1-4 Extrema and Average Rates of Change. 5–Minute Check 2 A.yes B.no Determine whether the function is continuous at x = 4.

1-4 Extrema and Average Rates of Change

A. yes

B. no

Determine whether the function is continuous at x = 4.

A. yes

B. no

Determine whether the function is continuous at x = 2.

Describe the end behavior off (x) = –6x 4 + 3x 3 – 17x

2 – 5x + 12.

A.

B.

C.

D.

Determine between which consecutive integers the real zeros of f (x) = x

3 + x 2 – 2x + 5 are located

on the interval [–4, 4].

A. –2 < x < –1

B. –3 < x < –2

C. 0 < x < 1

D. –4 < x < –3

You found function values. (Lesson 1-1)

• Determine intervals on which functions are increasing, constant, or decreasing, and determine maxima and minima of functions.

• Determine the average rate of change of a function.

As x increases, f(x) increases

As x increases, f(x) decreases

As x increases, f(x) stays the same

Analyze Increasing and Decreasing Behavior

A. Use the graph of the function f (x) = x 2 – 4 to

estimate intervals to the nearest 0.5 unit on which the function is increasing, decreasing, or constant. Support the answer numerically.

Analyze Increasing and Decreasing Behavior

Analyze Graphically

From the graph, we can estimate that f is decreasing on and increasing on .

Support Numerically – (for demonstration)

Create a table using x-values in each interval.

The table shows that as x increases from negative values to 0, f (x) decreases; as x increases from 0 to positive values, f (x) increases. This supports the conjecture.

Analyze Increasing and Decreasing Behavior

Answer: f (x) is decreasing on and increasing on .

Analyze Increasing and Decreasing Behavior

B. Use the graph of the function f (x) = –x 3 + x to

estimate intervals to the nearest 0.5 unit on which the function is increasing, decreasing, or constant. Support the answer numerically.

Analyze Increasing and Decreasing Behavior

Support Numerically

Create a table using x-values in each interval.

Analyze Graphically

From the graph, we can estimate that f is decreasing on

, increasing on , and decreasing on

.

Analyze Increasing and Decreasing Behavior

0.5 1 2 2.5 3

–6 –13.125 –24

Analyze Increasing and Decreasing Behavior

The table shows that as x increases to , f (x)

decreases; as x increases from , f (x) increases;

as x increases from , f (x) decreases. This supports

the conjecture.

Answer: f (x) is decreasing on and

and increasing on

Use the graph of the function f (x) = 2x 2 + 3x – 1 to

estimate intervals to the nearest 0.5 unit on which the function is increasing, decreasing, or constant. Support the answer numerically.

A. f (x) is increasing on (–∞, –1) and (–1, ∞).

B. f (x) is increasing on (–∞, –1) and decreasing on (–1, ∞).

C. f (x) is decreasing on (–∞, –1) and increasing on (–1, ∞).

D. f (x) is decreasing on (–∞, –1) and decreasing on (–1, ∞).

QUESTIONS?

Estimate and Identify Extrema of a Function

Estimate and classify the extrema to the nearest 0.5 unit for the graph of f (x).

Estimate and Identify Extrema of a Function

Analyze Graphically

It appears that f (x) has a relative minimum at

x = –1 and a relative maximum at x = 2. It also appears

that so we conjecture

that this function has no absolute extrema.

Answer: To the nearest 0.5 unit, there is a relative minimum at x = –1 and a relative maximum at x = 2. There are no absolute extrema

Use a Graphing Calculator to Approximate Extrema

GRAPHING CALCULATOR Approximate to the nearest hundredth the relative or absolute extrema of f (x) = x

4 – 5x 2 – 2x + 4. State the x-value(s)

where they occur.

f (x) = x 4 – 5x

2 – 2x + 4

Graph the function and adjust the window as needed so that all of the graph’s behavior is visible.

Use a Graphing Calculator to Approximate Extrema

Answer: relative minimum: (–1.47, 0.80); relative maximum: (–0.20, 4.20);absolute minimum: (1.67, –5.51)

Day 2

Average Rate of Change

Find Average Rates of Change

A. Find the average rate of change of f (x) = –2x

2 + 4x + 6 on the interval [–3, –1].

Use the Slope Formula to find the average rate of change of f on the interval [–3, –1].

Substitute –3 for x1 and –1 for x2.

Evaluate f(–1) and f(–3).

•Homework• Pg 40: 9-13, 21, 24, 28, 40,

42, 47, 54-56, 60-63

QUIZ TOMORROW!!!