AP Calculus Worksheet 3-1 Mr. James 1. Consider the function p ( x) x 3 ax , where a is constant and a > 0. a. Find the local maxima and minima of p(x). b. What effect does increasing the value of a have on the position of the maxima and minima? c. On the same set of axes, sketch and label the graphs of p for three positive values of a. 2. What effect does increasing the value of a have on the graph of q ( x) x 2 2ax ? Consider roots, maxima, and minima for both positive and negative values of a. 3. Consider the family of functions y f ( x) x k x , with k a positive constant and x 0. Show that the graph of f(x) has a local minimum at a point whose x-coordinate is ¼ the way between its x-intercepts. 4. Consider f ( x) x 4 ax 2 b . a. Find all critical points. b. Under what conditions of a and b will f(x) have exactly one critical point? What is the one critical point and is it a maximum, minimum, or neither? c. Under what conditions of a and b will f(x) have exactly three critical points? What are they and which are local maxima and which are local minima? d. Is it ever possible for this function to have two critical points? No critical points? More than three critical points? Give an explanation in each case. 5. Consider the family of functions f ( x) x 2 cos(kx) where k > 0. a. Sketch the graph of f for k = 0.5, 1, 3, and 5. Try to find the smallest value of k at which you see points of inflection in the graph of f. b. Explain why the graph has no points of inflection if k 2 and infinitely many points of inflection if k 2 . c. Explain why f has only a finite number of critical points, no matter what the value of k. A 6. Consider the family y . xB a. If B = 0, what is the effect of varying A on the graph? b. If A = 1, what is the effect of varying B? c. On one set of axes, graph the function for several values of A and B.
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