Calculus 4.3

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4.3 Using Derivatives for Curve Sketching
Old Faithful Geyser, Yellowstone National Park
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4.3 Using Derivatives for Curve Sketching
Yellowstone Falls, Yellowstone National Park
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4.3 Using Derivatives for Curve Sketching
Mammoth Hot Springs, Yellowstone National Park
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In the past, one of the important uses of
derivatives was as an aid in curve sketching.
Even though we usually use a calculator or
computer to draw complicated graphs, it is still
important to understand the relationships between
derivatives and graphs.
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First derivative
Curve is rising.
Curve is falling.
Possible local maximum or minimum.
Second derivative
Curve is concave up.
Curve is concave down.
Possible inflection point (where concavity
changes).
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Example
Graph
We can use a chart to organize our thoughts.
First derivative test
negative
positive
positive
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Example
Graph
First derivative test
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Example
Graph
NOTE On the AP Exam, it is not sufficient to
simply draw the chart and write the answer. You
must give a written explanation!
First derivative test
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Example
Graph
Or you could use the second derivative test
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Example
Graph
We then look for inflection points by setting the
second derivative equal to zero.
negative
positive
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Make a summary table
p