5.3 A – Curve Sketching

Goal

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

Increasing/decreasing functions

Definitions: Let f be differentiable on (a, b):

o If f’ > 0 at each point of (a, b), then f increases on [a, b]

o If f’ < 0 at each point of (a, b), then f decreases on [a, b]

o If f’ = 0 at each point of (a, b), then f is constant on [a, b] (called monotonic)

Increasing/decreasing functions

• Example: Find the open intervals on which

23

2

3)( xxxf is increasing or decreasing.

1.) Find critical points

033)(' 2 xxxf0)1(3 xx

03 x 01x0x 1x

Interval (-∞, 0) (0, 1) (1, ∞)

Test Value x = -1 x = ½ x = 2

f’(x) 6 > 0 -3/4 < 0 6 > 0

Conclusion Increasing Decreasing Increasing

First derivative:

y is positive Curve is increasing .

y is negative Curve is decreasing.

y is zero Possible local maximum or minimum.

We already know:

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)

First Derivative Test

At a critical point c:

1. 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.

3. If f’ does not change sign at c, then c is not a local min or max.

local min

f’<0 f’>0

no extreme

f’>0 f’>0

Hand out

• List of important terms for this section.

• Note cards?

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=13. 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

Homework

• 5.2- 14-22• 5.3 – 1,2,5,6