Exponential Functions and Models

Lesson 3.1

Contrast

LinearFunctions

Change at a constant rate Rate of change (slope) is a constant

ExponentialFunctions

Change at a changing rate Change at a constant percent rate

Contrast

Suppose you have a choice of two different jobs at graduation Start at $30,000 with a 6% per year increase Start at $40,000 with $1200 per year raise

Which should you choose? One is linear growth One is exponential growth

Which Job?

How do we get each nextvalue for Option A?

When is Option A better? When is Option B better?

Rate of increase a constant $1200

Rate of increase changing Percent of increase is a constant Ratio of successive years is 1.06

Year Option A Option B

1 $30,000 $40,000

2 $31,800 $41,200

3 $33,708 $42,400

4 $35,730 $43,600

5 $37,874 $44,800

6 $40,147 $46,000

7 $42,556 $47,200

8 $45,109 $48,400

9 $47,815 $49,600

10 $50,684 $50,800

11 $53,725 $52,000

12 $56,949 $53,200

13 $60,366 $54,400

14 $63,988 $55,600

Example

Consider a savings account with compounded yearly income You have $100 in the account You receive 5% annual interest

At end of year

Amount of interest earnedNew balance in

account

1 100 * 0.05 = $5.00 $105.00

2 105 * 0.05 = $5.25 $110.25

3 110.25 * 0.05 = $5.51 $115.76

4    

5    

View completed table

Compounded Interest

Completed table

At end of year

Amount of interest earned

New balance in account

0 0 $100.001 $5.00 $105.002 $5.25 $110.253 $5.51 $115.764 $5.79 $121.555 $6.08 $127.636 $6.38 $134.017 $6.70 $140.718 $7.04 $147.759 $7.39 $155.1310 $7.76 $162.89

Compounded Interest Table of results from

calculator Set Y= screen

y1(x)=100*1.05^x Choose Table (♦ Y)

Graph of results

Compound Interest

Consider an amount A0 of money deposited in an account Pays annual rate of interest r percent Compounded m times per year Stays in the account n years

Then the resulting balance An

0 1m n

n

rA A

m

Exponential Modeling

Population growth often modeled by exponential function

Half life of radioactive materials modeled by exponential function

Growth Factor

Recall formulanew balance = old balance + 0.05 * old balance

Another way of writing the formulanew balance = 1.05 * old balance

Why equivalent?

Growth factor: 1 + interest rate as a fraction

Decreasing Exponentials Consider a medication

Patient takes 100 mg Once it is taken, body filters medication out over

period of time Suppose it removes 15% of what is present in

the blood stream every hour

At end of hour Amount remaining

1 100 – 0.15 * 100 = 85

2 85 – 0.15 * 85 = 72.25

3

4

5

Fill in the rest of the

table

What is the growth factor?

Decreasing Exponentials Completed chart

Graph

At end of hour Amount Remaining1 85.002 72.253 61.414 52.205 44.376 37.717 32.06

Amount Remaining

0.00

20.00

40.00

60.00

80.00

100.00

0 1 2 3 4 5 6 7 8

At End of Hour

Mg

rem

ain

ing

Growth Factor = 0.85

Note: when growth factor < 1, exponential is a decreasing

function

Solving Exponential Equations Graphically

For our medication example when does the amount of medication amount to less than 5 mg

Graph the functionfor 0 < t < 25

Use the graph todetermine when

( ) 100 0.85 5.0tM t

General Formula

All exponential functions have the general format:

Where A = initial value B = growth rate t = number of time periods

( ) tf t A B

Typical Exponential Graphs

When B > 1

When B < 1

( ) tf t A B

Using e As the Base

We have used y = A * Bt

Consider letting B = ek

Then by substitution y = A * (ek)t

Recall B = (1 + r) (the growth factor)

It turns out that k r

Continuous Growth

The constant k is called the continuous percent growth rate

For Q = a bt k can be found by solving ek = b

Then Q = a ek*t

For positive a if k > 0 then Q is an increasing function if k < 0 then Q is a decreasing function

Continuous Growth

For Q = a ek*t Assume a > 0

k > 0

k < 0

Continuous Growth

For the functionwhat is thecontinuous growth rate?

The growth rate is the coefficient of t Growth rate = 0.4 or 40%

Graph the function (predict what it looks like)

0.43 tQ e

Converting Between Forms

Change to the form Q = A*Bt

We know B = ek

Change to the form Q = A*ek*t

We know k = ln B (Why?)

0.43 tQ e

94.5(1.076)tQ

Continuous Growth Rates

May be a better mathematical model for some situations

Bacteria growth Decrease of medicine

in the bloodstream

Population growth of a large group

Example A population grows from its initial level of

22,000 people and grows at a continuous growth rate of 7.1% per year.

What is the formula P(t), the population in year t? P(t) = 22000*e.071t

By what percent does the population increase each year (What is the yearly growth rate)? Use b = ek