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Session Length
~11 min
Adaptive Checks
10 questions
Transfer Probes
5
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches: differential calculus, which concerns instantaneous rates of change and slopes of curves, and integral calculus, which concerns accumulation of quantities and the areas under and between curves. Together, these branches are unified by the Fundamental Theorem of Calculus, one of the most important results in all of mathematics.
Developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, calculus is one of the most powerful mathematical tools ever created. It provides the language and framework for physics, engineering, economics, computer science, and virtually every quantitative discipline. Key concepts include limits, continuity, differentiation rules, integration techniques, infinite series, and multivariable extensions that generalize these ideas to higher dimensions.
From modeling planetary orbits to optimizing machine learning algorithms, calculus is the mathematical backbone of modern science and technology. Understanding derivatives and integrals unlocks the ability to analyze change, optimize systems, and solve problems that are impossible with algebra alone. Differential equations, which arise naturally from calculus, describe phenomena ranging from population growth and radioactive decay to fluid dynamics and electrical circuits.
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The value that a function approaches as the input approaches some value. Limits are the foundation of calculus, used to define both derivatives and integrals.
Example: As $x$ approaches 2, the function $f(x) = x^2$ approaches 4. We write: $\lim_{x \to 2} x^2 = 4$.
The derivative measures the instantaneous rate of change of a function. Geometrically, it gives the slope of the tangent line to a curve at any point.
Example: The derivative of position with respect to time gives velocity. If $s(t) = t^2$, then $v(t) = s'(t) = 2t$.
Integration is the reverse process of differentiation. Definite integrals compute the signed area under a curve, while indefinite integrals find antiderivatives.
Example: The integral of velocity gives displacement. $\int 2t\,dt = t^2 + C$.
Links differentiation and integration as inverse processes. Part 1: integration can be reversed by differentiation. Part 2: definite integrals can be computed using antiderivatives.
Example: $\int_a^b f(x)\,dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$.
A formula for computing the derivative of a composition of functions. If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.
Example: If $y = \sin(x^2)$, then $\frac{dy}{dx} = \cos(x^2) \cdot 2x$.
The derivative of $x^n$ is $nx^{n-1}$. This is the most frequently used differentiation rule.
Example: $\frac{d}{dx}(x^3) = 3x^2$, $\frac{d}{dx}(x^{-1}) = -x^{-2}$.
A technique for evaluating integrals of products. Based on the product rule in reverse: $\int u\,dv = uv - \int v\,du$.
Example: $\int x \cdot e^x\,dx$: let $u = x$, $dv = e^x\,dx$. Then $uv - \int v\,du = xe^x - e^x + C$.
An infinite sum of terms calculated from the derivatives of a function at a single point. Allows approximation of complex functions with polynomials.
Example: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$ (Taylor series centered at $x = 0$).
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