Introduction to Calculus
39 concepts ยท ordered by prerequisite depth
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 โ the derivative and the integral โ 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 order: 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.
Start here
Limit
The value a function gets closer and closer to as the input approaches a specific target value, without necessarily reaching it.
Open lesson
Derivative
The instantaneous rate of change of a function at a single point, defined as the limit of the slope of secant lines.
Open lesson
Integral
The reverse operation of differentiation; it also computes the exact area under a curve between two points.
Open lesson
Continue from here ยท 36 concepts
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.
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 $r$.
Infinity
A concept representing a quantity that grows without bound โ infinity is not a real number but a description of unbounded behavior.
Series
The result of adding all the terms of a sequence together, either finitely or infinitely many terms.
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 $\lim_{x \to a} f(x) = L$.
Types of Continuity and Discontinuity
Continuity types classify how a function can fail to be continuous at a point. A removable discontinuity (hole) occurs when the limit exists but doesn't equal f(a). A jump discontinuity occurs when left and right limits differ. An infinite discontinuity occurs when the function approaches ยฑโ.
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.
Differentiation Rules
A set of standard formulas for finding derivatives of common function types without using the limit definition each time.
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.
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 $f(c) = N$.
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 $\pm\infty$).
Optimization
The process of using derivatives to systematically find maximum or minimum values of a function over a domain.
Recursive vs Explicit Formulas
Two ways to define a sequence: recursive uses the previous term(s), explicit gives the $n$th term directly as a function of $n$.
Sigma Notation
Sigma notation uses the Greek letter ฮฃ to express the sum of many terms compactly. The expression $\sum_{i=1}^{n} a_i$ means 'add up $a_i$ for every integer $i$ from 1 to $n$.' For example, $\sum_{i=1}^{4} i^2 = 1 + 4 + 9 + 16 = 30$.
Tangent Line
A line that touches a curve at exactly one point and has the same slope as the curve there.
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.
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.
Definite Integral
An integral evaluated between specific bounds $a$ and $b$, yielding a single number: the signed area under the curve.
Fundamental Theorem of Calculus
The theorem stating that differentiation and integration are inverse operations, linking antiderivatives to definite integrals.
Implicit Differentiation
Finding $\frac{dy}{dx}$ when $y$ is defined implicitly by an equation like $F(x, y) = 0$, by differentiating both sides and solving for $\frac{dy}{dx}$.
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.
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.
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 $$f'(c) = \frac{f(b) - f(a)}{b - a}$$
Partial Fraction Decomposition
Breaking a rational expression into a sum of simpler fractions whose denominators are the factors of the original denominator.
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.
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 $A = \int_a^b |f(x) - g(x)|\,dx$.
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.
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.
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.
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.
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.
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.
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.