1. The First Derivative Test. Using first derivative
in graphing
4.3
Rita Korsunsky

2. p f (p) q f (q) f ’ is positive
A function f is increasing on an interval I if f (p) < f (q) for all p and q in I with p < q
f ’ is positive

3. p f (p) q f (q) f ’ is negative
A function f is decreasing on an interval I if f (p) > f (q) for all p and q in I with p < q
f ’ is negative

4. Examine the following graph:
f ’(x) = 0,
critical point
When f (x) is decreasing f ’(x) is negative,
When f (x) is increasing f ’(x) is positive,

5. _ + + Let Example 1: g(x) increases on: (–, –1] U [3 , ) -1 3
a) Find the intervals on which g is increasing and decreasing. _ + +
g(x) increases on:
(–, –1] U [3 , ) -1 3
g(x) decreases on [–1 , 3]
( -1, 27 ) -1 3
b) Sketch this graph
decreasing
increasing
(3, -5)
increasing

6. The first-derivative test
Let f be continuous at c and differentiable on the open interval containing c except possibly at c itself C
Local min C
Local max
For Example:
f ’(x) < 0 C
f ’(x) > 0 or
No local extremum
No local extremum

7. The critical numbers are x=0 and x=3
Example 2 3
min
The critical numbers are x=0 and x=3 3
(0 , 12)
(3 , – 15)

8. Find the local extrema of f and the intervals on which f is increasing or is decreasing, and sketch the graph of f. _
min
max 7 +
Test f’(x):
f’(x)
(0 , 2)
(7 , 2)
Critical numbers:

9. That’s all folks!