AP Calculus AB 2 — Questions and Answers
Question 1: What is the derivative of f(x) = x³ − 4x² + 7x − 2?
- 3x² − 8x + 7 (Correct answer)
- 3x² − 4x + 7
- x² − 8x + 7
- 3x³ − 8x² + 7x
Correct answer: 3x² − 8x + 7
Applying the power rule term by term gives 3x² − 8x + 7.
Question 2: Using the chain rule, what is d/dx[sin(x²)]?
- cos(x²)
- 2x·cos(x²) (Correct answer)
- 2x·sin(x²)
- cos(2x)
Correct answer: 2x·cos(x²)
The outer function is sin and the inner function is x²; by the chain rule, the derivative is cos(x²)·2x.
Question 3: What is the derivative of f(x) = e^(3x)?
- e^(3x)
- 3e^(3x) (Correct answer)
- 3x·e^(3x)
- e^(3x)/3
Correct answer: 3e^(3x)
By the chain rule, d/dx[e^u] = e^u·u′; here u = 3x and u′ = 3, giving 3e^(3x).
Question 4: If f(x) = x·sin(x), what is f′(x)?
- cos(x)
- x·cos(x)
- sin(x) + x·cos(x) (Correct answer)
- sin(x) − x·cos(x)
Correct answer: sin(x) + x·cos(x)
The product rule gives f′(x) = 1·sin(x) + x·cos(x) = sin(x) + x·cos(x).
Question 5: At what x-value does f(x) = x³ − 3x have a local maximum?
- x = −1 (Correct answer)
- x = 0
- x = 1
- x = 3
Correct answer: x = −1
Setting f′(x) = 3x² − 3 = 0 gives x = ±1; since f″(−1) = −6 < 0, x = −1 is the local maximum.
Question 6: What is d/dx[ln(x²)]?
- 1/x²
- 2/x (Correct answer)
- 2x
- 1/(2x)
Correct answer: 2/x
Using the chain rule, d/dx[ln(x²)] = (1/x²)·2x = 2/x; equivalently, ln(x²) = 2ln(x) → derivative = 2/x.
Question 7: A particle's position is s(t) = t³ − 6t² + 9t. At what times is the particle at rest?
- t = 1 only
- t = 3 only
- t = 1 and t = 3 (Correct answer)
- t = 2 and t = 4
Correct answer: t = 1 and t = 3
The particle is at rest when v(t) = s′(t) = 3t² − 12t + 9 = 3(t−1)(t−3) = 0, giving t = 1 and t = 3.
AP Calculus AB 2
What is the derivative of f(x) = x³ − 4x² + 7x − 2?