Calculus AB Flashcards

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

Answer: 3x² − 8x + 7

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

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

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.

What is the derivative of f(x) = e^(3x)?

Answer: 3e^(3x)

By the chain rule, d/dx[e^u] = e^u·u′; here u = 3x and u′ = 3, giving 3e^(3x).

If f(x) = x·sin(x), what is f′(x)?

Answer: sin(x) + x·cos(x)

The product rule gives f′(x) = 1·sin(x) + x·cos(x) = sin(x) + x·cos(x).

At what x-value does f(x) = x³ − 3x have a local maximum?

Answer: x = −1

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

What is d/dx[ln(x²)]?

Answer: 2/x

Using the chain rule, d/dx[ln(x²)] = (1/x²)·2x = 2/x; equivalently, ln(x²) = 2ln(x) → derivative = 2/x.

A particle's position is s(t) = t³ − 6t² + 9t. At what times is the particle at rest?

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.