s Disease

Example The chances of dying of influenza or pneumonia increase exponentially after age 55 according to the function defined by

where r is the risk (in decimal form) at age 55 and x is the number of years greater than 55. Compare the risk at age 75 with the risk at age 55.Solutionx = 75 – 55 = 20, soThus, the risk is almost fives times as great at age 75as at age 55.

( ) (1.082) , xf x r

20( ) (1.082) 4.84 f x r r

Copyright © 2011 Pearson Education, Inc. Slide 5.2-1 5.2 Exponential Functions Additional...

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