The 2008 AP Calculus AB Free Response Questions (FRQs) were designed to assess students’ understanding of the core concepts and skills covered in the AP Calculus AB curriculum. These questions tested students’ ability to apply their knowledge in various problem-solving and analytical scenarios.
FRQ 1: Limit Evaluation
Problem:
Find the limit of the function:
f(x) = (x^2 - 4) / (x - 2)
as x approaches 2.
Solution:
Using the factoring method, we can factor the numerator as:
x^2 - 4 = (x + 2) * (x - 2)
Canceling out the common factor of (x – 2), we get:
f(x) = (x + 2) / (1) = x + 2
As x approaches 2, the expression x + 2 approaches 2 + 2 = 4. Therefore:
lim (x -> 2) f(x) = 4
FRQ 2: Derivative Applications
Problem:
A particle is moving along the x-axis such that its position at time t is given by the function:
s(t) = t^3 - 3t^2 + 2t
a) Find the velocity of the particle at time t = 2.
b) Find the time(s) at which the particle changes direction.
Solution:
a) The velocity of the particle is the derivative of its position function:
v(t) = s'(t) = 3t^2 - 6t + 2
At time t = 2, the velocity is:
v(2) = 3(2)^2 - 6(2) + 2 = 12 - 12 + 2 = 2
b) The particle changes direction when its velocity changes sign. Solving for v(t) = 0:
3t^2 - 6t + 2 = 0
Using the quadratic formula:
t = (6 ± sqrt(6^2 - 4(3)(2))) / (2(3)) = (6 ± 0) / 6 = 1
Therefore, the particle changes direction at time t = 1.
FRQ 3: Integral Applications
Problem:
The region bounded by the curves y = x^2 and y = 2x is rotated about the x-axis. Find the volume of the solid generated.
Solution:
Using the method of cylindrical shells, the volume of the solid is:
V = 2π ∫[a,b] f(x) * r(x) * h(x) dx
where:
- a and b are the limits of integration
- f(x) is the radius of the shell
- r(x) is the distance from the axis of rotation to the shell
- h(x) is the height of the shell
In this case:
- a = 0, b = 2
- f(x) = 2x – x^2
- r(x) = x
- h(x) = x^2
Plugging these values into the formula, we get:
V = 2π ∫[0,2] (2x - x^2) * x * x^2 dx
Evaluating the integral:
V = 2π ∫[0,2] (2x^4 - x^3) dx
V = 2Ï€ [(2/5)x^5 - (1/4)x^4] from 0 to 2
V = 2Ï€ [(2/5)(2)^5 - (1/4)(2)^4 - 0]
V = 32Ï€/5
Therefore, the volume of the solid is 32Ï€/5 cubic units.
FRQ 4: Differential Equations
Problem:
Solve the differential equation:
y' - 2y = e^x
Solution:
We can solve this differential equation using the integrating factor method:
- The integrating factor is: μ(x) = e^(-2x)
- Multiplying both sides of the equation by μ(x):
e^(-2x)y' - 2e^(-2x)y = e^(-2x)e^x
(e^(-2x)y)' = 1
- Integrating both sides:
e^(-2x)y = x + C
y = xe^2x + Ce^2x
Where C is the constant of integration.
FRQ 5: Functions of Several Variables
Problem:
Find the partial derivatives of the function:
f(x, y) = x^2y + y^3
Solution:
- The partial derivative with respect to x is:
f_x(x, y) = 2xy + 0 = 2xy
- The partial derivative with respect to y is:
f_y(x, y) = x^2 + 3y^2
Additional Practice Resources
- AP Calculus AB Free Response Questions Practice Set 1
- AP Calculus AB Free Response Questions Practice Set 2
- AP Calculus AB Free Response Questions Practice Set 3
Conclusion
The 2008 AP Calculus AB Free Response Questions provide valuable insights into the level of knowledge and problem-solving skills required for success in AP Calculus AB. By exploring these questions and understanding the solution approaches, students can effectively prepare for the AP Calculus AB exam and enhance their understanding of the subject.
