1-4 Extrema and Average Rates of Change

1-4 – Extrema and Average Rates of Change Name ______

Objectives: 1. Determine intervals on which functions are increasing, constant, or decreasing, and determine maxima and minima of functions. 2. Determine the average rate of change of a function.

Increasing and Decreasing Behavior An analysis of a function can also include a description of the ______on which the function is ______/______/______.

We do this by following the graph from ______to ______.  As the graph goes up, we say the graph is ______.  If the graph is flat, we say the graph is ______.  If the graph goes down, we say the graph is ______.  Example: Use the graph of each function to estimate intervals to the nearest 0.5 unit on which the function is increasing, decreasing, or constant. Support the answer numerically.

Analyze Graphically Support Numerically

 Example: Use the graph of each function to estimate intervals to the nearest 0.5 unit on which the function is increasing, decreasing, or constant. Support the answer numerically.

Analyze Graphically Support Numerically Critical Points  Critical points of a function are those points at which a line drawn ______to the curve is ______or ______.  Extrema are critical points at which a function changes its ______.  At these points, the function has a ______or a ______value, either relative or absolute.  A point of inflection is a critical point at which the graph doesn’t change its increasing/decreasing behavior, but is where the curve changes from ______.  Example: Estimate and classify the extrema for the graph of f(x). Support answers numerically.

Analyze Graphically Support Numerically x -100 -1 -0.5 0 0.5 1 1.5 100 f(x)

 Example: Estimate and classify the extrema for the graph of f(x). Support answers numerically.

x -100 -1 -0.5 0 0.5 1 1.5 100 f(x)

CW/HW Assignment: ______Average Rates of Change In algebra, you learned that the slop between any two points on the graph of a linear function represents a constant rate of change. For a nonlinear function, the slope changes between different pairs of points, so we can only talk about the average rate of change between any two points.

 When the average rate of change over an interval is positive, the function ______on average over that interval. When the average rate of change is negative, the function ______on average over that interval.

 Example: Find the average rate of change of f(x) = -x3 + 3x on each interval: A. [-2, -1] B. [0, 1]

To the right is the graph of f(x). Does this support your answers from A and B? Why or why not? Average rate of change has many real-world applications. One common application involves the average speed of an object traveling over a distance d or from a height h in a given period of time t. Because speed is distance traveled per unit time, the average speed of an object cannot be negative.  Example (Physics): The height of an object that is thrown straight up from a height of 4 feet above ground is given by h(t) = -16t2 + 30t + 4, where t is the time in seconds after the object is thrown. Find and interpret the average speed of the object from 1.25 to 1.75 seconds.

Below is a graph of h(t). Does this confirm your answer? Why or why not?

 Example (Physics): If wind resistance is ignored, the distance d(t) in feet an object travels when dropped from a high place is given by d(t) = 16t2, where t is the time in seconds after the object is dropped. Find and interpret the average speed of the object from 2 to 4 seconds.

CW/HW Assignment: ______