**Links:**
Home
Calendar
Syllabus
*Terminology*
Homework

## Math 142B Spring 2024

Terminology and Theorems

#### Updated 3/21/24

**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'HÃ´pital's rule

**Section 5.31**
- Taylor series for
*f* about *c*
*n*^{th} remainder, *R*_{n}(x)
- Taylor's theorem
- Taylor's theorem (Cauchy integral remainder)
- Cauchy's form of
*R*_{n}
- 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,
*f*_{n} converges pointwise
- uniform convergence,
*f*_{n} 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

**Links:**
Home
Calendar
Syllabus
*Terminology*
Homework