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.