Exponential Functions

Lesson 2.4

Aeronautical Controls

Exponential Rate • Offers servo travel that is not directly

proportional to stick travel. • Control response is milder below half-stick, but

becomes increasing stronger as stick travel approaches 100%. Great for aerobatics and trouble situations.

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What airplane is this?

What airplane is this?

General Formula

All exponential functions have the general format:

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

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( ) tf t A B

Typical Exponential Graphs

When B > 1

When B < 1

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( ) tf t A B

Exponential Equations

Given

• What could you say about x and y?

If the two quantities are equal and the base value for the exponential expression are the same . . .• Then the exponents must be the same

Use to solve exponential equations

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x ya a

9 27x

Simple Interest

If you start with an amount P, the principal• and receive interest rate at r%• for time t

• Then the interest earned is I, the product of P, r (as a decimal) and t

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I P r t

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 t years

Then the resulting balance At

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0 1m t

n

rA A

m

Compound Interest

What happens when we increase the number of compounding periods?

Try $1000 at 3.5% for 6 years• Compounded yearly?• Quarterly• Monthly• Weekly• Daily• For every hour? every minute? every second?

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The Irrational Number e

As the number of compounding periods increase• The change in the end result becomes less• We reach a limit

Can be shown

• Where e ≈ 2.71828

Note Page 90, 919

lim 1m t

r t

m

rP P e

m

Continuous Compounding

Try our $1000 at 3.5% for 6 years using

Compare to with large m

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r tP e

0 1m t

n

rA A

m

Assignment

Lesson 2.4

Page 106

Exercises 3 – 47 EOO

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