1 Starting with f x, we compute

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1/ Curve Sketching
A General Approach to Curve Sketching
1) Starting with f (x), we compute
2) Next, we locate all relative maximum and
relative minimum points and make a partial sketch.
3) We study the concavity of f (x) and locate
all inflection points.
4) We consider other properties of the graph,
such as the intercepts, and complete the sketch.
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1/ Locating Relative Extreme Points
a / Critical Values
A number a in the domain of f such that
f(a)0 is called critical number or critical
value of f (we also call a value a (in the
domain of) a critical value id f(a) does not
exist). If a is a critical value then (a,f(a))
is called critical point.
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b/ First Derivative Test
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b/ First Derivative Test
EXAMPLE
Find the local maximum and minimum points of
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b/ First Derivative Test
Set up the chart

• Divide the real line into intervals with the
  critical values as endpoints.
• Since the sign of depends on the signs of
  its two factors 9x 3 and 2x 1, determine the
  signs of the factors of over each
  interval. Usually this is done by testing the
  sign of a factor at points selected from each
  interval.
• In each interval, use a plus sign if the factor
  is positive and a minus sign if the factor is
  negative. Then determine the sign of over
  each interval by multiplying the signs of the
  factors and using

• A plus sign of corresponds to an
  increasing portion of the graph f and a minus
  sign to a decreasing portion. Denote an
  increasing portion with an upward arrow and a
  decreasing portion with a downward arrow. The
  sequence of arrows should convey the general
  shape of the graph and, in particular, tell you
  whether or not your critical values correspond to
  extreme points.

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c/ Second Derivative Test
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c/ Second Derivative Test
EXAMPLE
Locate all possible relative extreme points on
the graph of the function
Check the concavity at these points and use this
information
to sketch the graph of f (x).
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c/ Second Derivative Test
The following is a sketch of the function.
(-3, 0)
(-1, -4)
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3/ Find and Test the Inflection Points
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EXAMPLE
Sketch the graph of
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(0, 2)