3.1 Increasing and Decreasing Functions; Relative Extrema

General Discussion about topics in Chapter 3

Four Sections

Information from f ‘

Information from f ‘’

Absolute Maximum and Minimum

APPLICATIONS

Increasing and Decreasing Functions

If f(x) is a function defined on the interval (a,b), and are two numbers in the interval, then

f(x) is increasing on the interval if

f(x) is decreasing on the interval if

Transparency – page 195

Identify where increasing, decreasing, relative max and min, absolute max and min

Transparency for problems #2 and #4

Procedure of Using Derivative to find where Increase/Decrease

1. Find all values of x for which f’(x)=0 or f’(x) is not continuous

and mark on a number line to create open intervals

2. Choose a test point, c, from each interval to evaluate f’’(c)

If f’ is positive, the function is increasing on the interval

If f’ is negative, the function is decreasing on the interval

Find the intervals of increase and decrease for the given function

#10

#20

Relative Extrema

Relative Maximum is larger than “nearby” points.

Relative Maximum at x = c if f(c) > f(x) in a small interval near c

Relative Minimum is smaller than “nearby” points.

Relative Minimum at x = c if f(c) < f(x) in a small interval near c

Critical Numbers and Critical Points

A number c in the domain of f(x) is called a critical number if either

a) f ‘(c) = 0 or

b) f ‘(c) does not exist.

The corresponding point (c, f(c)) on the graph is called a critical point

Determining if critical point is relative max, min or neither:

First Derivative Test – pg 197

If c is a critical number, and the sign of f’ changes at c, a relative extrema occurs.

If f’ changes at c from + to -, c is a relative maximum.

If f’ changes at c from – to +, c is a relative minimum

Use a number line with critical numbers and test points

Determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither.

#26

#28.

#34

Critical points for which the derivative is undefined:

FIGURE 1 FIGURE 2 FIGURE 3

“ corner “ “ cusp” “vertical tangent”

Sketch the graph of a function – Procedure listed on page 198

1) Domain

2) Find f ‘ and critical points

3) Use sign graph to determine where increase & decrease

4) Determine extrema points

5) Include intercepts and above info to sketch

Use calculus to sketch the graph of the given function

#36

#38 Same as #28 above

#42

Applications

#54 MARGINAL ANALYSIS

The total cost of producing x units of a certain commodity is given by . Sketch the cost curve and find the marginal cost. Does marginal cost increase or decrease with increasing production?