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## Math 142A Fall 2017

Terminology and Theorems

#### Updated 10/3/17

**Note:** The following list is a minimal collection of the important terms and theorems for this course.
You simply *must* be familiar with them and how they are used: in the case of theorems, you should
be familiar with how they are proved. This basic knowledge is analogous to the vocabulary of a language: it
is impossible to speak the language without it.

**Chapter 1**
- inductive set
- principle of mathematical induction
- bounded above/below, bounded
- supremum
- infimum
- completeness axiom
- distribution of the integers
- distribution of the rational numbers
- archimedean property
- dense
- the set of rational numbers is dense
- the set of irrational numbers is dense
- triangle inequality
- difference of powers formula
- geometric sum formula
- binomial formula

**Chapter 2**
- sequence
- converge
- comparison lemma
- sum, product and quotient properties
- linearity and polynomial properties
- bounded sequence
- sequentially dense set
- closed set
- monotonically increasing/decreasing, monotone
- monotone convergence theorem
- nested interval theorem
- subsequence
- every sequence has a monotone subsequence
- every bounded sequence has a convergent subsequence
- sequentially compact
- sequential compactness theorem

**Chapter 3**
- continuous at
*x*_{0}
- continuous
- composition
*f o g*
- image
- maximum value
- maximizer
- minimum value
- minimizer
- extreme value theorem
- intermediate value theorem
- bisection method
- convex set
- uniformly continuous
*ε-δ* criterion at a point
*ε-δ* criterion on the domain
- monotonically increasing/decreasing
- monotone
- strictly increasing/decreasing
- strictly monotone
- one-to-one
- inverse
- distinct from
- limit point
- limit (of a function)

**Chapter 4**
- tangent line
- differentiable
- derivative
- differentiation rules
- sum rule
- product rule
- quotient rule

- derivative of inverse function
- derivative of the composition: chain rule
- Rolle's theorem
- mean value theorem
- identity criterion
- criterion for strict monotonicity
- local maximizer, local minimizer
- Cauchy mean value theorem
- Leibnitz notation

**Links:**
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