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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 𝑉 of water in a tank with respect to time, 𝑡 is
directly proportional to the square root of the volume. Which of the following
is a differential equation that describes this relationship?
(A) 𝑉(𝑡) = 𝑘√𝑡
(B) 𝑉(𝑡) = 𝑘√𝑉
𝑑𝑉
(C) 𝑑𝑡 = 𝑘√𝑉
(D)
2.
𝑑𝑉
𝑑𝑡
=
𝑘
√𝑉
Which of the following is the solution to the differential equation
the initial condition 𝑦(3) = −2?
2𝑥 3
(A) 𝑦 = −√
(B) 𝑦 = −2𝑒
3
𝑑𝑦
𝑑𝑥
=
𝑥2
𝑦
with
− 14
−9+𝑥3
3
2𝑥 3
(C) 𝑦 = √
3
2𝑥 3
(D) 𝑦 = √
3.
3
− 14
Water flows continuously from a large tank at a rate proportional to the amount
𝑑𝑦
of water remaining in the tank; that is, 𝑑𝑡 = 𝑘𝑦. There was initially 10,000 cubic
feet of water in the tank, and at time t  4 hours, 8,000 cubic feet of water
𝑑𝑦
remained. What is the value of k in the equation 𝑑𝑡 = 𝑘𝑦? You may use your
calculator for this problem.
(A)
(B)
(C)
(D)
4.
−0.050
−0.056
−0.169
−0.200
Population 𝑦 grows according to the equation
𝑑𝑦
𝑑𝑡
= 𝑘𝑦, where 𝑘 is a constant and 𝑡
is measured in years. It the population double every 10 years, then the value of
is 𝑘? (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
𝑑𝑦
𝑑𝑥
=
3𝑥 2
𝑒 2𝑦
.
1
(a)
Find a solution 𝑦 = 𝑓(𝑥) to the differential equation satisfying 𝑓(0) = 2.
(b)
Find the domain and range of the function 𝑓 found in part (a).
1
2. The function 𝑓 is differentiable for all real numbers. The point (3, 4) is on the graph of
𝑦 = 𝑓(𝑥), and the slope at each point (𝑥, 𝑦) on the graph is given by
𝑑2 𝑦
𝑑𝑥
= 𝑦 2 (6 − 2𝑥).
1
(a)
Find
(b)
Find 𝑦 = 𝑓(𝑥) by solving the differential equation
𝑑𝑥 2
𝑑𝑦
and evaluate it at the point(3, 4).
1
condition 𝑓(3) = 4.
𝑑𝑦
𝑑𝑥
= 𝑦 2 (6 − 2𝑥) with the initial
3. Consider the differential equation
𝑑𝑦
𝑑𝑥
= (3 − 𝑦) cos 𝑥. Let 𝑦 = 𝑓(𝑥) be the particular
solution to the differential equation with the initial condition 𝑓(0) = 1. The function 𝑓 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 𝑓(0.2).
(c)
Find 𝑦 = 𝑓(𝑥), the particular solution to the diffenetial equation with the initial
condition 𝑓(0) = 1.