Introductory calculus

Builds on

Required prerequisites, all the way down: 24

You will learn to

• Estimate limits from tables and graphs, and compute simple limits algebraically.
• Differentiate polynomials and use derivatives to find tangent lines, rates of change and extrema.
• Approximate areas under curves with rectangles, and relate integrals to antiderivatives through the fundamental theorem.

Definition

• The derivative of f at a is f′(a) = lim_{h→0} (f(a + h) − f(a))/h, when this limit exists.
• The definite integral ∫_a^b f(x) dx is the limit of Riemann sums: a signed area.
• An antiderivative of f is a function F with F′ = f.

Theorem

• (xⁿ)′ = nxⁿ⁻¹ for every positive integer n.
• At an interior maximum or minimum of a differentiable function, f′ = 0.
• Fundamental theorem of calculus: if f is continuous on [a, b] and F′ = f there, then ∫_a^b f(x) dx = F(b) − F(a).

Common misconceptions

• Thinking f′(a) = 0 guarantees a maximum or minimum; x³ at 0 is a counterexample.
• Believing the limit at a point is always the value of the function there.
• Thinking an integral always gives a positive area; integrals are signed.

Leads to

Optional discoveries

• Real analysis Making limits precise took until the nineteenth century; the ε–δ definition is due largely to Cauchy and Weierstrass.
• Numerical methods When no antiderivative can be written in closed form, integrals are computed numerically, for example by the trapezoidal rule or Simpson's rule.
• ← Estimation and rounding
• ← Graphs of change
• ← Quadratic equations and functions

Learning pathways

• Conventional K–12 progression
• Accelerated secondary progression
• Applied mathematics

Further reading

• Silvanus P. Thompson, Calculus Made Easy (1910)