Introduction to Calculus
39 concepts in Math
Calculus is the mathematics of change and accumulation. It gives students precise tools for answering questions like "How fast is something changing right now?" and "How much total change has occurred over time?" The two central ideas — the derivative and the integral — turn out to be deeply connected through the Fundamental Theorem of Calculus. Derivatives measure instantaneous rates of change, which is essential for modeling velocity, growth rates, and optimization problems. Integrals measure accumulated quantities such as area, volume, and total distance. Although calculus has a reputation for difficulty, it builds naturally on algebra and geometry concepts students already know. Understanding limits, the conceptual gateway to calculus, is the key first step.
Suggested learning path: Begin with an intuitive understanding of limits and continuity, then move to derivatives and their applications before exploring integrals and the Fundamental Theorem of Calculus.
Limit
The value a function gets closer and closer to as the input approaches a specific target value, without necessarily reaching it.
Derivative
The instantaneous rate of change of a function at a single point, defined as the limit of the slope of secant lines.
Differentiation Rules
A set of standard formulas for finding derivatives of common function types without using the limit definition each time.
Chain Rule
The derivative of a composite function $f(g(x))$ equals $f'(g(x)) \cdot g'(x)$: the derivative of the outer function evaluated at the inner, times the derivative of the inner.
Integral
The reverse operation of differentiation; it also computes the exact area under a curve between two points.
Definite Integral
An integral evaluated between specific lower and upper bounds, yielding a single numerical value rather than a function.
Fundamental Theorem of Calculus
The theorem stating that differentiation and integration are inverse operations, linking antiderivatives to definite integrals.
Optimization
The process of using derivatives to systematically find maximum or minimum values of a function over a domain.
Rate of Change
A measure of how quickly one quantity changes with respect to another; the ratio of the change in output to the change in input.
Tangent Line
A line that touches a curve at exactly one point and has the same slope as the curve there.
Infinity
A concept representing a quantity that grows without bound — infinity is not a real number but a description of unbounded behavior.
Sequence
An ordered list of numbers generated by a rule, where each number has a specific position (first, second, third, ...).
Arithmetic Sequence
A sequence where each term is obtained from the previous by adding a fixed constant called the common difference $d$.
Geometric Sequence
A sequence where each term is obtained from the previous by multiplying by a fixed non-zero constant called the common ratio $r$.
Series
The result of adding all the terms of a sequence together, either finitely or infinitely many terms.
Riemann Sums
A method of approximating the definite integral $\int_a^b f(x)\,dx$ by dividing the interval $[a, b]$ into subintervals and summing the areas of rectangles (or trapezoids) whose heights are determined by the function.
u-Substitution
An integration technique where you substitute $u = g(x)$ and $du = g'(x)\,dx$ to transform a complicated integral into a simpler one. It is the reverse of the chain rule for differentiation.
Integration by Parts
An integration technique based on the product rule: $\int u\,dv = uv - \int v\,du$. Used when the integrand is a product of two functions.
Area Between Curves
The area of the region enclosed between two functions $f(x)$ and $g(x)$ from $x = a$ to $x = b$, computed as $A = \int_a^b |f(x) - g(x)|\,dx$.
Volumes of Revolution
Finding the volume of a three-dimensional solid formed by rotating a two-dimensional region around an axis. The disc/washer method uses circular cross-sections perpendicular to the axis; the shell method uses cylindrical shells parallel to the axis.
Recursive vs Explicit Formulas
Two ways to define a sequence: a recursive formula defines each term using the previous term(s), while an explicit (closed-form) formula gives the $n$th term directly as a function of $n$.
Sigma Notation
A compact way to write the sum of many terms using the Greek letter $\Sigma$ (sigma). $\sum_{i=m}^{n} a_i$ means add up $a_i$ for every integer $i$ from $m$ to $n$.
Infinite Geometric Series
The sum of all terms of a geometric sequence with common ratio $|r| < 1$. The infinite sum converges to $\frac{a}{1-r}$, where $a$ is the first term.
Convergence and Divergence
A series converges if the sequence of its partial sums approaches a finite limit. A series diverges if the partial sums grow without bound or oscillate without settling.
Types of Continuity and Discontinuity
A function is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$. Discontinuities are classified as removable (limit exists but doesn't equal $f(a)$), jump (left and right limits exist but differ), or infinite (function blows up to $\pm\infty$).
Squeeze Theorem
If $g(x) \leq f(x) \leq h(x)$ near $x = a$, and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.
Intermediate Value Theorem
If $f$ is continuous on the closed interval $[a, b]$ and $N$ is any value between $f(a)$ and $f(b)$, then there exists at least one $c$ in $(a, b)$ such that $f(c) = N$.
Implicit Differentiation
A technique for finding $\frac{dy}{dx}$ when $y$ is defined implicitly by an equation $F(x, y) = 0$ rather than explicitly as $y = f(x)$. Differentiate both sides with respect to $x$, treating $y$ as a function of $x$, then solve for $\frac{dy}{dx}$.
Related Rates
Problems where two or more quantities change with time and are related by an equation. Differentiate the equation with respect to time $t$ and use known rates to find an unknown rate.
L'Hopital's Rule
If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is an indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$ provided the right-hand limit exists (or is $\pm\infty$).
Mean Value Theorem
If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists at least one point $c$ in $(a, b)$ where $$f'(c) = \frac{f(b) - f(a)}{b - a}$$
Curve Sketching
Using the first and second derivatives to determine a function's behavior: intervals of increase/decrease, local maxima/minima, concavity (up/down), and inflection points, then combining this information to sketch an accurate graph.
Partial Fraction Decomposition
A technique for breaking a rational expression $\frac{P(x)}{Q(x)}$ into a sum of simpler fractions whose denominators are the factors of $Q(x)$. This makes integration of rational functions possible.
Improper Integrals
Integrals where the interval of integration is infinite (Type I: $\int_a^{\infty} f(x)\,dx$) or the integrand has an infinite discontinuity on the interval (Type II: $\int_a^b f(x)\,dx$ where $f$ blows up at some point in $[a, b]$). Evaluated as limits of proper integrals.
Introduction to Differential Equations
An equation that contains an unknown function and one or more of its derivatives. Solving a DE means finding the function(s) that satisfy the equation.
Slope Fields
A graphical representation of a first-order DE $\frac{dy}{dx} = f(x, y)$. At each point $(x, y)$ in the plane, draw a short line segment with slope $f(x, y)$. The resulting pattern of segments shows the direction solutions must follow.
Separation of Variables
A method for solving first-order DEs of the form $\frac{dy}{dx} = f(x) \cdot g(y)$: rearrange to $\frac{dy}{g(y)} = f(x)\,dx$, then integrate both sides.
Taylor Series
A representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$ When $a = 0$, it's called a Maclaurin series.
Power Series
An infinite series of the form $$\sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots$$ where $c$ is the center and $a_n$ are the coefficients. A power series defines a function of $x$ wherever it converges.