Calculus Flashcards

For f(x) = e^(x²), what is f′(x)?

Answer: 2x·e^(x²)

By the chain rule, d/dx[e^(x²)] = e^(x²) · 2x.

The area enclosed between y = x² and y = x over [0, 1] is:

Answer: 1/6

∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6.

If h(x) = sin(3x), what is h″(x)?

Answer: −9 sin(3x)

h′(x) = 3 cos(3x) and h″(x) = −9 sin(3x).

Which theorem guarantees that a continuous function on a closed interval [a, b] attains a maximum and a minimum?

Answer: Extreme Value Theorem

The Extreme Value Theorem states that a continuous function on a closed, bounded interval attains its maximum and minimum.

Evaluate lim_{x→∞} (3x² + 1)/(x² − 5).

Answer: 3

Dividing numerator and denominator by x² gives (3 + 1/x²)/(1 − 5/x²) → 3 as x → ∞.

Using the substitution u = x² + 1, evaluate ∫ 2x(x² + 1)⁴ dx.

Answer: (x² + 1)⁵/5 + C

With u = x² + 1, du = 2x dx, so the integral becomes ∫ u⁴ du = u⁵/5 + C = (x² + 1)⁵/5 + C.

A particle moves with velocity v(t) = 6t − t². Its acceleration at t = 2 is:

Answer: 2

a(t) = v′(t) = 6 − 2t; at t = 2, a(2) = 6 − 4 = 2.