Calculus

Introduction to Calculus

Calculus is a section of mathematics dealing with continuous change. It encompasses several fundamental concepts: limits, derivatives, integrals, and infinite series. These ideas work together to create a powerful mathematical framework.

The core components of calculus include:
Limits - examining the behavior of functions as they approach specific values
Differential calculus - studying rates of change through derivatives
Integral calculus - analyzing accumulation and total change
Infinite series - representing functions as sums of infinite terms

Differential calculus allows us to find instantaneous rates of change and optimize functions, while integral calculus provides tools for calculating areas, volumes, and accumulated quantities. The connection between these two branches, established by the Fundamental Theorem of Calculus, creates a unified system for analyzing continuous change.

Applications of calculus extend throughout science, engineering, and economics. In physics, it models motion and energy; in engineering, it optimizes designs and processes; in economics, it analyzes rates of growth and market behavior. The subject's precise mathematical framework makes it essential for understanding and describing natural phenomena.

Calculus Formulas

Visit Calculus formulas page.

Definition of a Limit

\lim_{x \to a} f(x) = L

Limit Laws - Sum Rule

\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)

Limit Laws - Difference Rule

\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)

Limit Laws - Product Rule

\lim_{x \to a} [f(x) \cdot g(x)] = \left(\lim_{x \to a} f(x)\right) \cdot \left(\lim_{x \to a} g(x)\right)

Limit Laws - Quotient Rule

\lim_{x \to a} \left( \frac{f(x)}{g(x)} \right) = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, provided \lim_{x \to a} g(x) \ne 0

Limit Laws - Constant Multiple Rule

\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)

Special Limit of Sine over x

\lim_{x \to 0} \frac{\sin x}{x} = 1

Special Exponential Limit

\lim_{x \to 0} \frac{e^x - 1}{x} = 1

Limit of (1 + 1/n)^n

\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e

Limit of a Constant Function

\lim_{x \to a} c = c

Limit of Identity Function

\lim_{x \to a} x = a

Definition of the Derivative

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

Power Rule

\frac{d}{dx}[x^n] = n x^{n-1}

Constant Rule

\frac{d}{dx}[c] = 0

Constant Multiple Rule

\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)

Sum and Difference Rules

\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)

Product Rule

\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)

Quotient Rule

\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

Chain Rule

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Derivative of Sine Function

\frac{d}{dx}[\sin x] = \cos x

Derivative of Cosine Function

\frac{d}{dx}[\cos x] = -\sin x

Derivative of Tangent Function

\frac{d}{dx}[\tan x] = \sec^2 x

Derivative of Exponential Function

\frac{d}{dx}[e^{x}] = e^{x}

Derivative of a^x

\frac{d}{dx}[a^{x}] = a^{x} \ln a

Derivative of Natural Logarithm

\frac{d}{dx}[\ln x] = \frac{1}{x}

Derivative of Logarithm Base a

\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

Derivative of Inverse Sine

\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1 - x^2}}

Derivative of Inverse Cosine

\frac{d}{dx}[\arccos x] = \frac{-1}{\sqrt{1 - x^2}}

Derivative of Inverse Tangent

\frac{d}{dx}[\arctan x] = \frac{1}{1 + x^2}

Indefinite Integral (Antiderivative)

\int f(x) \, dx = F(x) + C

Definite Integral

\int_a^b f(x) \, dx = F(b) - F(a)

Basic Integration Rule - Power Rule

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, where n \ne -1

Basic Integration Rule - Constant Multiple

\int c \cdot f(x) \, dx = c \cdot \int f(x) \, dx

Basic Integration Rule - Sum and Difference

\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx

Integral of Sine Function

\int \sin x \, dx = -\cos x + C

Integral of Cosine Function

\int \cos x \, dx = \sin x + C

Integral of Exponential Function

\int e^{x} \, dx = e^{x} + C

Integral of a^x

\int a^{x} \, dx = \frac{a^{x}}{\ln a} + C

Integral of Reciprocal Function

\int \frac{1}{x} \, dx = \ln |x| + C

Fundamental Theorem of Calculus - Part 1

\int_a^b f(x) \, dx = F(b) - F(a)

Fundamental Theorem of Calculus - Part 2

\frac{d}{dx} \left( \int_a^x f(t) \, dt \right) = f(x)

Integration by Substitution (Reverse Chain Rule)

If u = g(x), then \int f(g(x)) g'(x) \, dx = \int f(u) \, du

Integration by Parts

\int u \, dv = u v - \int v \, du

Partial Fractions Decomposition

Decompose \frac{P(x)}{Q(x)} into simpler fractions before integrating

Trigonometric Integrals

Use identities to simplify \int \sin^n x \cos^m x \, dx

Improper Integral

\int_a^{\infty} f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx

Average Value of a Function

f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x) \, dx