Download Lesson20 - Purdue Math

Lesson 20 Sections 4.1 and 4.2
Exponential Functions
We have discussed powers where the exponents are integers or rational numbers. There
also exists powers such as 2 3 . You can approximate powers on your calculator using
the power key. On most one-liner scientific calculators, the power key looks like
x
y
Enter the base into the calculator first, press the power key, enter the exponent, and press
enter or equal.
Ex 1: Approximate the following powers to 4 decimal places.
a) 2 3 2 
b)
( 2.3) 4.8 
If b  0 and b  1 , the function y  b x defines a function. Since x can be any number, the
domain is the set of all real numbers. Since the base b is positive, the value of y will
always be positive, so the range is [0,  ) .
Exponential Function: An exponential function is defined by the equation
f ( x)  b x or y  b x (b  0, b  1, and x is a real number).
The domain of any exponentail function is (, ) and the range is [0, ).
Ex 1: Graph each exponential function.
a)
3
y 
2
x
The following are not
exponential functions. Why?
f ( x)  x 3
f ( x)  1x
f ( x)  (4) x
f ( x)  x x
1
1
2
1
b)
1
f ( x)   
 3
x
8
4
2
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Generalizations about graphs of exponential functions of the form y = bx:
1. The y-intercept will always be (0, 1) because any base to the 0 power is 1.
2. If the base is greater than 1, the graph is increasing. If the base is between 0 and
1, the graph is decreasing.
3. The graph lies above the x-axis, because the range is only positive values.
4. The x-axis is an asymptote of the graph. An asymptote is a line that the graph
approaches but never touches.
5. The point (1, b) is on the graph.
Ex 2: Determine which of these graphs could be an exponential function of the form
y  bx .
 2
1, 
 3
Ex 3: Approximate a possible equation for graph (a) above. (Remember, the point
(1, b) will be on the graph.)
Ex 4: Lynn made a graph of a basic exponential function. It contains the points
1

(0,1) and  1,  . What is the equation of the function?
4

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COMPOUND INTEREST
If a bank account earns interest on a regular basis and the interest earned is added to the
total before the next interest is figured, this is called compound interest. Some banks
figure interest monthly, quarterly, semi-annually, etc.
kt
r

The formula used to figure compound interest is A  P1   , where A is the final
 k
amount in the account, P is the initial deposit or amount, r is the annual interest rate
earned (as a decimal), k is the number of times in a year that interest is figured, and t is
time in years. As k gets very large, so that interest is compounded constantly, we say
interest is compounded continuously. What happens to the formula when k becomes very
large? It becomes the following formula, A  Pe rt . The A, P, r, and t represent the same
as in the regular compound interest formula.
x
 1
What is the number e? If the quantity 1   is evaluated for very large values of x, it
x

becomes approximately 2.71828182845904... . As x becomes infinity, this number is
called e.
You will notice that your calculator has e x above a key marked ln on your scientific
calculator. The base e is approximately 2.71828... .
To find these powers:
1. Enter the power in your calculator.
2. Because the e power is above the ln key, you must press the 2nd
then the
key.
ln
key first and
3. The value is approximately the power.
Ex 5: Approximate each power to 4 decimal places.
a) e 3 
b)
e 0.024 
c)
e

2
3

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Compound Interest Formula: If P dollars are deposited in an account earning interest
at an annual rate r, compounded continuously, the amount A after t years is given by the
formula A  Pe rt .
Ex 6: Jim deposited $4500 at 5.4% annual interest rate in an account compounded
continuously. How much is in Jim's account at the end of 5 years? (Assume he
made no more deposits into the account.)
Always convert
percent rates to
decimals in these
types of formulas.
Ex 7: Lily's parents deposited an amount in her account on her day of birth. The
account earned 6% annual interest compounded continuously and on her 18th
birthday the account was $40,000. How much was the initial deposit by her
parents?
Ex 8: (optional) Suppose Jim could not invest in an account compounded continuously
(see example 6 above). He still had $4500 at 5.4% annual interest for 5 years, but
the money was compounded quarterly. How much would be in the account?
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Other application problems use formulas that contain an exponential expression as part of
the formula, some with base e, and others with different bases. Several of these formulas
are found in lesson 4.2 in your textbook. We will use some of them in these problems.
The half-life of an element is the amount of time necessary for the element to decay to
half the original amount. Uranium is an example of an element that has a half-life. The
half-life of radium is approximately 1600 years. The formula used to find the amount of
radioactive material present at time t, where A0 is the initial amount present (at t = 0), and

t
h = half-life of the element is A  A0 2 h or A  A0 2 t / h .
Ex 9: Tritium, a radioactive isotope of hydrogen, has a half-life of 12.4 years. If an
initial sample has 50 grams, how much will remain after 100 years?
A subscript of 0, such as
A0 , means the amount
initially or amount at time 0.
Another formula represents the population growth of lots of cities, towns, or countries.
The formula is P  P0 e kt where P is the final population, P0 is the initial population at
time 0, t is time in years, and k = b - d (b is birth rate and d is death rate). k is a percent
converted to a decimal. Note: In order for a country to grow k must be positive.
Ex 10: The population of a city is 45,000 in 2008. The birth rate is 11 per 1000 and the
death rate is 10 per 1000. What will be the population in 2018?
11 per 1000 =
1.1% or 0.011
10 per 1000 = 1%
or 0.01
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The intensity of light I (in lumens) at a distance of x meters below the surface of water is
represented by I  I 0 k x , where I0 is the intensity of light above the water and k is a
constant that depends on the clarity of the water.
Ex 11: At the center of a certain lake, the intensity of light above the water is 10 lumens
and the value of k is 0.8. Find the intensity of light at 3 meters below the surface.
Ex 12: The population of Eagle River is growing exponentially according to the model,
P  375(1.3) t , where t is years from the present date. Find the population in 6
years.
Ex 13: For one individual the percent of alcohol absorbed into the bloodstream after
drinking two shots of whiskey is given by P  0.3(1  e 0.05t ) , where t is in
minutes. Find the percent of alcohol in the bloodstream after ½ hour.
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