sec. 4.3 using derivatives for curve sketching. in the left hand column are graphs of several...
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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: