1. 1.4 – Extrema and Rates of Change

2. Warm-Up: 1. Determine whether the function is continuous at x = 2.
2. Describe the end behavior of
f (x) = –6x4 + 3x3 – 17x2 – 5x + 12. A. B. C. D.

4. Example 1:
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.

5. Example 2:
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.
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, ∞).

6. Maxima:
Relative Maximum/Minimum: the greatest/least value f(x) can attain on SOME interval of the domain. Absolute Maximum/Minimum: the greatest/least value f(x) can attain over its ENTIRE domain.

7. Maxima:

8. Example 3:
Estimate and classify the extrema to the nearest 0.5 unit for the graph of f (x). Support the answers numerically.

9. Example 4:
Estimate and classify the extrema to the nearest 0.5 unit for the graph of f (x). Support the answers numerically. (Coordinate, x-value, or y-value?)

10. Example 5:
GRAPHING CALCULATOR Approximate to the nearest hundredth the relative or absolute extrema of f (x) = x 4 – 5x 2 – 2x + 4. Give the coordinates.

11. Example 6:
GRAPHING CALCULATOR Approximate to the nearest hundredth the relative or absolute extrema of f (x) = x 3 + 2x 2 – x – 1. Give the coordinates.

12. Example 7:
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?

13. Example 8:
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.

14. 𝑚 𝑠𝑒𝑐 = 𝑓 𝑥 2 −𝑓( 𝑥 1 ) 𝑥 2 − 𝑥 1

15. Example 9:
a. Find the average rate of change of f (x) = –2x 2 + 4x + 6 on the interval [–3, –1]. b. Find the average rate of change of f (x) = –3x 3+ 2x + 3 on the interval [–2, –1].

16. Example 10:
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 1 to 2 seconds.