Math 311 Final Exam Review

Dangello and Seyfried

• Definitions in chapters 6 and 7
• Proof of Theorems 6.7, 6.8, 6.9 and 6.10
• Statement and proof of
  • Theorem 6.11 Fundamental Theorem of Calculus part 1
  • Theorem 6.12 Fundamental Theorem of Calculus part 2
• Statements of
  • Theorem 7.5 Limit Comparison Test
  • Theorem 7.6 Alternating Series Test
  • Theorem 7.7 Ratio Test
• Proof of Theorem 7.6 and part 1 of Theorem 7.7
• Proof of Theorem 7.14 Merten's Theorem
• Homework problems 6.3#10, 6.4#7 and 6.5#5
• Homework problems 7.1#7, 7.2#2 and 7.2#8

Folland

• In Rⁿ the definitions
  • ball of radius r about a
  • bounded set
  • interior point
  • boundary point
  • open and closed
  • closure
  • limit and continuity
  • connected
  • disconnected
  • differentiable
  • convex set
  • compact set
  • class C^k
  • multi-index notation
  • partition
  • upper and lower sum
  • Riemann integral
  • characteristic function
  • zero content
  • Jordan measurable
  • rectifiable
  • smooth curve
  • arc length
  • regular region
• Proof of
  • The Bolzano-Weierstrass theorem in Rⁿ.
  • A continuous function on a closed and bounded set in Rⁿ is uniformly continuous and attains its maximum.
  • The characterstic function of a bounded measurable set is Riemann integrable.
  • Green's Theorem on a rectangle.
• Statement of
  • Theorem 2.45 interchange of the order of partial derivatives.
  • Theorem 2.68 Taylor's theorem for several variables using multi-index notation with only the remainder term (2.72). Do not memorize (2.70) or (2.71).
  • The change of variables formula.
  • Theorem 3.1 the implicit function theorem.
  • Theorem 5.12 Greens Theorem.
• All assigned homework problems.