Project 11
MAC 2311
1. Examine the function
f
(
x
) =
x
3

3
x
. From precalculus, what should be the two distinct
possibilities for the number of turning points of any cubic (degree 3) polynomial function?
In order for a
smooth
graph to “turn around”, what value would the slope of the curve need
to be at the turning point? (Illustrate with a picture.) Is the reverse true (can a graph have
that slope without turning around)?
Find the values of
x
at which
f
(
x
)
has horizontal tangent lines.
Find the zeros (
x
intercepts) of
f
(
x
)
.
Calculate the slope of
f
(
x
)
at EACH of its zeros.
How many horizontal tangent lines does the function
g
(
x
) =
x
3
+ 3
x
have? Can it turn
around? How many zeros does it have?
Sketch the functions
f
(
x
)
and
g
(
x
)
, incorporating the above information and labelling the
points where the horizontal tangent lines occur.
Based on your discoveries so far, how does the derivative of a cubic function determine the
number of turning points that it has?
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 Spring '08
 ALL
 Calculus, Derivative, 3 years, θ, horizontal tangent lines

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