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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In the past, one of the important uses of
derivatives was as an aid in curve sketching. We
usually use a calculator of 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
First derivative test
negative
positive
positive
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Example
Graph
First derivative test
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Example
Graph
Or you could use the second derivative test
negative
concave down
local maximum
positive
concave up
local minimum
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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
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Make a summary table
p