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Photo by Vickie Kelly, 1995. Greg Kelly, Hanford High School, Richland, Washington. 4.3 Using Derivatives for Curve Sketching. Old Faithful Geyser, Yellowstone National Park. Photo by Vickie Kelly, 2007. Greg Kelly, Hanford High School, Richland, Washington. 4.3

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Photo by Vickie Kelly, 1995

Greg Kelly, Hanford High School, Richland, Washington

4.3

Using Derivatives for Curve Sketching

Old Faithful Geyser, Yellowstone National Park


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Photo by Vickie Kelly, 2007

Greg Kelly, Hanford High School, Richland, Washington

4.3

Using Derivatives for Curve Sketching

Yellowstone Falls, Yellowstone National Park


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Photo by Vickie Kelly, 2007

Greg Kelly, Hanford High School, Richland, Washington

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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is positive 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.

is negative

is zero

is positive

is negative

is zero

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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There are roots at and . 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.

Possible extreme at .

Set

Example:

Graph

We can use a chart to organize our thoughts.

First derivative test:

negative

positive

positive


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There are roots at and . 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.

Possible extreme at .

Set

maximum at

minimum at

Example:

Graph

First derivative test:


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There is a local maximum at 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.(0,4) because for all x in

and for all x in (0,2) .

There is a local minimum at (2,0) because for all x in

(0,2) and for all x in .

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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There are roots at and . 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.

Possible extreme at .

Because the second derivative at

x =0 is negative, the graph is concave down and therefore (0,4) is a local maximum.

Because the second derivative at

x =2 is positive, the graph is concave up and therefore (2,0) is a local minimum.

Example:

Graph

Or you could use the second derivative test:


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Possible 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. inflection point at .

There is an inflection point at x =1 because the second derivative changes from negative to positive.

inflection point at

Example:

Graph

We then look for inflection points by setting the second derivative equal to zero.

negative

positive


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rising, concave down 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.

local max

falling, inflection point

local min

rising, concave up

Make a summary table:

p


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