Calculus

College/University

Definition

Is the mathamatical study of change. Calculus can be split into differential calculus and integral calculus. Differential calculus studies derivatives, which is the rate of change of functions. Integrals are the inverse of derivatives, and lets you find the area and volume of things.

Worked examples

\(\frac{d}{dx}(x^2) = 2x\)

Differential calculus finds the derivative — the rate at which \(x^2\) changes with respect to \(x\).

\(\int 2x\,dx = x^2 + C\)

Integral calculus reverses differentiation and finds area under curves; \(C\) is the constant of integration.

Common mistakes

\(\int f'(x)\,dx = f(x)\) → \(\int f'(x)\,dx = f(x) + C\) Always include the constant of integration \(C\) when computing indefinite integrals.
The derivative of \(x^3\) is \(3x^2\), so the integral of \(3x^2\) is \(x^3\). → \(\int 3x^2\,dx = x^3 + C\) Integrals produce a family of functions; never forget \(+C\).
Calculus is only about finding slopes of lines. → Calculus studies rates of change (derivatives) and accumulation (integrals) of any function. It applies to curves, motion, growth, area, volume, and much more — not just straight lines.

Where you'll use it next

Calculus is the foundation for physics (motion, energy), engineering, economics (marginal cost), biology (population models), and advanced math courses like multivariable calculus, differential equations, and real analysis.

Found in 1 StudyPug lesson

Unlock the Power of Calculus: Understanding Limits and Derivatives

Calculus 1

Dive into the world of calculus with our expert-led lessons on limits and derivatives. Learn to analyze function behavior, solve real-world problems, and build a strong foundation for advanced mathematics.

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