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Math 142B Winter 2023
Terminology and Theorems
Updated 2/8/23
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.
- Section 5.29
- Rolle's theorem
- mean value theorem
- increasing, strictly increasing
- decreasing, strictly decreasing
- intermediate value theorem for derivatives
- derivative of inverse function
- Section 5.30
- generalized mean value theorem
- L'Hôpital's rule
- Section 5.31
- Taylor series for f about c
- nth remainder, Rn(x)
- Taylor's theorem
- Taylor's theorem (Cauchy integral remainder)
- Cauchy's form of Rn
- Newton's method
- secant method
- Section 6.32
- partition of [a,b]
- upper Darboux sum U(f,P) of f with respect to P
- lower Darboux sum L(f,P) of f with respect to P
- upper Darboux integral U(f) of f over [a,b]
- lower Darboux integral L(f) of f over [a,b]
- integrable
- Riemann (Darboux) integral
- mesh of a partition P, mesh(P)
- Section 6.33
- monotonic functions are integrable
- continuous functions are integrable
- linearity (constant multiple, additivity) (33.3)
- absolute integrability (33.5)
- piecwise integrability (33.6, 33.8)
- intermediate value theorem for integrals (mean value theorem for integrals)
- Section 6.34
- fundamental theorem of Calculus I
- integration by parts
- fundamental theorem of Calculus II
- change of variable
- Section 4.24
- pointwise convergence, fn converges pointwise
- uniform convergence, fn converges uniformly
- the uniform limit of continuous functions is continuous
- Section 4.25
- uniformly Cauchy on a set S ⊆ ℝ
- Weierstrass M-test
- Section 4.23
- rules of convergence for power series (23.1)
- Section 4.26
- differentiation and integration of power series
- Abel's theorem
- Section 4.27
- Bernstein polynomial for the function f
- Weierstrass's approximation theorem
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