exponential functions

Exponential FunctionsSection 1.3

Let be a positive real number other than 1. The function

( )

is the .

x

a

f x a

a

exponential function with base

Exponential Functions

The function , 0, is a model for

if 1, and a model for if 0 1.

xy k a k

a a

exponential growth

exponential decay

What real-world situations can be modeledwith exponential functions???

Rules for Exponents

If 0 and 0, the following hold for all real numbers and .

1. 4.

2. 5.

3.

xx y x y x x

xx xx y

y x

y xx y xy

a b x y

a a a a b ab

a a aa

ba b

a a a

The Number eMany natural, physical and economic phenomena are best modeled

by an exponential function whose base is the famous number , which is

2.718281828 to nine decimal places.

We can define to be the numbe

e

e 1r that the function 1

approaches as approaches infinity.

x

f xx

x

Basic Practice Problems

Graph the function. State its domain, range, and intercepts.

3 4xy e

4y : ,D

: , 4R y-int: 0,1

x-int: 0.288,0

Basic Practice Problems

Rewrite the exponential expression to have the indicated base.

532 ,x base 2

55 532 2xx 252 x

31

,625

x

base 5

3 3

4

1 1

625 5

x x

345x 125 x

Basic Practice Problems

Solve the given equations graphically.

2 1xe

Did you remember to sketch your graphs?

No solution!

14 3 8 0x 1.631x

Multiple correct graphs for the second one?

Application ProblemsThe population of Flagstaff was 58,154 in 2005, and assumethat the population is growing exponentially at a rate of0.24% annually. What was the population in 2012?Approximately when (in what year) will the population be 70,000?

Hint: Let t represent the number of years since 2005

The model: 58154 1.0024t

P t

77 58154 1.0024P 59138 people

58154 1.0024 70000t

Solve graphically:

77.343t According to this model, the population of Flagstaff

will reach 70,000 in the year 2082.

Application ProblemsThe half-life of the radioactive element Proctorium-34 is 39 days. If Proctor needs at least 1 gram of the element to properly teach calculus, and he has 892 grams on the first day of school, for how long will he be able to teach calculus?

The model: 39892 0.5

tA t

Solve graphically: 382.235t Proctor will be able to teach calculus through the end of the school year!!!

39892 0.5 1

t

Application ProblemsDetermine how much time is required for an investment to triple in value if interest is earned at the rate of 6.1% compounded quarterly.

0 1kt

rA t A

k

Do you remember the equation for compound interest?

In this case, we want to solve:

4

0 0

0.0611 3

4

t

A A

18.147t years

Application ProblemsDetermine how much time is required for an investment to quadruple in value if interest is earned at the rate of 8.719% compounded continuously.

0rtA t A e

Do you remember the equation for continuous compounding?

In this case, we want to solve:0.08719

0 04tA e A15.900t years