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Lecture Notes on Differential Equation of Calculus II - Fall 2005 | MATH 236, Study notes of Calculus

Material Type: Notes; Professor: Pruett; Class: CALCULUS II; Subject: Mathematics; University: James Madison University; Term: Unknown 1989;

Typology: Study notes

Pre 2010

Uploaded on 02/13/2009

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Math 236 Calculus II/Spring 2005/Pruett
TEST III REVIEW SHEET
1 Differential Equations
1. What does it mean for a function to satisfy an ODE?
2. Integrating factors [8.8]
3. Separation of variables [8.9]
2 Sequences, L’Hˆopital’s Rule, and Improper Integrals
1. Boundedness of sets; least upper and greatest lower bounds; least upper bound axiom [10.1]
2. Sequences as sets; sequences as functions; increasing and decreasing sequences [10.2]
3. Definition of the limit of a sequence; convergence and divergence; properties of limits; limit
theorems including pinching theorem [10.3]
4. Some important limits to know [10.4]
(a) {1
nα},α > 0
(b) {xn},|x|<1
(c) {ln n
n}
(d) {x1
n},x > 0
(e) {n1
n}
(f) {xn
n!}
(g) {(1 + 1
n)n}
5. L’Hˆopital’s Rule: appropriate and inappropriate uses of it [10.5]
6. Indeterminate forms other than 0
0[10.6]
(a)
, 1 · ,∞−∞, etc.
(b) Not to be confused with determinate forms: 0
c,c
0, etc
7. Improper integrals and comparison tests [10.7]
3 General Hints
1. The test will cover sections 8.8-8.9 on ODEs and sections 10.1-10.7 on sequences and limits.
2. The very first problem will be to ask you to state the formal definition of a convergent sequence.
3. The extra-credit problem will ask for an Kproof for a convergent sequence.
4. There may be an optional help session Sunday, April 10, if there is sufficient interest.
5. There will be a mixture of problem types, including short answer, T/F, and problems involving
the rigorous evaluation of limits including possibly those requiring integration [10.7].
6. Finally, do NOT try to cram the night before. Study well in advance.

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Math 236 Calculus II/Spring 2005/Pruett

TEST III REVIEW SHEET

1 Differential Equations

  1. What does it mean for a function to satisfy an ODE?
  2. Integrating factors [8.8]
  3. Separation of variables [8.9]

2 Sequences, L’Hˆopital’s Rule, and Improper Integrals

  1. Boundedness of sets; least upper and greatest lower bounds; least upper bound axiom [10.1]
  2. Sequences as sets; sequences as functions; increasing and decreasing sequences [10.2]
  3. Definition of the limit of a sequence; convergence and divergence; properties of limits; limit theorems including pinching theorem [10.3]
  4. Some important limits to know [10.4] (a) { (^) n^1 α }, α > 0 (b) {xn}, |x| < 1 (c) {lnn^ n } (d) {x 1 n^ }, x > 0 (e) {n 1 n^ } (f) {x nn! } (g) {(1 + (^1) n )n}
  5. L’Hˆopital’s Rule: appropriate and inappropriate uses of it [10.5]
  6. Indeterminate forms other than 00 [10.6] (a) ∞∞ , 1 · ∞, ∞ − ∞, etc. (b) Not to be confused with determinate forms: (^0) c , 0 c , etc
  7. Improper integrals and comparison tests [10.7]

3 General Hints

  1. The test will cover sections 8.8-8.9 on ODEs and sections 10.1-10.7 on sequences and limits.
  2. The very first problem will be to ask you to state the formal definition of a convergent sequence.
  3. The extra-credit problem will ask for an  − K proof for a convergent sequence.
  4. There may be an optional help session Sunday, April 10, if there is sufficient interest.
  5. There will be a mixture of problem types, including short answer, T/F, and problems involving the rigorous evaluation of limits including possibly those requiring integration [10.7].
  6. Finally, do NOT try to cram the night before. Study well in advance.