AP Calculus AB Chapter 2 Test Review #1
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1 AP Calculus AB Chapter Test Review # Open-Ended Practice Problems:. Nicole just loves drinking chocolate milk out of her special cone cup which has a radius of inches and a height of 8 inches. Nicole pours milk into her cone cup at the constant rate of.5 in /sec. The total amount of milk that this cone cup can hold is.50 in. a. Find the radius and height of the milk when the cone cup is half full. b. How fast is the height and radius of the milk changing at the instant the cone cup is half full of milk? c. The milk leaves a ring of chocolate inside the cup as it is poured. How fast is the milk ring moving up the sides of the cone cup at the instant the cone is half full?. The position equation of a particle moving along the x-axis is given by x ( t) t t, t 0. a. Describe the motion of the particle. b. Find the position of the particle when its instantaneous velocity is zero. c. Find the speed of the particle when the position is zero. d. Is the speed of the particle increasing or decreasing at t. e. Find the average velocity of the particle during its first three seconds of travel.
2 . Consider the curve given by y dy y a. Show that dx y x. xy. b. Find all points ( x, y ) on the curve where the line tangent to the curve has slope. c. Show that there are no points ( x, y ) on the curve where the line tangent to the curve is horizontal. d. Let x and y be functions of time t that are related by the equation y xy. At time dy t 5, the value of y is and 6 dt dx. Find the value of at time t 5. (No Units!) dt. Show that the slope of every line tangent to the curve f ( x) ( x) is positive. 5. A six foot tall man walks away from a 5 foot tall lamp post at 5 ft/sec. How fast is his shadow lengthening? How fast is the shadow s tip moving?
3 6. At a given moment, a plane passes directly above a radar station at an altitude of 6 km. The plane s speed is 800 km/h. a. How fast is the distance between the plane and the station changing half a minute later? b. How fast is the distance between the plane and the station changing when the plane passes directly above the station? c. Let θ be the angle that the line through the radar station and the plane makes with the horizontal. How fast is θ changing minutes after the plane passes over the radar station? 7. Let s( t) t 6t 9t be the position function, in feet, for a projectile s path along the y- axis for t 0, for t seconds. a. What is the acceleration of the particle at 8 seconds? b. What is the average velocity of the particle during the first four seconds? c. Describe the motion of the particle. d. What is the speed of the particle when the acceleration is zero? e. What is the displacement of the particle during the first seconds? f. Is the speed of the particle increasing or decreasing when t 8?
4 8. Sand is falling from a rectangular box container whose base measures 0 inches by 0 inches at a constant rate of 00 cubic inches per minute. a. How fast is the depth of the sand in the box changing? (Exact answers only) b. As the sand falls from the rectangular box it forms a conical pile on the ground. At a particular moment, the pile is inches high and the diameter of the base is 6 inches. The radius of the base at this moment is increasing at inches per minute. i. At this moment, how fast is the area of the circular base of the cone increasing? (Exact answers only) ii. Write an equation for the rate of change of the volume in terms of the radius and height. Determine how fast the height of the pile increasing at this moment. (Round your answer to nearest thousandth.)
5 Multiple-Choice Practice Problems: cos h 9. lim h0 h a. b. c. d. e. 0. tan 5x lim x x a. 0 b. c. 5 d. 0 e. nonexistent. Find the derivative of cos sin f. cos( b. sin c. sin d. sin cos a. ) e. cos( ). Let f x x, then f a. b. 0 c. d. e. nonexistent Use the following table to answer questions #-6. x x x x. If H x F x, then H F F F G x G x G x a. 0 b. 0 c. 5 d. 0 e. 00 F. If x H x, then H Gx a. 7 b. c. 0 d. 7 e. 5. If H x GF x, then H a. 6 b. 6 c. d. 8 e. 6. If H x GF x, then H a. b. 0 c. 6 d. 56 e. 88
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