Calculus Flashcards

The volume of the solid formed by rotating y = √x about the x-axis from x = 0 to x = 4 is:

Answer: 8π

V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π[x²/2]₀⁴ = 8π.

Which of the following is an antiderivative of 1/(1 + x²)?

Answer: arctan(x)

d/dx[arctan(x)] = 1/(1 + x²), so arctan(x) + C is the antiderivative.

Using implicit differentiation, find dy/dx if x² + y² = 25.

Answer: −x/y

Differentiating: 2x + 2y·(dy/dx) = 0, so dy/dx = −x/y.

lim_{h→0} [f(x+h) − f(x)]/h defines:

Answer: The derivative f′(x)

This is the limit definition of the derivative f′(x) at a point x.

f(x) = x⁴ − 8x² + 3 has local minima at:

Answer: x = 2 and x = −2

f′(x) = 4x(x² − 4) = 0 at x = 0, ±2; f″(±2) = 32 > 0 confirms minima at x = ±2.

The product rule for differentiating u(x)·v(x) gives:

Answer: u′·v + u·v′

The product rule states d/dx[u·v] = u′·v + u·v′.

∫ (2x + 3)⁵ dx = ?

Answer: (2x + 3)⁶/12 + C

With u = 2x + 3, du = 2 dx; ∫ u⁵ du/2 = u⁶/12 + C = (2x + 3)⁶/12 + C.