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Chapter Three: Section Four Concavity and the Second Derivative Test Chapter Three: Section Four In the last section we talked about using the first derivative to make distinctions about when a function is increasing or decreasing. We are working on our ability to make accurate sketches of complex functions. We know that graphs can look different even when they are both increasing or both decreasing. Look at the graph on the next slide carefully and try to describe what looks different about the regions where the function is increasing. Chapter Three: Section Four y x 4 12 x3 48x 2 64 x Chapter Three: Section Four I want you to complete a sign chart for this function the way that we have been for the first derivative test. After that, repeat the process for the second derivative and see what happens to the second derivative over the regions where the function is increasing. Remember, the equation we are examining is y x 4 12 x3 48x 2 64 x Chapter Three: Section Four Compare the results of the two sign charts and look again at the graph below; Chapter Three: Section Four The word we will use to describe the different behaviors of the increasing regions of the function is the word concavity. When the second derivative of a function is positive, we say that the function is concave up. What this means physically is that the movement of the graph has positive acceleration. When the second derivative is negative then the function is said to be concave down and this means that the acceleration of the graph is negative. Chapter Three: Section Four To summarize the past two sections; If both f ‘ and f “ are positive, then both velocity and acceleration are positive. The function is rising and concave up. If f ‘ is negative and f “ is positive, then velocity is negative and acceleration is positive. The function is falling and concave up. If f ‘ is positive and f “ is negative, then both velocity is positive and acceleration is negative. The function is rising and concave down. If both f ‘ and f “ are negative, then both velocity and acceleration are negative. The function is falling and concave down. Chapter Three: Section Four Can you sketch functions (just a sketch, not an equation) that matches up with these four pieces? Be prepared to share your ideas in class.