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Lesson Menu Five-Minute Check Then/Now New Vocabulary Key Concept:Increasing, Decreasing, and Constant Functions Example 1:Analyze Increasing and Decreasing Behavior Key Concept:Relative and Absolute Extrema Example 2:Estimate and Identify Extrema of a Function Example 3:Real-World Example: Use a Graphing Calculator to Approximate Extrema Example 4:Use Extrema for Optimization Key Concept:Average Rate of Change Example 5:Find Average Rates of Change Example 6:Real-World Example: Find Average Speed

Then/Now 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.

Vocabulary increasing decreasing constant maximum minimum extrema average rate of change secant line

Key Concept 1

Example 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 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 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, ∞).

Key Concept 2

Example 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). Support the answers numerically.

Example 2 Estimate and classify the extrema to the nearest 0.5 unit for the graph of f (x). Support the answers numerically. A.There is a relative minimum of 2 at x = –1 and a relative maximum of 1 at x = 0. There are no absolute extrema. B.There is a relative maximum of 2 at x = –1 and a relative minimum of 1 at x = 0. There are no absolute extrema. C.There is a relative maximum of 2 at x = –1 and no relative minimum. There are no absolute extrema. D.There is no relative maximum and there is a relative minimum of 1 at x = 0. There are no absolute extrema.

Example 4 Use Extrema for Optimization FUEL ECONOMY Advertisements for a new car claim that a tank of gas will take a driver and three passengers about 360 miles. After researching on the Internet, you find the function for miles per tank of gas for the car is f (x) =  0.025x x + 240, where x is the speed in miles per hour. What speed optimizes the distance the car can travel on a tank of gas? How far will the car travel at that optimum speed?

Example 4 VOLUME A square with side length x is cut from each corner of a rectangle with dimensions 8 inches by 12 inches. Then the figure is folded to form an open box, as shown in the diagram. Determine the length and width of the box that will allow the maximum volume. A.6.43 in. by in. B.4.86 in. by 8.86 in. C.3 in. by 7 in. D.1.57 in. by 67.6 in.

Key Concept3 Average rate of change is slope between two points!!!!

Example 5 Find Average Rates of Change A. Find the average rate of change of f (x) = –2x 2 + 4x + 6 on the interval [–3, –1].

Example 5 Find Average Rates of Change B. Find the average rate of change of f (x) = –2x 2 + 4x + 6 on the interval [2, 5]. Use the Slope Formula to find the average rate of change of f on the interval [2, 5]. Substitute 2 for x 1 and 5 for x 2. Evaluate f(5) and f(2).

Example 6 B. GRAVITY The formula for the distance traveled by falling objects on the Moon is d (t) = 2.7t 2, where d (t) is the distance in feet and t is the time in seconds. Find and interpret the average speed of the object for the time interval of 2 to 3 seconds. Find Average Speed Substitute 2 for t 1 and 3 for t 2. Evaluate d(3) and d(2). Simplify.

End of the Lesson