Single-Variable Calculus Summary

GRE Math Subject · Axiom Academy

SUMMARY Single-Variable Calculus Comprehensive review of all single-variable calculus topics for the GRE Math Subject Test -- approximately 40% of the exam Direct substitution -- always try first. Factoring/cancellation -- for 0/0 forms with polynomials. Rationalization -- multiply by conjugate for radicals. L'Hopital's Rule: For 0/0 or / forms: f is continuous at a if: (1) f(a) is defined, (2) exists, (3) . Intermediate Value Theorem: If f is continuous on [a, b] and N is between f(a) and f(b), then there exists c in (a, b) with f(c) = N. Extreme Value Theorem: If f is continuous on [a, b], then f attains its absolute maximum and minimum on [a, b]. Implicit and Logarithmic Differentiation Implicit: Differentiate both sides with respect to x, treating y as y(x). Collect dy/dx terms. Logarithmic: For : take ln of both sides, differentiate, solve for y'. IV. Derivatives -- Applications f'(x) > 0: f is increasing. f'(x) < 0: f is decreasing. f''(x) > 0: f is concave up. f''(x) < 0: f is concave down. Critical points: where f'(x) = 0 or f'(x) DNE. Inflection points: where f'' changes sign. Mean Value Theorem: If f is continuous on [a, b] and differentiable on (a, b), there exists c in (a, b) with: Rolle's Theorem: Special case where f(a) = f(b), so f'(c) = 0 for some c in (a, b). 1. Find critical points: set f'(x) = 0. 2. Second derivative test: f''(c) > 0 means local min, f''(c) < 0 means local max. 3. For absolute extrema on [a, b]: compare f at critical points and endpoints.

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