Calculus

The mathematics of change and accumulation—calculus is essential for modeling dynamic systems, optimizing processes, and understanding continuous behavior. From rates to areas, it’s the engine behind much of science, engineering, and data analysis. (Stewart 2012) 🔍

Learning Objectives

Learning objectives of the Calculus section.

Summary Table

Summary of the Single-Variable section.

Concept Description Notation
Derivative Instantaneous rate of change; slope of the tangent line f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
Constant Rule Derivative of a constant function \frac{d}{dx}[c] = 0
Power Rule Derivative of a power function \frac{d}{dx}[x^n] = nx^{n-1}
Sum/Difference Rule Derivative of a sum or difference \frac{d}{dx}[g(x) \pm h(x)] = g'(x) \pm h'(x)
Product Rule Derivative of a product \frac{d}{dx}[g(x)h(x)] = g'(x)h(x) + g(x)h'(x)
Quotient Rule Derivative of a quotient \frac{d}{dx}\left[\frac{g(x)}{h(x)}\right] = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}
Chain Rule Derivative of a composite function \frac{d}{dx}[g(h(x))] = g'(h(x)) \cdot h'(x)
Definite Integral Net area under a curve from a to b \int_a^b f(x)\, dx
Indefinite Integral General antiderivative \int f(x)\, dx = F(x) + C
Constant Rule (Int.) Integral of a constant \int c\,dx = cx + C
Power Rule (Int.) Integral of a power function (when n \neq -1) \int x^n\,dx = \frac{x^{n+1}}{n+1} + C
Sum/Difference Rule (Int.) Integral of a sum or difference \int [g(x) \pm h(x)]\,dx = \int g(x)\,dx \pm \int h(x)\,dx
Exponential Rule (base e) Integral of e^x \int e^x\,dx = e^x + C
Exponential Rule (base a) Integral of a^x \int a^x\,dx = \frac{a^x}{\ln a} + C
Logarithmic Rule Integral of reciprocal \int \frac{1}{x}\,dx = \ln|x| + C
Chain Rule (u-sub) Substitution technique \int f(g(x))g'(x)\,dx = \int f(u)\,du,\ u = g(x)
Integration by Parts Product of functions technique \int u(x)v'(x)\,dx = u(x)v(x) - \int u'(x)v(x)\,dx
Fundamental Theorem Derivative of accumulation function F(x) = \int_a^x f(t)\,dt \Rightarrow F'(x) = f(x)
FTC Evaluation Total accumulation from a to b \int_a^b f(x)\,dx = F(b) - F(a)
Optimization Find extrema using derivatives f'(x) = 0, f''(x) \gtrless 0
First Derivative Test Sign change test for local extrema f'(x): + \to - (max), - \to + (min), no change = none
Second Derivative Test Concavity test for local extrema f''(x) > 0: min, f''(x) < 0: max, f''(x) = 0: inconclusive