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)
• uniform continuity
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