Problem sets and exams

Almost every other Thursday, there will be an exam during the last hour of the class period, closely based on an associated problem set. (However, there is no exam in the first week of the course or on Thanksgiving, and there is an additional final exam on Thursday in the last week.) Unless otherwise specified, all problems are from the 10th Edition of Algebra & Trigonometry written by Sullivan and published by Prentice-Hall (Pearson).

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

1. 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.
2. 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.
3. 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.
4. 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 + yi, then the principal power is written in polar coordinates as
(r cis θ)x+yi = (rx eyθ) cis (y ln r + xθ),
where e ≈ 2.718 and ln is the natural logarithm (so ln means loge). Show at least one intermediate step for each of these:
1. If the base is positive (θ = 0) and the exponent is real (y = 0), then what is the principal power rx?
2. 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?
3. 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 nth root of −r, in contrast to the real nth root of −r that you should know from Algebra.)
4. If the base is e (r = e and θ = 0), then what is the principal power ex+yi? (For this reason, ln a + yi is called the principal natural logarithm of a cis y.)
5. If the base is e and the exponent is purely imaginary (x = 0), then what is the principal power eiy? (This result is called Euler's formula.)
6. What is e + 1? (This result is called Euler's identity.)
• 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.
That's it!
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