Math 414: Analysis I
Final Exam Topics
1. The Completeness Property (Existence of a sup/inf) (See in book: 1.2.9, 1.2.10,
1.1.2, 1.1.4, 1.1.5, 1.1.6, 1.1.9, HW 2 #4-6)
2. The Archimedean Property/ The Density Theorem (See in book: 1.2.1, 1.2.2, 1.2.8)
3. Sequences (including what it means for a sequence to be bounded/unbounded),
Definition of lim(xn ) (See in book: 2.1.1–2.1.7, , HW 4: #3#4)
4. Monotone Sequences, Monotone Convergence Theorem (See in book: 2.1.9–2.1.12,
HW 4: #2 )
5. Subsequences (See in Book: 2.1.14, 2.1.15, 2.1.17, HW 4: #1)
6. Limit Laws Associated with Sequences, Squeeze Theorem, Bolzano-Weierstrass Theorem (See in Book: 2.2.1–2.2.7, HW 5: #1–#5)
7. Cauchy Sequences (See in book: 2.4.1, 2.4.2, 2.4.4, HW 6: #6–#9)
8. Series (See in book: 2.5.1–2.5.4, 2.5.11)
9. Cluster Points/Limits of Functions (See in book: 3.1.1(c), 3.1.2-3.1.5, 3.1.8-3.1.10,
HW 8 #1–#8)
10. Continuous Functions (See in book: 3.2.1–3.2.6, 3.2.9–3.2.12, HW 9 #7)
11. Intermediate Value Theorem/ Location of Roots Theorem (See in book: 3.3.1, 3.3.2,
3.3.7, 3.3.10, HW 9 #8)
12. Uniform Continuity/ Lipschitz Continuous Functions (See in book: 3.4.3, 3.4.4,
3.4.7, 3.4.8, HW 10 #3–#7)
13. The Derivative (See in book: 4.1.1. 4.1.2, 4.1.3, 4.1.5, 4.1.6, 4.1.9, 4.1.11 HW 10
#8-9)
14. The Mean Value Theorem (See in book: 4.2.3–4.2.8, HW 11 #4–#6)
15. Integration (See in book: 5.1.1–5.1.3, 5.1.5, Integration Problems Worksheet)
Additional Details
• The final exam will occur in class on Tuesday, May 6 from 9:45-11:45 in our
usual classroom.
• In addition to going over the homework, please refer to the examples done in class,
as parts of the exam may resemble these examples.
• No notes or electronic devices may be used on the exam.
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