Analysis II – Spring Term, 2012: Calendar of lectures and topics
Chapter 1: Continuity
• Lecture 1: Introduction
• Lecture 2: Definition and examples
• Lecture 3: Continuity and sequential continuity
• Lecture 4: Algebra of continuous functions — sum, product, composite of continuous functions are continuous; quotient of continuous functions is continuous where the denominator
is non-zero. But cannot yet say anything about power series.
• Lecture 5: Continuity of special functions: trigonometric functions, exponential function.
Intermediate value theorem.
• Lecture 6: Intermediate value theorem continued; applications to solution of equations, fixed
point theorem. (Statement of Brouwer’s Fixed Point Theorem)
• Lecture 7: A continuous function on a closed bounded interval is bounded and attains its
bounds — so the image of a closed bounded interval under a continuous real-valued function
is a closed bounded interval. Examples to show boundedness, closedness and continuity are
required.
• Lecture 8: Existence and continuity of inverse functions.
Chapter 2: Limits of functions
• Lecture 9:
discontinuity.
Left limits, right limits, continuity in terms of limits. Different kinds of
• Lecture 10: Limits at ∞ and infinite limits.
Chapter 3: Differentiation
• Lecture 11: Examples, calculations.
• Lecture 12: Differentiability implies continuity. Algebra of differentiable functions: sums,
products, quotients, composites of differentiable functions.
• Lecture 13: Chain rule. Mean value theorem, Rolle’s theorem.
• Lecture 14: Inverse function theorem. Local maxima and minima.
• Lecture 15: Higher order derivatives.
• Lecture 16: The algebras C k ([a, b]). Notion of distance in these “spaces”.
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Chapter 4: Power series
• Lecture 17: Definitions, examples. Some theorems of Euler.
• Lecture 18:
convergence.
Lim sup, lim inf, radius of convergence. Theorem of Hadamard on radius of
• Lecture 19: Differentiation of power series. Lagrange’s theorem on term by term differentiation.
• Lecture 20: More examples. More on exponential function. Applications to solution of
differential equations.
Chapter 5: Taylor’s Theorem
• Lecture 21: MacLaurin’s problem. Uniqueness of power series representation.
• Lecture 22: Cauchy’s Mean Value Theorem.
• Lecture 23: Taylor’s Theorem with Lagrange remainder.
• Lecture 24: Applications of Taylor’s theorem. Analytic and non-analytic functions.
• Lecture 25: The algebra of analytic functions.
Chapter 6: L’Hopitâl’s Rule
• Lecture 26: L’Hopitâl’s rule; examples of application.
• Lecture 27: L’Hopitâl’s rule with infinite limits. Hard examples.
• Lecture 28: Examples, revision, overspill
• Lecture 29: Examples, revision, overspill
• Lecture 30: Examples, revision, overspill
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