Download Here is the final list of topics for the final exam.

Information about the final exam
The final exam will be on Monday, December 19, from 9 a.m. until noon in our usual classroom.
For roughly 80 percent of the exam, the focus will be on the definitions, theorems (and their proofs), and
examples we have done in class (listed below). The remaining 20 percent will ask you to build on these to
prove some new results or define some new concepts (much like the Inquiries we have done in class). Note:
for each lemma, proposition, or theorem below, you should also learn the proof unless it is explicitly stated
otherwise.
1. Definition of the limit of a function (Definition 1 in section 2)
2. Theorem 2.2: parts (a) and (b) [Limits distribute over sums and products]
3. Definition of a function being continuous at a point.
4. Give an example of a function that is everywhere discontinuous (and prove that it is everywhere
discontinuous). You may use the results of HW3 that both the rational and irrational numbers are
dense.
5. Theorem 3.3 (a continuous function that is positive must be positive on an interval)
6. Definition of a function being bounded above (section 4), definition of a set being bounded above
(section 5)
7. Definition of the least upper bound (sup) and greatest lower bound (inf). Statement of the Completeness Axsiom.
8. Theorem 4.1 (proof in section 5)
9. Every positive real number has an nth root (for any positive integer n). See Theorem 4.8.
10. Definition of the derivative (section 6)
11. Give an example of a function that is continuous but not differentiable at a point(and prove it).
12. Give an example of a function that is differentiable but whose derivative is not continuous at a
point(and prove it).
13. Theorem 6.1 (differentiable functions are continuous)
14. Theorem 6.3 (the product rule) and be able to apply it to some examples.
15. Statement of the chain rule (not the proof), and be able to apply it to some examples.
16. Definition of local maximum/minimum point/value. Definition of critical point. (section 7)
17. Theorem 7.1 (if a point is a local max/min, it is a critical point)
18. Theorem 7.3 (Rolle’s Theorem)
19. Theorem 7.4 (MVT)
20. Corollary 7.1 (if a function has everywhere zero derivative, then it must be constant)
21. Corollary 7.2
22. Corollary 7.3
23. Statement of the Inverse Function Theorem (not the proof) + how to apply it to some examples
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24. Definition of a partition, lower and upper sums for a partition (section 8)
25. Understand how lower and upper sums compare across partitions (statement of Theorem 8.1 and the
lemma preceeding it, not the proofs)
26. Definition of integrability/the integral.
27. Give an example of a function that is not integrable (and prove it).
28. Proof of Theorem 8.7
29. Homework 7, problem 1
30. Homework 7, problem 2
31. Theorem 9.1 and its corollary (the FTC)
32. Theorem 9.3 (integration by parts) and be able to apply it to some examples.
33. Theorem 9.4 (substitution) and be able to apply it to some examples.
34. Definitions of cos x, sin x, and exp(x), calculating their derivatives. (section 10)
35. Definition of the limit of a sequence.
36. Theorem 10.2: every sequence that is bounded above and nondecreasing converges.
37. Definition of a sequence being summable (in other words, of a series converging).
38. Analyzing convergence of the geometric series (for what r does it converge? what does it converge
to?).
39. Theorem 10.5 (The Ratio Test)
40. Statements of the Vanishing Criterion, the Comparison Test, the Integral Test (without proofs)
41. Applying
P thep above tests of convergence to the examples we did in class (e.g. geometric series, the
series
1/n , etc.)
42. Definition of Taylor polynomial, remainder term.
43. Statement of Taylor’s Theorem (without proof)
44. Computing Taylor polynomials
45. Using Taylor’s Theorem to estimate remainders, show they go to zero.
46. Using Taylor’s Theorem to approximate the values of various functions to desired degrees of accuracy.
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