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 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.


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


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



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

.


0.5 1 2 2.5 3

–6 –13.125 –24


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


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!!!

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.

Documents

function f x

f x decreases

end behavior of f x

real zeros of f x

graph of f

function estimate

function values

average rates of change