# 1-4 Extrema and Average Rates of Change

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

Determine whether the function is continuous at x = 2. A. yes B. no Describe the end behavior of

f (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

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 on . and increasing

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

. Support Numerically Create a table using x-values in each interval. 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 decreases; as x increases from as x increases from , f (x) , f (x) increases; , f (x) decreases. This supports

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

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, ). QUESTION S?

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 graphs 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!!!

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