Section 1.5 Exponential Functions:
An exponential function is a function that has the variable in the exponent , for
example:
General Form of an Exponential Function:
, and a
When 0<a<1, the graph will be exponential decay and the graph
falls from left to right.
When a>1, the graph will be exponential growth and
the graph rises from left to right.
 |
|
a>1:exponential growth |
0<a<1: exponential decay |
Applications of Exponential Functions: Compound Interest :
To generalize:
For P dollars invested at an interest rate of r compounded m times per year, the
value after t years,
denoted by A is 
De preciation by a Fixed Percentage
Depreciation by a fixed percentage, such as for a refrigerator, or a car, (say
10% of its value) is like
compounding interest, only with a negative interest
rate. So to compute this depreciation, we
use m = 1 (since the item loses value
only once per year) and with r being negative.
Continuous Compounding
We looked at compounding annually, quarterly, and monthly. But you could also
compound weekly,
daily, or even minute by minute. Ultimately, you can compound
continuously. The formula for
continuous compounding is:
For $P invested at a rate r compound continuously for t years
Section 1.6 Logarithmic Functions
The result of a log is the exponent
Common Logarithms
Logs with base 10 are called common logarithms. Usually, we omit the base with
the base 10
understood.
is equivalent to
General form of a logarithm:
is equivalent to
Natural Logarithm:
is equivalent to
Properties of Logs :
|
Logs with base 10 |
Logs with base e |
 |
Problems for Review:
1. Write each interval in set notation and graph it on the real line :
a. [0,4)
b. [-3,3]
2. Write each inequality in set notation:
a.
b. x > 3
3. Write an equation of the line satisfying the fol lowing conditions . If
possible write your
answer in the form y = mx + b
i. Slope -1 and passing through (4,3)
ii. Horizontal line passing through (1/2, ¾)
iii. Vertical line passing through (1/2, ¾)
iv. Passing through the points (3 ,-1) and (6, 0)
4. A newspaper buys a printing press for $800,000 and estimates its useful life
to be 20
years, after which its scrap value will be $60,000. Find a formula for
the value V of the press after
10 years.
5.
Simplify:
a. 
b. 
c. 
6.
Does this graph represent the graph of a function?
7. Graph the following:
a. 
b. f(x)=3x2-6x-9
c. 
d. 
8. A company manufactures bicycles at a cost of $55 each. If the company’s fixed
costs are
$900, express the company’s cost as a linear function of x, the number
of bicycles produced.
9. A company that installs car alarm systems finds that if it installs x systems
per week, its costs
will be 
and its revenue will be
(both in dollars).
i. find the company’s break-even points.
ii. Find the number of installations that will maximize profit and the maximum
profit.
10. For the following functions
i. evaluate the given expression
ii. Find the domain of the function
a. 
b. 
11. Solve
a. 
b. 
12.
For each pair f(x) and g(x) find f(g(x)) and g(f(x)):
13.For f(x)=3x2-5x+2 find
14. The following function expresses an income tax that is 15% for incomes up to
$6,000,
and otherwise is $900 plus 40% of income in excess of $6,000.
Calculate the tax on an income of
a.$3,000
b.$6,000
c.$10,000
15. The Black-Scholes formula for pricing options involves continuous
compounding. If an
option is now worth $10,000 and its value grows at an
interest rate of 6.1% compounded
continuously, what will be its value in 5
years?
16.Evaluate without using a calculator
a. 
b. 
17. Simplify using properties of logarithms
