1-4 Extrema and Average Rates of Change

Determine whether the function is continuous at x = 4. A. yes B. no 5–Minute Check 2

Determine whether the function is continuous at x = 2. A. yes B. no 5–Minute Check 3

Describe the end behavior of f (x) = –6x 4 + 3x 3 – 17x 2 – 5x + 12. 5–Minute Check 4

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 5–Minute Check 5

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. Then/Now

Key Concept 1

As x increases, f(x) increases Key Concept 1

As x increases, f(x) decreases Key Concept 1

As x increases, f(x) stays the same Key Concept 1

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. Example 1

Support Numerically – (for demonstration) 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. Example 1

Answer: f (x) is decreasing on and increasing on . Analyze Increasing and Decreasing Behavior Answer: f (x) is decreasing on and increasing on . Example 1

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. Example 1

Create a table using x-values in each interval. 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. Example 1

Analyze Increasing and Decreasing Behavior 0.5 1 2 2.5 3 –6 –13.125 –24 Example 1

Answer: f (x) is decreasing on and and increasing on 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 Example 1

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

QUESTIONS?

Key Concept 2

Key Concept 2

Estimate and Identify Extrema of a Function Estimate and classify the extrema to the nearest 0.5 unit for the graph of f (x). Example 2

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 Example 2

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. Example 3

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) Example 3

Day 2 Average Rate of Change

Key Concept3

Substitute –3 for x1 and –1 for x2. 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). Example 5

Homework QUIZ TOMORROW!!! Pg 40: 9-13, 21, 24, 28, 40, 42, 47, 54-56, 60-63 QUIZ TOMORROW!!!