1. 5.3 A – Curve Sketching

2. Goal
Determine if a function is increasing or decreasing, and use the first derivative test to determine if critical points are maxs or mins.

3. Increasing/decreasing functions
Definitions: Let f be differentiable on (a, b):
If f’ > 0 at each point of (a, b), then f increases on [a, b]
If f’ < 0 at each point of (a, b), then f decreases on [a, b]
If f’ = 0 at each point of (a, b), then f is constant on [a, b] (called monotonic)

4. Increasing/decreasing functions
Example: Find the open intervals on which
is increasing or decreasing.
1.) Find critical points
Interval
(-∞, 0)
(0, 1)
(1, ∞)
Test Value
x = -1
x = ½
x = 2
f’(x)
6 > 0
-3/4 < 0
Conclusion
Increasing
Decreasing

5. We already know: First derivative: is positive Curve is increasing .
is negative
Curve is decreasing.
is zero
Possible local maximum or minimum.

6. First Derivative Test
Used to determine if a critical point is a relative max or min.
We already leaned how to find critical points (places where the derivative equals 0)

7. First Derivative Test At a critical point c:
If f’ changes sign from pos to
neg at c, then f has a local
max at c.
local max
f’>0
f’<0
2. If f’ changes sign from neg to
pos at c, then f has a local
min at c.
local min
f’<0
f’>0
If f’ does not change sign
at c, then c is not a local min or max.
no extreme
f’>0

8. Hand out
List of important terms for this section.
Note cards? 

9. Example
Use the First Derivative Test to find the local extreme values of g(x) = (x2 – 3)ex. 1. Find g’(x) 2. Find the zeros of g’. x=-3 and x=1 3. Put zeros on a number line to test. 4. Pick a value in the interval and see g’ is + or – at that value. 5. Use first derivative test to determine max and mins. -3= local max and 1= local min

10. Homework
5.3 – 1,2,5,6