SEC. 4.3 USING DERIVATIVES FOR CURVE SKETCHING. IN THE LEFT HAND COLUMN ARE GRAPHS OF SEVERAL FUNCTIONS. IN THE RIGHT- HAND COLUMN – IN A DIFFERENT ORDER.

S E C . 4 . 3

USING DERIVATIVES FOR CURVE SKETCHING

IN THE LEFT HAND COLUMN ARE GRAPHS OF SEVERAL

FUNCTIONS. IN THE RIGHT-HAND COLUMN – IN A

DIFFERENT ORDER – ARE GRAPHS OF THE ASSOCIATED

DERIVATIVE FUNCTIONS. MATCH EACH FUNCTION WITH ITS DERIVATIVE. [NOTE: THE SCALES ON THE GRAPHS ARE

NOT ALL THE SAME.]

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

p

Graph Example: