Calculus

Calculus is the mathematics of change. It has two main ideas. The derivative tells us how fast something changes. The integral tells us how small changes add up to a total. Both ideas are built on limits.

These lessons cover Calculus I. Start with Chapter 1.

Limits and continuity

What a limit is; one-sided limits; computing limits with the limit laws, direct substitution, and algebra; infinite limits and limits at infinity; the Squeeze Theorem; and continuity and the Intermediate Value Theorem.

The derivative

Tangent lines and rates of change; the derivative at a point and what it means; and the derivative as a function, its notation, its graph, and where it does not exist.

Differentiation rules

The power, product, quotient, and chain rules; derivatives of trigonometric, exponential, and logarithmic functions; implicit differentiation; and higher derivatives.

Applications of derivatives

Related rates; linear approximation; maximum and minimum values; the Mean Value Theorem; the shape of a graph and curve sketching; optimization; and L'Hôpital's rule.

Integrals

Antiderivatives; areas and Riemann sums; the definite integral; the Fundamental Theorem of Calculus; indefinite integrals and net change; and the substitution rule.

Applications of integrals

Areas between curves; volumes by slicing, with disks and washers; volumes by cylindrical shells; and the average value of a function.