Mathematics 133 -- Intensive Calculus for Science 1
Discussion 3 -- Polynomial and Rational Functions, etc.
September 26, 2001
Background
A polynomial function is one defined by a formula
f(x) = anxn + an-1xn-1
+ ... + a1x + a0
where n is a non-negative integer, and the ai
are real numbers. The highest power occurring with a nonzero coefficient
in f(x) is called the degree of the polynomial.
A polynomial function of degree n has
- at most n real zeros -- x-axis intercepts
of its graph
- at most n - 1 turning points where the function
changes from increasing to decreasing or vice versa.
A rational function is one defined by a formula
f(x) = p(x)/q(x), where p(x),q(x) are polynomials.
Zeroes of p(x) give x-axis intercepts of the
graph of f(x); zeroes of q(x) give vertical asymptotes
of the graph. We can also determine whether f(x) has
horizontal asymptotes by analyzing its behavior as x -> +/- infinity.
Discussion Questions
- A) From Section 1.6 in text: Exercise 5
- B) From Section 1.6 in text: Exercise 6
- C) From Section 1.6 in text: Problem 32 (Don't use a graphing
calculator on this one -- try to match properties of the graphs
with features of the formulas. Some algebraic rearrangements
of the formulas may provide useful information.)
- D) (Preview of next week) If we have a graph of a function
such as a polynomial of degree 2 or larger, then we can estimate
the slope of the graph at a given point by drawing a
tangent line to the graph going through that point
(that is a line ``just touching the graph'' at the point) and
finding the slope of that line. Consider the function
f(x) = x2 - 1. On graph paper, make an accurate
sketch of the graph y = x2 - 1
for -3 <= x <= 3. Draw in tangent lines to this graph
at x = -3,-2,-1,0,1,2,3. Estimate the slopes of these
lines. Then use those values to plot the ``slope function'' for
f(x) (the function whose value at x = -3 is the slope of
y = f(x) at x = -3, etc.). Can you guess
a formula for the slope function?
Assignment
One write-up per group of solutions for these problems,
due Tuesday, October 2.