Sections Practice AP Calculus AB Name

Transcription

1 Sections Practice AP Calculus AB Name Be sure to show work, giving written explanations when requested. Answers should be written exactly or rounded to the nearest thousandth. When the calculator is used to find an answer, the written notation must be mathematical, not calculator-ese. Include units where applicable. 1. Let h be a function defined for all x 0 such that h (4) 3 and the derivative of h is given by h'( x) x 2 x 2 for all x 0. a. Find all values of x for which the graph of h has a horizontal tangent, and determine whether h has a local maximum, a local minimum, or neither at each of these values. Justify your answers. b. On what intervals, if any, is the graph of h concave up? Justify your answer. c. Write an equation for the line tangent to the graph of h at x = 4.

2 d. Does the line tangent to the graph of h at x = 4 lie above or below the graph of h for all x > 4? Why? 2. Let f (x) = x 3 + px 2 + qx, where p and q are constants.. a. Find the values of p and q so that f (-1) = -8 and f (-1) = 12. b. Find the values of p and q so that the line y = 20x + 20 is tangent to the graph of f at x = -2.

3 c. Find the value of p so that the graph of f changes concavity at x = Suppose that the function f has a continuous second derivative for all x, and that f (0) 2, f '(0) 3, and f "(0) 0. Let g be a function whose derivative is given by g'( x) e 2 x (3 f ( x) 2 f '( x)) for all x. a. Write an equation of the line tangent to the graph of f at the point where x = 0. b. Is there sufficient information to determine whether or not the graph of f has a point of inflection when x = 0? Explain your answer.

4 c. Given that g (0) 4, write an equation of the line tangent to the graph of g at the point where x = 0. 2x d. Show that g"( x) e ( 6 f ( x) f '( x) 2 f "( x)). Does g have a local maximum at x = 0? Justify your answer.

5 4. The graph of f, the derivative of a differentiable function f, is shown below. a. State the sub-interval(s) of [-4, 4] on which f is increasing. y = f b. State the x-coordinate of any relative extrema of f on the interval [-4,4]. Identify them as minima or maxima and explain. c. Estimate f (1). d. To the nearest integer, state the x-coordinate(s) of the inflection point(s) of f on the interval [-4, 4]. Explain.

6 e. To the nearest integer, state the sub-interval(s) of [-4, 4] on which f is concave down. Explain. 5. Let f be the function defined by f ( x) ln(2 sin x) for x 2. a. Find the absolute maximum and absolute minimum values of f on x 2 the analysis that leads to your conclusion.. Show b. Find the x-coordinate of each inflection point on the graph of f for x 2. Justify your answer.

7 CALCULATOR ACTIVE! 6. Suppose the derivative of a polynomial function p(x) is p (x) = (x + 1)(x 1)(x 2) 2 (x 4) 3. a. What is the degree of p? b. What is the instantaneous rate of change of p at x = 6? c. Identify the critical points of p and classify them as relative minima, relative maxima, or neither. d. Find the intervals on which the graph of p is increasing. Explain. e. Find the intervals on which the graph of p is concave down. Explain.

8 7. Two points, A and B, are 275 feet apart. At a given instant, a balloon is released at B and rises vertically at a constant rate of 2.5 ft/sec. At the same instant, a cat starts running from A to B at a constant rate of 5 ft/sec. a. After 40 seconds, is the distance between the cat and the balloon decreasing or increasing? At what rate? b. Describe what is happening to the distance between the cat and the balloon at time t = 50 seconds. Be specific.

9 8. Two runners, A and B, run on a straight racetrack for 0 t 10 seconds. The graph above shows the velocity, in meters per second, of Runner A [Note: the graph consists of two line segments: between (0, 0) and (3, 10) and between (3, 10) and (10, 10)]. The velocity, in 24t meters per second, of Runner B is given by the function defined by v ( t). 2t 3 a. Find the velocity of Runner A and the velocity of Runner B at time t = 2 seconds. Indicate units of measure. b. Find the acceleration of Runner A and the acceleration of Runner B at time t = 2 seconds. Indicate units of measure. c. Find the total distance covered by Runner A over the time interval 0 t 10 seconds. Indicate units of measure.