AP Calculus AB 2

Question 1

What is the derivative of f(x) = x³ − 4x² + 7x − 2?

Accepted Answer

3x² − 8x + 7

Answer

Applying the power rule term by term gives 3x² − 8x + 7.

Question 2

Using the chain rule, what is d/dx[sin(x²)]?

Accepted Answer

2x·cos(x²)

Answer

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)?

Accepted Answer

3e^(3x)

Answer

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)?

Accepted Answer

sin(x) + x·cos(x)

Answer

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?

Accepted Answer

x = −1

Answer

Setting f′(x) = 3x² − 3 = 0 gives x = ±1; since f″(−1) = −6 < 0, x = −1 is the local maximum.

AP Practice Test