Calculus Flashcards

Find the derivative of f(x) = x·ln(x) − x.

Answer: ln(x)

Using the product rule, d/dx[x·ln(x)] = ln(x) + 1, so f′(x) = ln(x) + 1 − 1 = ln(x).

Evaluate the definite integral ∫₀¹ x·eˣ dx.

Answer: 1

Integrating by parts with u = x, dv = eˣ dx gives [x·eˣ − eˣ]₀¹ = (e − e) − (0 − 1) = 1.

A function g satisfies g′(x) = 3x² − 6x and g(0) = 4. What is g(2)?

Answer: 0

Integrating gives g(x) = x³ − 3x² + 4; then g(2) = 8 − 12 + 4 = 0.

Which of the following best describes lim_{x→0} (sin x)/x?

Answer: The limit is 1

The standard result lim_{x→0} (sin x)/x = 1, provable by L'Hôpital's rule or the squeeze theorem.

The curve y = x³ − 3x has a local maximum at x = ?

Answer: x = −1

Setting y′ = 3x² − 3 = 0 gives x = ±1; y″(−1) = −6 < 0 confirms a local maximum at x = −1.

∫ sec²(x) dx = ?

Answer: tan(x) + C

The derivative of tan(x) is sec²(x), so the antiderivative of sec²(x) is tan(x) + C.

If f(x) = x⁴ − 2x² + 1, what is f′(x)?

Answer: 4x³ − 4x

Applying the power rule term by term: f′(x) = 4x³ − 4x.