1. 5.3 Using Derivatives for Curve Sketching
AP Calculus AB
5.3 Using Derivatives for Curve Sketching

2. 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.

3. First derivative:
is positive
Curve is rising.
is negative
Curve is falling.
is zero
Possible local maximum or minimum.
Second derivative:
is positive
Curve is concave up.
is negative
Curve is concave down.
is zero
Possible inflection point
(where concavity changes).

4. Example:
Graph
There are roots at and
Possible extreme at
We can use a chart to organize our thoughts.
Set
First derivative test:
negative
positive
positive

5. Example:
Graph
There are roots at and
Possible extreme at
Set
First derivative test:
maximum at
minimum at

6. 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:
There is a local maximum at (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

7. Example:
Graph
There are roots at and
Possible extreme at
Or you could use the second derivative test:
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.

8. Example:
Graph
We then look for inflection points by setting the second derivative equal to zero.
Possible inflection point at
negative
positive
inflection point at
There is an inflection point at x = 1 because the second derivative changes from negative to positive.

9. Make a summary table: rising, concave down local max
falling, inflection point
local min
rising, concave up

10. Example:
Use the concavity test to determine intervals on which the graph of the function is (a) concave up, and (b) concave down.
First, find the first and second derivatives.
Notice that at x = 0, y '' is undefined. That means something is happening at x = 0.
So, check values on both sides of x = 0.

11. Example:
Find all points of inflection of the function.
First, find the first and second derivatives.
x = -2 is a point of inflection because the graph changes from concave down to concave up at x = -2.

12. Example:
Use the graph of the function f to estimate where (a) f ' and (b) f '' are 0, positive, and negative.
(a) f ' = 0 at any maximum or minimum values; or x = -1 and x = 1.
f ' > 0 where f is increasing; (-∞, -1) and (1, ∞)
f ' < 0 where f is decreasing; (-1, 1)

13. Example:
Use the graph of the function f to estimate where (a) f ' and (b) f '' are 0, positive, and negative.
(b) f '' = 0 at any points of inflection; or x = 0.
f '' > 0 where f is concave up; (0, ∞)
f '' < 0 where f is concave down; (- ∞, 0) p