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How to find the equations of horizontal and vertical asymptotes for rational functions and introduces the concept of exponential functions with their basic properties and graphs. It includes examples and application problems.
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(2.4) To find the equation of a horizontal asymptote (H.A.):
For the rational function ( )
d x
n x f x , there
are three cases:
Case 1: If degree n(x) < degree d(x), then H.A. is y = 0;
Case 2: If degree n(x) = degree d(x), the H.A. is y = a/b, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.
Case 3: If degree n(x) > degree d(x), then there is no H.A.
Example 4 For the rational functions:
2
x
x f x and 2 4
3
x
x x g x , find
the equations of horizontal and vertical asymptotes if any.
the equations of all horizontal asymptotes, if any;
Use the results from part 1) to 4) and additional points if necessary to sketch the graph.
Application Problems
Example 6 (Average Cost) A company manufacturing surfboards has fixed costs of $300 per day and total cost of $5, per day at a daily output of 20 boards.
A) Assuming that the total cost per day, C( x ), is linearly related to the total output per day, x , write an equation for the cost function.
B) The average cost per board for an output of x boards is given by C ( x ) C ( x )/ x ,Find C ( x ).
Section 2-5: Exponential Functions
(a calculator is needed for some hw problems in this section and 2-6)
Exponential Functions
y x^2 is a quadratic function;
If we switch x and 2, we get
y 2 x , called exponential function of base
Definition: Exponential function is the form:
y bx , where b is called ‘base’, b > 0 and b 1.
See the graphs when the base is 2, 3, ½, and 1/3:
Other properties of exponential functions:
x
x x x x x
x y x y xy y
x x y x y
b
a b
a ab a b
a a a a
a a a a
For example, x
x 5
2
3
2. If a 1 , then a x^ ay if and only if x = y
For example, if 65 t ^1 63 t ^3 , then
For example, if a^4 24 , then
A special exponential function: base = e, where e is an irrational number: e 2. 718281828459 ... See graph:
Example 1 Solve for x :
(^2) x x
