<p>1-4 – Extrema and Average Rates of Change Name ______</p><p>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.</p><p>Increasing and Decreasing Behavior An analysis of a function can also include a description of the ______on which the function is ______/______/______. </p><p>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.</p><p>Analyze Graphically Support Numerically</p><p> 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.</p><p>Analyze Graphically Support Numerically</p><p> 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.</p><p>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.</p><p>Analyze Graphically Support Numerically x -100 -1 -0.5 0 0.5 1 1.5 100 f(x)</p><p> Example: Estimate and classify the extrema for the graph of f(x). Support answers numerically.</p><p>Analyze Graphically Support Numerically</p><p> x -100 -1 -0.5 0 0.5 1 1.5 100 f(x)</p><p>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.</p><p> 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.</p><p> Example: Find the average rate of change of f(x) = -x3 + 3x on each interval: A. [-2, -1] B. [0, 1]</p><p>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.</p><p>Below is a graph of h(t). Does this confirm your answer? Why or why not?</p><p> 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.</p><p>CW/HW Assignment: ______</p>