Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
AP Calculus 6.4 AP Problems Name ___________________ Date _____ Directions: The following are past AP Exam problems! Part I: Multiple Choice: NO CALCULATOR 1. The rate of change of the volume π of water in a tank with respect to time, π‘ is directly proportional to the square root of the volume. Which of the following is a differential equation that describes this relationship? (A) π(π‘) = πβπ‘ (B) π(π‘) = πβπ ππ (C) ππ‘ = πβπ (D) 2. ππ ππ‘ = π βπ Which of the following is the solution to the differential equation the initial condition π¦(3) = β2? 2π₯ 3 (A) π¦ = ββ (B) π¦ = β2π 3 ππ¦ ππ₯ = π₯2 π¦ with β 14 β9+π₯3 3 2π₯ 3 (C) π¦ = β 3 2π₯ 3 (D) π¦ = β 3. 3 β 14 Water flows continuously from a large tank at a rate proportional to the amount ππ¦ of water remaining in the tank; that is, ππ‘ = ππ¦. There was initially 10,000 cubic feet of water in the tank, and at time t ο½ 4 hours, 8,000 cubic feet of water ππ¦ remained. What is the value of k in the equation ππ‘ = ππ¦? You may use your calculator for this problem. (A) (B) (C) (D) 4. β0.050 β0.056 β0.169 β0.200 Population π¦ grows according to the equation ππ¦ ππ‘ = ππ¦, where π is a constant and π‘ is measured in years. It the population double every 10 years, then the value of is π? (You may use your calculator) (A) 0.069 (B) 0.200 (C) 0.301 (D) 3.322 Part II: Free-Response: NO CALCULATOR 1. Consider the differential equation ππ¦ ππ₯ = 3π₯ 2 π 2π¦ . 1 (a) Find a solution π¦ = π(π₯) to the differential equation satisfying π(0) = 2. (b) Find the domain and range of the function π found in part (a). 1 2. The function π is differentiable for all real numbers. The point (3, 4) is on the graph of π¦ = π(π₯), and the slope at each point (π₯, π¦) on the graph is given by π2 π¦ ππ₯ = π¦ 2 (6 β 2π₯). 1 (a) Find (b) Find π¦ = π(π₯) by solving the differential equation ππ₯ 2 ππ¦ and evaluate it at the point(3, 4). 1 condition π(3) = 4. ππ¦ ππ₯ = π¦ 2 (6 β 2π₯) with the initial 3. Consider the differential equation ππ¦ ππ₯ = (3 β π¦) cos π₯. Let π¦ = π(π₯) be the particular solution to the differential equation with the initial condition π(0) = 1. The function π is defined for all real numbers. (a) A portion of the slope field of the differential equation is given below. Sketch the solution curve through the point (0,1). (b) Write an equation for the line tangent to the solution curve in part (a) at the point (0,1). Use the equation to approximate π(0.2). (c) Find π¦ = π(π₯), the particular solution to the diffenetial equation with the initial condition π(0) = 1.