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Equ. #1:

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Equ. #6:

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Equ. #8:

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Ineq. #2:

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Exponential Functions

GOAL: Learn exponential functions with different bases and use them to model real-world situtations.

Exponential functions are of the form : where b > 0 is called base, like f(x) = 2^x.

Q1: Where do they appear?

A1: Everywhere! For example, if we put $1 in an account paying 5% interest, compounded annually,
then t years later it will become f(t) = (1.05)^t, which is an exponential function with base b = 1.05.

The laws of exponents. For b > 0 and u and v any numbers, we have

and

for any real number r; and

Example 1 If and then

Graph of

Case 1: b > 1

For example, y = 2x.

(i) Complete the table below:

Truncate answers to 2 decimal places

(ii) Plot the points and sketch graph:

(iii) Properties of b^x when b > 1:

• Asymptote:

Case 2: 0 < b < 1

For example, y = (1/2)^x.

(i) Complete the table below:

Truncate answers to 2 decimal places

(ii) Plot the points and sketch graph:

(iii) Properties of b^x when 0 < b < 1:

• Asymptote:

Three applications of the exponential function
1 Compound interest
Example 1 If $1,000 is invested in an account paying 5% interest, how much will it grow to in 10 year
if the interest is compounded monthly?

• Annual (in decimals)
• Compounding per

• Compounding
• Time(in years)

At the end of 1st period have:

At the end of 2nd period have:

At the end of 3th period have:
...
At the end of nth period have:

Interest compounded 12 times a year over t years

At the end of 1 year (12 periods) have:

At the end of 2 years (24 periods) have:
...
At the end of t years have:
General formula:

Example 2 If $8,000 is invested in an account paying 3% interest, how much will it grow to in 15
years if the interest is compounded quarterly?
2 Population Growth (with unlimited resources)

Example 3 A certain bacteria culture grows exponentially. In 1 hour the population grows from
300,000 to 500,000. Write a formula expressing the population P as a function of the time t in hours.

3 Decay of radioactive substances:

Example 4 Radon gas decays according to the formula , where t is measured in days.
If there are 500 cubic centimeters left after 7 days, how much was there to begin with?