Introduction
The 2004 Advanced Placement Calculus AB Free Response Question (FRQ) was a challenging yet rewarding experience for students. This article provides a comprehensive analysis of the question, along with step-by-step solutions and insightful explanations. By studying this material, students can gain a deeper understanding of the concepts tested on the exam and improve their performance on future assessments.
Question 1: Limits and Continuity
(a) Find the limit of the function:
lim (x -> 0) (sin(x) - x) / x^3
(b) Use the result from part (a) to determine whether the function:
f(x) = (x - sin(x)) / x^3
is continuous at x = 0.
Solution:
(a) Using L’Hopital’s Rule:
lim (x -> 0) (sin(x) - x) / x^3 = lim (x -> 0) (cos(x) - 1) / 3x^2 = -1/3
(b) Since the limit in part (a) exists and is equal to -1/3, and the function is defined at x = 0, f(x) is continuous at x = 0.
Question 2: Derivatives and Applications
Model: A population of rabbits grows at a rate proportional to its size. At time t = 0, there are 100 rabbits, and at time t = 3, there are 200 rabbits.
(a) Find the differential equation that models the population growth.
(b) Solve the differential equation to find the population function.
(c) How many rabbits will there be at time t = 8?
Solution:
(a) The differential equation is:
dP/dt = kP
where P is the population size and k is the growth constant.
(b) Solving using separation of variables:
∫ (dP/P) = k dt
ln(P) = kt + C
P = e^(kt+C) = Ce^(kt)
Using the initial condition P(0) = 100, we get:
100 = Ce^(k⋅0)
C = 100
Therefore, the population function is:
P(t) = 100e^(kt)
To find k, we use the second initial condition:
200 = 100e^(k⋅3)
k = ln(2)/3
So, the population function is:
P(t) = 100e^(ln(2)⋅t/3) = 100⋅2^(t/3)
(c) At time t = 8, there will be:
P(8) = 100⋅2^(8/3) ≈ 800 rabbits
Question 3: Related Rates
Model: A spherical balloon is being inflated at a rate of 10 cubic centimeters per second.
(a) Find the rate of change of the radius of the balloon when the radius is 5 centimeters.
(b) Use the result from part (a) to determine whether the volume is increasing or decreasing faster at that instant.
Solution:
(a) The volume of a sphere is:
V = (4/3)πr^3
Differentiating both sides with respect to time:
dV/dt = 4πr^2 dr/dt
Given dV/dt = 10 when r = 5, we get:
10 = 4π(5)^2 dr/dt
dr/dt = 1/(100π) ≈ 0.00318 cm/s
(b) The volume is increasing faster.
Additional Resources
Conclusion
The 2004 AP Calc AB FRQ was a challenging yet rewarding experience for students. By carefully analyzing the question, using appropriate mathematical techniques, and applying insightful reasoning, students can develop a deep understanding of the concepts tested on the exam. This article provides a comprehensive analysis of the question, along with step-by-step solutions and insightful explanations, serving as a valuable resource for students preparing for future assessments.