4.3Using Derivatives for Curve Sketching

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1995

Old Faithful Geyser, Yellowstone National Park

4.3Using Derivatives for Curve Sketching

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007

Yellowstone Falls, Yellowstone National Park

4.3Using Derivatives for Curve Sketching

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007

Mammoth Hot Springs, Yellowstone National Park

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.

First derivative:

y is positive Curve is rising.

y is negative Curve is falling.

y is zero Possible local maximum or minimum.

Second derivative:

y is positive Curve is concave up.

y is negative Curve is concave down.

y is zero Possible inflection point(where concavity changes).

Example:Graph 23 23 4 1 2y x x x x

There are roots at and .1x 2x

23 6y x x

0ySet

20 3 6x x

20 2x x

0 2x x

0, 2x

First derivative test:

y0 2

0 0

21 3 1 6 1 3y negative

21 3 1 6 1 9y positive

23 3 3 6 3 9y positive

Possible extreme at .0, 2x

We can use a chart to organize our thoughts.

Example:Graph 23 23 4 1 2y x x x x

There are roots at and .1x 2x

23 6y x x

0ySet

20 3 6x x

20 2x x

0 2x x

0, 2x

First derivative test:

y0 2

0 0

maximum at 0x

minimum at 2x

Possible extreme at .0, 2x

Example:Graph 23 23 4 1 2y x x x x

23 6y x x First derivative test:

y0 2

0 0

NOTE: On the AP Exam, it is not sufficient to simply draw the chart and write the answer. You must give a written explanation!

There is a local maximum at (0,4) because for all x in and for all x in (0,2) .

0y( ,0) 0y

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

0y(2, )0y

Because the second derivative atx = 0 is negative, the graph is concave down and therefore (0,4) is a local maximum.

Example:Graph 23 23 4 1 2y x x x x

There are roots at and .1x 2x

23 6y x x Possible extreme at .0, 2x

Or you could use the second derivative test:

6 6y x

0 6 0 6 6y

2 6 2 6 6y Because the second derivative atx = 2 is positive, the graph is concave up and therefore (2,0) is a local minimum.

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

Example:Graph 23 23 4 1 2y x x x x

6 6y x

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

0 6 6x

6 6x

1 x

Possible inflection point at .1x

y1

0

0 6 0 6 6y negative

2 6 2 6 6y positive

Make a summary table:

x y y y

1 0 9 12 rising, concave down

0 4 0 6 local max

1 2 3 0 falling, inflection point

2 0 0 6 local min

3 4 9 12 rising, concave up