Calculus
The Calculus domain examines limits, continuity, derivatives, and integrals, emphasizing conceptual meaning and standard applications.
Limits and continuity
A limit exists when the left- and right-hand limits agree. L'Hôpital's rule applies only to the indeterminate forms 0/0 or ∞/∞. A function is continuous at x = a when f(a) exists, the limit exists, and the two are equal.
Derivatives
The derivative is the limit of the difference quotient and measures instantaneous rate of change and tangent slope. Master the power, product, quotient, and chain rules; the chain rule is the most frequently penalized. Critical points occur where f′ = 0 or is undefined. The second derivative identifies concavity and classifies extrema.
Integrals
A definite integral is a signed area, computed by the Fundamental Theorem: ∫ₐᵇ f(x) dx = F(b) − F(a) for an antiderivative F. Area between curves is ∫ (top − bottom).
Worked example
Differentiate f(x) = x² sin x. The product rule gives f′(x) = 2x sin x + x² cos x. At x = 0 this is 0 + 0 = 0, consistent with a horizontal tangent there. For an integral check, ∫₀¹ (3x² + 2x) dx = [x³ + x²]₀¹ = 2.
Traps
Omitting the inner derivative in the chain rule; dropping the constant of integration on indefinite integrals; invoking L'Hôpital on a determinate form; reversing the sign when the upper curve and lower curve swap.