Curve Sketching Today we will look at the graph of a function, and then the graph of the derivative of a function and draw conclusions about important characteristics of these functions using the knowledge you have gained from creating sign studies.

The graph of f(x) is shown below At which labeled points is f’(x) positive? Give a reason for your selection.

The graph of f(x) is shown below Between which pairs of labeled points does f(x) have a stationary point? (A stationary point occurs where the first derivative is zero) Give a reason for your selection.

The graph of f(x) is shown below Between which pairs of labeled points does f(x) have an inflection point? Give a reason for your selection.

The graph of f(x) is shown below Between which pairs of labeled points does f’(x) achieve its minimum value? Give a reason for your selection.

The graph of the derivative of a function f is shown below. [Note: the graph of f is NOT shown.] Where does f have stationary points? Explain.

The graph of the derivative of a function f is shown below. [Note: the graph of f is NOT shown.] Where does f have local maxima? Minima? Inflection points? Explain.

Matching Functions with their Derivatives Today you will look at the graphs of several functions and try to match each function with the graph of its derivative and second derivative by using the characteristics (such as relative extrema, concavity, increasing & decreasing, etc.) defined by the sign studies.

Function – Derivative Matching Activity Each of you randomly received a card with either the graph of a function, the graph of a first derivative, or the graph of the second derivative. Your task is to walk around the room and try to find the people who have the matching function, first derivative, and second derivative graphs by using the characteristics of f(x), f’(x), and f”(x) that you have discovered using the sign studies. When you think you have found your partners, check with me to see if you are correct.

Summary Sign Study Zeros (Critical Pts)+ sign study- sign study x-intercepts of graph lies above x-axis lies below x-axis Relative Extrema (if sign changes) (Max + to -) (Min - to +) is increasing is decreasing Points of inflection (if sign changes) is concave up is concave down

In the left hand column are graphs of several functions. In the right-hand column – in a different order – are graphs of the associated derivative functions. Match each function with its derivative. [Note: The scales on the graphs are not all the same.]

Jigsaw Activity Now we are going to look at some number line sign studies that illustrate the characteristics of f(x), f’(x), and f”(x). Work with your team to try to draw a possible graph of f(x), given these sign studies.

Assignment HW H Quiz tomorrow!