1. Find the area bounded by the curve and the x axis between 5 and Find the area bounded by and the x axis between 3 and 3.

Transcription

1 AP Calculus November 15, Find the area bounded by the curve and the x axis between 5 and Find the area bounded by and the x axis between 3 and Find the area bounded by the curve graphed below and the x axis between 5 and 5. Nov 26 5:49 PM 1

2 November 16, Find an estimate for the area bounded by, the x axis, x = 4, and x = 8 using a midpoint Reimann sum. Remember, do not round until the end of the problem. Round to three decimal places. 2. Kathleen skates on a straight track. She starts at rest from the starting line at time. For seconds, her velocity k, measured in feet per second, is increasing and differentiable. Values of at various times are given in the table. t (seconds) (a) Use the data to estimate Kathleen's accelration at time t = 4. Show work. k(t) (ft/sec) (b) Use a right Reimann sum with four subintervals to estimate the distance traveled by Kathleen from time t = 0 to time t = 12. Include units. Would this be an overestimate or an underestimate. Explain. Nov 26 5:53 PM 2

3 November 27, Use a right Reimann sum with 3 equal subintervals to estimate the area bounded by the curve and the x axis from x = 0 to x = 9. Is the estimation an overestimate or underestimate? Explain. 2. Ruth rode her bicycle on a straight track. She recorded her velocity v(t) in miles per hour for selected values of t over as shown in the table. For,. t (hours) (a) Approximate Ruth's v(t) (mph) acceleration at time t = 1.4 hr. Show computations and include units. (b) Use a midpoint Reimann sum with three subintervals of equal length to approximate Ruth's distance traveled over. Include units. Nov 26 5:08 PM 3

4 AP Calculus November 28, If f (x) = 5x, then the relationship between x and y would be described graphically by (A) a parabola (B) a circle (C) an ellipse (D) a straight line 2. Given f (x) = 3x 2 + 2x 3, find f(x). 3. Given f''(x) = 32, find f(x). 4. On May 7, 1992, the space shuttle Endeavor was launched on a mission to install a new kick motor in a communications satellite. The table below gives velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. Using 6 subintervals, find upper and lower estimates (left and right endpoint Reimann sum) for the area under the velocity curve 62 seconds after liftoff. time (s) velocity (ft/s) Nov 26 5:45 PM 4

5 November 29, An object thrown from the top of a 100 meter tall building will have an acceleration of 9.8 m/s 2. If the object is thrown with an initial velocity of 25 m/s, what is the position function of the object? 2. Find g(x), given that 3. A function f has relative extreme values at x = 2 and x = 3. Also, the y intercept of the function is 5. Find f. se Dec 1 1:43 PM 5

6 November 30, Find the following: (a) (b) (c) (d) 2. Given that and the graph of y = f(x) contains the point ( 1,5), find f(x). 3. On the moon, the acceleration due to gravity is 1.6 m/s 2. A stone is dropped from a cliff on the moon and hits the surface of the moon 20 seconds later. How far did it fall? What was its velocity on impact? 4. If, find the value of k. Nov 30 6:41 AM 6

7 December 4, Evaluate the following: (a) (b) 2. The marginal cost of manufacturing x yards of a certain fabric is C'(x) = x x 2 (in dollars per yard). Find the increase in cost of the production level is raised from 2000 yards to 4000 yards. 3. Suppose a volcano is erupting and readings of the rate r(t) at which solid materials are spewed into the atmosphere are given in the table. Estimate the value of using a right Reimann sum with 6 subintervals. What does the value of represent in the context of the problem? t (seconds) r(t) (metric tons per second) Dec 12 1:36 PM 7

8 December 5, The velocity of a particle is given by the function v(t) = 6 4t. (a) Find the displacement of the object over the interval 0 t 5. (b) Find the distance traveled over the same interval. 2. The acceleration of a particle is given by the function a(t) = 2t 4. The initial velocity of the function is v(0) = 3. Find the total distance traveled over the interval 0 t If f (x) = 6x 2 and f(2) = 1, then (A) 22 (B) 16 (C) 2 (D) 8 (E) 18 Nov 28 3:47 PM 8

9 December 6, A particle with an initial velocity of 3 m/s has an acceleration given by 8x 8. Find the displacement of the particle in the first 4 seconds of motion. Then find the total distance traveled in the first 4 seconds. 2. If p > 0 and q > 0, (A) (B) p (C) (D) (E) 3. A bacteria population starts with 400 bacteria and grows at a rate of bacteria per hour. How many bacteria will there be after three hours? Set up the integral and use your calculator to evaluate. Dec 7 6:37 AM 9