Calculus of one variable

前置知识

34 项必需的前置知识（一直追溯到底）

你将学会

• Differentiate and integrate elementary functions using the chain rule, substitution and integration by parts.
• State and apply the mean value theorem and L'Hôpital's rule.
• Test series for convergence and compute Taylor polynomials with error bounds.

定义

• f is continuous at a if for every ε > 0 there is δ > 0 such that |f(x) − f(a)| < ε whenever |x − a| < δ.
• The Taylor series of f at a is Σ f⁽ⁿ⁾(a)(x − a)ⁿ/n!.

定理

• Mean value theorem: if f is continuous on [a, b] and differentiable on (a, b), then f′(c) = (f(b) − f(a))/(b − a) for some c in (a, b).
• Taylor's theorem with remainder.
• eˣ = Σ xⁿ/n! for every real x.
• An absolutely convergent series converges.

常见误解

• Assuming every smooth function equals its Taylor series; f(x) = e^(−1/x²) with f(0) = 0 is a counterexample at 0.
• Applying L'Hôpital's rule to limits that are not indeterminate forms.

通向

可选发现

• Complex analysis Why does the Taylor series of 1/(1 + x²) at 0 converge only for |x| < 1, although the function is smooth on all of ℝ? Its complex poles at ±i explain it.

学习路径

• Accelerated secondary progression
• Undergraduate mathematics major
• Applied mathematics

延伸阅读

• Michael Spivak, Calculus (1967)
• Tom M. Apostol, Calculus