6.1 Exponential Functions

8/4/2019 6.1 Exponential Functions

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Transcendental

Functions6.1 Exponential Functions

RA Idoy

MATH17



• Theorems

• Simple interest

• Compound interest

• The number e

• Exponential Function with baseb

• Natural Exponential Function

• Application



Theorems

Theorem 1.

If r and s are rational

numbers, then

(i) if b>1, r<s implies br <bs,

(ii)if 0<b<1, r<s implies br>bs



Theorems

Theorem 2.

If a and b are any positive

numbers, and x and y are any

real numbers, then

)

)

)

x y x y

x

x y y

y x xy

i a a a

aii aa

iii a a

)

)

x x x

x x

x

iv ab a b

a av

b b



Simple Interest

Definition:

Simple interest is interest

due only on the original amount

that is borrowed. It is given

by the formula

where

P dollars is the investment, i is the

simple interest rate, n is the years

1 A P Pni P ni



Simple Interest

1 A P Pni P ni



Simple Interest

Example:

A loan of $500 is made

for a period of 90 daysat a simple interest rate

of 16 percent annually.

Determine the amount tobe repaid at the end of

90 days.



Simple Interest

Solution:

Let P = 500, i=0.16, n=90/360

Therefore, the amount to be repaid at

the end of 90 days is $520.

1

1500 1 0.16

4

520

A P ni



Compound Interest

When the interest for each

period is added to the

principal and itself earns

interest, we have compound interest.

1

mt

i A Pm



Compound Interest

Example:

Suppose that $400 isdeposited into a saving account

that pays 8 percent per yearcompounded semiannually. If nowithdrawals and no additionaldeposits are made, what is theamount on deposit at the end of3 years?



Compound Interest

Solution: Let P = 400, i=0.08, m=2,

t=3

The amount of deposit at the end of 3

years is therefore $506.13.

2 3

6

1

0.08400 1

2

400 1.04

506.13

mt i

A Pm



Compound Interest

Let P=1,

i=1, and

t=1.

1

11

mt

m

i A P

m

m



The number e

Continuous compound interest

2.7182818e

it A Pe



Exponential Function with

base b

If b>0 and b≠1, then the

exponential function with base

b is the function f defined by

Note:

x f x b

Dom f

Rng f





Natural Exponential

Function

The natural exponential function

is the function f defined by

Note:

x

f x e

Dom f

Rng f



Applications

• Exponential Growth

• Exponential Decay

•

Bounded Growth



Exponential Growth

Usually applied to:

Bacteria growth

Population growth

Given by the function:

0kt f t Be t



Exponential Growth

Graph: 0kt f t Be t

B

t

f(t)



Exponential Growth

Example: In a particular bacteria

culture, if f(t) bacteria are

present at t minutes, then

where B is a constant. If there are

1500 bacteria present initially,how many bacteria will be present

after 1 hour?

0.04t f t Be



Exponential Growth

Solution:

Given: At time

t=0, f(0)=1500. After one hour…

0.04

0.04 0

0

0

1500

1500

t f t Be

f Be

Be

B

0.041500

t f t e

0.04 60

2.4

60 1500

1500

1500 11.023

16535

f e

e



Exponential Growth

Conclusion:

Therefore, there are 16535

bacteria present in the culture

after 1 hour.



Exponential Decay

Usually applied to:

Radioactivity

Sales


0kt f t Be t



Exponential Decay

Graph: 0kt f t Be t

B

t

f(t)



Exponential Decay

Example: If V(t) dollars is thevalue of a certain piece ofequipment t years after itspurchase, then

where B is a constant. If the

equipment was purchased for$8000, what will be its value in2 years?

0.20t V t Be



Exponential Decay

Solution:

Given: At time

t=0, V(0)=8000. After 2 years…

0.20

0.20 0

0

0

8000

8000

t V t Be

V Be

Be

B

0.208000

t V t e

0.20 2

0.4

2 8000

8000

8000 0.670320

5362.56

V e

e



Exponential Decay

Conclusion:

Therefore, the value of the

equipment in 2 years will be

$5362.56.



Bounded Growth


1 0

kt

f t A e t



Bounded Growth

Graph:

A

t

f(t)

1 0kt f t A e t



Bounded Growth

Example: A typical worker at a certainfactory can produce f(t) units perday after t days on the job, where

(a)How many units per day can theworker produce after 7 days on thejob?

(b)How many units per day can theworker eventually be expected toproduce?

0.34

50 1t

f t e



Bounded Growth

Solution for a:

We wish to find

f(7)

Conclusion:

The worker can

produce 45 units

per day after 7days on the job.

0.34 7

2.38

7 50 1

50 1

50 1 0.093

45

f e

e





Bounded Growth

Bounded growth is also described

by a function defined by

kt

f t A Be

0,t A B

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6.1 Exponential Functions