# MTHT 530 Analysis for Teachers II

## Key Concepts

You should be comfortable with all of the key definitions.

You should be able to state and apply all of the key results.

You should be able to sketch the proof of the key results marked with (*).

### Review

** Key Concepts** (Chapters 5,6,7,8) - least upper bounds
- Completeness of
**R** - limits
- continuity

**Key Theorems** - Nested Interval Theorem (*)
- Algebraic Properties of Limits
- Intermediate Value Theorem (*)
- Bounding Theorem (*)
- Extreme Value Theorem (*)

### Sequences

** Key Concepts** (Chapter 22) - sequences, covergence and divergence
- monotone sequences

**Key Theorems** - convergent sequences are bounded (*)
- Monotone Covergence Theorem (*)
- f is continuous at a if and only if f(a_n)->f(a) for every sequence a_n->a (*)
- Bolzano-Weierstrass Theorem (*)

### The Derivative

**Key Concepts** (Chapters 9,10,11,12) - differentialbility
- derivatives
- inverse functions

**Key Results** - algebraic rules for derivatives
- chain rule(*)
- differentiable functions are continuous(*), but their derivatives need not be.
- Interior Extemum Theorem(*)
- Darboux's Theorem
- Rolle's Theorem (*)
- Mean Value Theorem(*)
- Second Derivative Test (*)
- Cauchy's Mean Value Theorem
- L'Hospital's Rule
- Tangent approximation
- Newton's Method
- Continuity and differentiability of inverse functions (*)

### Uniform Continuity

**Key Concepts** (Chapter 8 appendix) **Key Results** - continuous functions on [a,b] are uniformly continuous

### Integration

**Key Concepts** (Chapters 13,14,18)
- Upper and Lower Riemann sums
- Integrable functions
- integral
- uniform continuity
- natural logarithm and exponential function

**Key Results** - continuous functions are integrable (*)
- properties of integrals
- Fundamental Theorem of Calculus(*)
- Properties of Logarithms and Exponentials (*)

### Series and Sequences of Functions

**Key Concepts** (Chapters 23,24)
- Cauchy Sequences
- convergent series
- absolutely convergent series
- pointwise convergence of sequenences of functions
- convergent series of fuctions
- power series

**Key Results**
Convergence of the geometric series (*)
Divergence of the harmonic series
A sequence converges if and only if it is Cauchy (*)
If a series converges, the limit of the terms is 0 (*)
Comparision Test (*)
Ratio Test
Integral Test
Alternating Series Test
Absolutely convergent series are convergent
Continuity of limits of uniformly convergent series of functions (*)
Integrability and differentiability of of limits of uniformly convergent series
Weierstrass M-test
If the series sum a_nR^n converges, then the power series sum a_nx^n converges
uniformly on [-a,a] for all 0 continuity, differentiability and integrability of power series.

Last Updated 3/29/06