What is a limit? — A limit describes the value a function approaches as the input approaches a given value. Written as lim[x→a] f(x) = L. The function does not have to actually reach L — just approach it.

What is the definition of a derivative? — f'(x) = lim[h→0] (f(x+h) − f(x)) / h — the instantaneous rate of change of f(x). It gives the slope of the tangent line at any point on the curve.

What is the power rule for differentiation? — If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. Multiply by the exponent, then reduce the exponent by 1. Example: d/dx(x⁵) = 5x⁴.

What is the constant rule for differentiation? — The derivative of any constant is 0. d/dx(c) = 0. Example: d/dx(7) = 0. Constants have no rate of change.

What is the sum/difference rule for differentiation? — d/dx[f(x) ± g(x)] = f'(x) ± g'(x). Differentiate each term separately and add or subtract.

What is the product rule for differentiation? — If h(x) = f(x)·g(x), then h'(x) = f'(x)g(x) + f(x)g'(x). Memory: "first times derivative of second plus second times derivative of first."

What is the quotient rule for differentiation? — If h(x) = f(x)/g(x), then h'(x) = [f'(x)g(x) − f(x)g'(x)] / [g(x)]². Memory: "low d-high minus high d-low over low squared."

What is the chain rule for differentiation? — If h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x). Differentiate the outer function (keeping the inner unchanged) then multiply by the derivative of the inner function.

What is the derivative of sin(x), cos(x), tan(x)? — d/dx(sin x) = cos x. d/dx(cos x) = −sin x. d/dx(tan x) = sec²x.

What is the derivative of eˣ and ln(x)? — d/dx(eˣ) = eˣ — the only function that is its own derivative. d/dx(ln x) = 1/x.

What is the derivative of aˣ and logₐ(x)? — d/dx(aˣ) = aˣ ln a. d/dx(logₐ x) = 1/(x ln a). These are the generalized forms of d/dx(eˣ) and d/dx(ln x).

What does the first derivative tell you about a function? — f'(x) > 0: function is increasing. f'(x) < 0: function is decreasing. f'(x) = 0: stationary point (possible maximum, minimum, or inflection).

What does the second derivative tell you? — f''(x) > 0: function is concave up (cup shape) — local minimum. f''(x) < 0: function is concave down (cap shape) — local maximum. f''(x) = 0: possible inflection point.

What is the second derivative test for maxima/minima? — At a stationary point where f'(x)=0: f''(x) < 0 → local maximum. f''(x) > 0 → local minimum. f''(x) = 0 → inconclusive — use another method.

What is an indefinite integral? — The reverse of differentiation: ∫f(x)dx = F(x) + C where F'(x) = f(x) and C is the constant of integration. Finding all antiderivatives of f(x).

What is the power rule for integration? — ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (for n ≠ −1). Add 1 to the exponent, then divide by the new exponent. Example: ∫x³ dx = x⁴/4 + C.

What is the Fundamental Theorem of Calculus? — ∫[a to b] f(x) dx = F(b) − F(a) where F is any antiderivative of f. It connects differentiation and integration — they are inverse operations.

What is a definite integral and what does it represent? — ∫[a to b] f(x) dx — the signed area between the curve y=f(x) and the x-axis from x=a to x=b. Area above x-axis is positive, area below is negative.

What is integration by substitution? — Let u = g(x), then du = g'(x)dx. Rewrite the integral in terms of u, integrate, then substitute back. Used to simplify composite functions. Analogous to the chain rule in reverse.

What is the L'Hôpital's Rule? — If lim[x→a] f(x)/g(x) gives 0/0 or ∞/∞, then lim[x→a] f(x)/g(x) = lim[x→a] f'(x)/g'(x) — differentiate numerator and denominator separately, then take the limit.

What is the area between two curves? — A = ∫[a to b] [f(x) − g(x)] dx where f(x) ≥ g(x) on [a,b]. The upper curve minus the lower curve, integrated over the interval.

What does continuity of a function require? — A function is continuous at x=a if: 1. f(a) is defined. 2. lim[x→a] f(x) exists. 3. lim[x→a] f(x) = f(a). All three must be true — no holes, jumps, or asymptotes.

What is implicit differentiation? — Differentiate both sides of an equation with respect to x, treating y as a function of x. Whenever you differentiate a term with y, multiply by dy/dx. Then solve for dy/dx.

What is the derivative of inverse trig functions? — d/dx(arcsin x) = 1/√(1−x²). d/dx(arccos x) = −1/√(1−x²). d/dx(arctan x) = 1/(1+x²). These appear frequently in integration problems.