AP Calc Notes: ID – 2 Graphs of f and f’
Graph the derivative
http://www.ltcconline.net/greenl/java/other/derivativegraph/classes/derivativegraph.html
Which is the function, which is the derivative?
http://rowdy.mscd.edu/~talmanl/MTH1410U08/Pictures_080529/
Geogebra
y
Ex: Let f be the continuous function graphed at right.
Find the value of f ' at each point a – g and
Determine the sign of f ' on each interval.
Definition: A critical point of f is a point where
a. f' = 0 or
b. f' undefined
0
f'
a
c d
e
+ 0 - 0 - 0 +U - U
1. If f ' = 0 (stationary point), then
a. f has a horizontal tangent line
b. f MAY have a relative maximum or relative minimum:
If f' changes from + to -, f has a rel. max
If f' changes from – to +, f has a rel min.
2. If f ' is undefined, then
a. f MAY have a vertical tangent line
b. f MAY have a “corner” or “cusp”
c. f MAY have a relative extremum
SAME RULES AS ABOVE.
3. On remaining intervals (intervals between critical points)
a. If f' > 0, then f is increasing on the interval.
b. If f' < 0, then f is decreasing on the interval.
b
+
x
f
g
U + U +
Ex: The graph of the derivative g' of a continuous function g is shown.
a. On what intervals is g increasing?
y = g'(x)
y
On [b, e] because g' > 0 on (b, d) and
(d, e)
°
a
b
c
d
b. Where does g have a relative minimum?
x
e
°
At x = b because g' changes from – to +
there.
c. Where does g have a relative maximum?
At x = e because g' changes from + to - there.
d. Does g have a stationary point that is neither a relative maximum nor minimum? (terrace point)
At x = d, g' = 0 but does not change sign.
Ex: Draw the graph of the derivative of each of the following functions:
y
y
x
y
y
x
y
x
x
y
x
x