EXTREMA and average rates of change

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EXTREMA and average rates of change LESSON 1-4 EXTREMA and average rates of change

Increasing and Decreasing Functions Increasing Functions If x1 < x2, then f(x1) < f(x2) Example y = x3 Decreasing Functions If x1 < x2, then f(x1) > f(x2) Example y = -2x + 1 Constant Functions f(x1) = f(x2) Example y = 3

Example 1 Graph y = -x5 + 3x3 on your graphing calculator and determine on which intervals the graph is increasing, decreasing, or constant. Answer: Increasing : (-1.34, 1.34) Decreasing: (-∞, -1.34) U (1.34,∞)

Extrema Absolute minimum the smallest value of the function over its entire domain Absolute maximum the largest value of the function over its entire domain Relative minimum the smallest value of the function on some interval of the domain Relative maximum the largest value of the function on some interval of the domain

Estimate the Extrema

Average Rate of Change The average rate of change between any two points on a graph is the slope of the line through those points (the line is called a secant line) msec = f(x2) – f(x1) x2 – x1 Can you draw a picture of this?

Find the average rate of change of f(x) = -2x2 + 4x + 6 on each interval. a) [-3,-1] Answer: 12 b) [2,5] Answer: -10 c) [0,2] Answer: 0

The distance traveled by falling objects on the moon is d(t) = 2 The distance traveled by falling objects on the moon is d(t) = 2.7t2 where d(t) is the distance traveled in feet and t is the time in seconds. What is the average speed of the object from 1 sec to 2 sec? From 2 sec to 3 sec? Answer: 8.1 feet/sec and 13.5 feet/second

Using only the table, describe the intervals on which the function is increasing/decreasing and list and label the extrema. y = (x – 1)3(x+2)4 , use [-3,3] by .1 rel min (-.3,-18.4) rel max (-2,0) inc (-∞, -2) U (-.3, ∞) dec (-2, -.3) b) y = 4x4 + 21x3 + 25x2 - 5x + 3 , use [-5,2] by .1 abs min (-2.8, -2.1) rel max (-1.3,17) rel min (.1,2.77) dec (-∞, -2.8) U (-1.3, .1) inc(-2.8, -1.3)U(.1, ∞)