Here are the exams and their associated problem sets (Exam 1, Exam 2, Exam 3, Exam 4):

- Exam 1:
- Date taken: October 20 Thursday.
- Problems from Section 2.1 (pages 154–157): 2, 12, 19, 23, 41.
- Problems from Section 2.4 (pages 186–188): 9, 11, 15, 23, 25, 39.
- Extra credit (essay): Explain your background in mathematics and what you are going to use this course for.
- Problems from Section 7.1 (pages 514–519): 15, 21, 27, 29, 31, 33, 39, 41, 43, 45, 73, 83, 97, 99.
- Problems from Section 7.2 (pages 527–531): 8, 9, 11, 13, 15, 21, 25, 29, 31, 35, 37, 39, 41, 43, 55, 63.
- Problems from Section 7.3 (pages 538–542):
- Give exact answers involving radicals if necessary: 7, 19, 21, 23, 25, 27, 29;
- Optional, using a calculator: 31, 33, 35;
- Give exact answers involving radicals or trigonometric operations if necesary, or use a calculator to get the answers in the back of the book: 73, 75, 81.

- Problems from Section 7.4 (pages 551–553): 3, 7, 8, 11, 15, 17, 21, 31, 33, 35, 51, 53, 57, 59, 61, 75, 83, 87, 93.
- Problems from Section 7.5 (pages 561–564): 11, 13, 17, 19, 23, 25, 29, 35, 39, 49, 81.

- Exam 2:
- Date taken: November 3 Thursday.
- Problems from Section 7.5 (pages 561–564): 63, 65, 69.
- Problems from Section 7.6 (pages 574–578): 11, 12, 17, 19, 25, 35, 39, 51, 57, 61, 85.
- Problems from Section 7.7 (pages 584–586): 7–16, 21, 23.
- Problems from Section 7.8 (pages 594–597):
- Optional: 9, 11, 15, 17;
- Extra credit: 12.

- Problems from Section 8.1 (pages 616–619):
- Give exact answers: 11, 23, 25;
- Optional, using a calculator: 27, 29, 31;
- Give exact answers: 39, 41, 43, 45, 51, 53, 55, 57, 59, 61.

- Problems from Section 8.2 (pages 623–625):
- Give exact answers: 11, 23, 25, 27, 29, 31, 39, 41, 43;
- Optional, using a calculator: 45, 51, 53, 55.

- Exam 3:
- Date taken: November 17 Thursday.
- Problems from §8.5: 7, 8, 9, 10, 19, 21, 27, 31, 35, 39, 41, 49, 61, 77, 81, 87, 95.
- Problems from §8.6: 4, 5, 6, 9, 13, 21, 23, 53, 71.
- Problems from §8.4: 1, 2, 6, 8, 11–19 odd, 29, 55, 71, 95.
- Problems from §8.3: 5, 8, 9, 13, 23, 25, 27, 37, 59, 61, 71, 107, 113.
- Problems from §8.7 (optional):
- 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 43;
- Extra credit: 44.

- Problems from §9.1: 9, 11, 13, 15, 17, 19, 21, 33.

- Exam 4:
- Date taken: December 8 Thursday.
- Problems from §9.2: 6, 7, 9, 11, 13, 15, 25, 27, 29, 31, 33, 35.
- Problems from §9.3: 3, 4, 5, 9, 11, 13, 15.
- Problems from §9.4: 1, 2, 3, 7, 9, 11, 13, 15, 17, 19, 23, 25, 35, 39, 47.
- Problems from §9.5 (optional): 1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23.
- Problems from §10.1: 1, 3, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 43, 45, 47, 49, 51, 57, 61.
- Problems from §10.2: 3, 15, 17, 19, 21, 23, 31, 33, 37.
- Problems from §10.3 (optional): 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 15, 17, 19, 21, 25–63 odd.
- Extra credit:
(For brevity,
I will write cis
*θ*to abbreviate cos*θ*+ i sin*θ*.) In the 1740s, Leonhard Euler defined a general rule for raising any nonzero complex number to the power of any other complex number, called the*principal power*. If you write the base in polar coordinates as*r*cis*θ*, with*r*> 0 and −π <*θ*≤ π, and the exponent in rectangular coordinates as*x*+*y*i, then the principal power is written in polar coordinates as(

where e ≈ 2.718 and ln is the natural logarithm (so ln means log*r*cis*θ*)^{x+yi}= (*r*^{x}e^{−yθ}) cis (*y*ln*r*+*x**θ*),_{e}). Show at least one intermediate step for each of these:- If the base is positive (
*θ*= 0) and the exponent is real (*y*= 0), then what is the principal power*r*^{x}? - If the base is negative (
*θ*= π) and the exponent is 1/2 (*x*= 1/2 and*y*= 0), then what is the principal power (−*r*)^{1/2}? - If the base is negative
and the exponent is 1/
*n*for an odd natural number*n*, then what is the principal power (−*r*)^{1/n}? (This is called the*principal**n*th root of −*r*, in contrast to the*real**n*th root of −*r*that you should know from Algebra.) - If the base is e (
*r*= e and*θ*= 0), then what is the principal power e^{x+yi}? (For this reason, ln*a*+*y*i is called the*principal natural logarithm*of*a*cis*y*.) - If the base is e and the exponent is purely imaginary (
*x*= 0), then what is the principal power e^{iy}? (This result is called*Euler's formula*.) - What is e
^{iπ}+ 1? (This result is called*Euler's identity*.)

- If the base is positive (
- Problems from §10.4: 1, 3, 11, 13, 15, 17, 29, 31, 39, 43, 45, 49, 51, 61, 67.
- Problems from §10.5: 2, 5, 9, 15, 21, 25.

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