Polynomial and Rational Functions

1
Chapter 5 Polynomial and Rational Functions
5.1 Quadratic Functions and Models 5.2 Polynomial
Functions and Models 5.3 Rational Functions and
Models
A linear or exponential or logistic model either
increases or decreases but not both.
Life, on the other hand gives us many instances
in which something at first increases then
decreases or vice-versa. For situations like
these, we might turn to polynomial models.
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Rational Functions and Models A rational function
is a quotient or ratio of two polynomials.

• Vertical Asymptote at x k
• k is not in the domain of f
• the values of f increase (or decrease) without
  bound as x approaches k
• near x k, the graph of f resembles a vertical
  line

• The quotient of leading terms determines the
  asymptotic (global) behavior of a rational
  function.
• Horizontal Asymptote at y m
• global behavior tends toward a constant value m
• graph resembles a horizontal line for x large in
  magnitude.

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43/259 Write a formula for the function that
could represent the graph. Give your reasoning.
y intercept f(0) 0
x intercept 0 f(x) only when x 0.numerator
0 only when x 0.
vertical asymptotes at x -1. denominator
contains (x1)
horizontal asymptote at y 1. f(x) ??/(x1)
1
4
37/259 After the engine of a moving motorboat is
cut off, the boats velocity decreases according
to the model, where t is elapsed time in seconds
and v is the velocity in feet per second.
a) Sketch a graph of the abstract function v(t).
b) Is the domain continuous or discrete?
c) How fast was the boat moving when the engine
was cut off?
d) After how many seconds did the velocity reach
10 ft/ sec ?
e) Find the acceleration rate of change of
velocity of the boat after 5 seconds.
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3350/259 The cost C (dollars) of operating a
studio on a day in which x pots are produced is
given by the function C(x) 0.01x3 0.65x2
14x 20. Let A(x) be the average cost of
producing each ceramic pot on a day when x pots
are made.
a) Use the formula for C(x) to find a formula
for A(x).
b) Sketch the graph of the abstract function
A(x).
c) Is the domain continuous or discrete?
d) Find the coordinates of the local minimum.
(33.3, 4)
e) To minimize the average cost per pot, how
many should the studio make in a given day and
what would be the average cost of each?
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3350/259 The cost C (dollars) of operating a
studio on a day in which x pots are produced is
given by the function C(x) 0.01x3 0.65x2
14x 20. Let A(x) be the average cost of
producing each ceramic pot on a day when x pots
are made.
Describe global behavior for A(x).
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HW Page 255 33-50
PROJECT Lab 5A The Doormats LabDUE Wednesday
April 23, 2008
Report should start with a well-written summary
of each of the three models as outlined on the
bottom of page 283 with graphs and asymptotic
analysis for each model as necessary for support
of your summary.