Problem sets and quizzes

Here are the quizzes and their associated problem sets (Quiz 1, Quiz 2, Quiz 3, Quiz 4, Quiz 5, Quiz 6, Quiz 7, Quiz 8, Quiz 9):

1. Introductory topics:
   • Date due: January 12 Thursday.
   • Problems from §5 Practice Exercises (pages 342–344): 9, 45, 51, 55, 65, 73, 75, 77, 85, 87, 95, 105, 121, 125.
   • Extra-credit essay question: Explain your background in mathematics and what you are going to use this course for.
   • Problems from §6.5 (pages 380–382): 5, 8, 9, 11, 13, 15, 16, 21, 23, 25.
   • Problems from §6.6 (pages 389&390): 3, 6, 7, 11, 13, 15, 25.
   • Problems from §7.2 (pages 409–411):
     • Where the textbook writes y′, it really means dy/dx (which is f′(x)): 1, 3;
     • Check that f(x) = (x − 2) e^−x2 is a solution of the initial-value problem f′(x) = e^−x2 − 2xf(x) with f(2) = 0 (based on Exercise 6);
     • Check that f(x) = (cos x)/x is a solution of the initial-value problem xf′(x) + f(x) = −sin x with f(π/2) = 0 (based on Exercise 7);
     • Solve for the quantity y (which the answers in the back of the textbook do not finish most of the time): 11, 13, 17, 18, 21;
     • 35, 37, 38, 41.
2. Integration techniques:
   • Date due: January 20 Friday.
   • Problems from §8.1 (pages 427–429):
     • 3, 6, 11, 17, 18, 25, 41, 47, 49;
     • Assume that n is a constant n ≥ 1: 63, 66;
     • 71.
   • Additional extra-credit problem: Find a general formula for ∫xⁿ ln x dx (the indefinite integral of xⁿ ln x with respect to x), where n is a constant natural number. (Use integration by parts at least once, show at least one step for each use of integration by parts, and indicate at least what u and v are for each use of integration by parts.)
   • Problems from §8.2 (pages 434&435): 7, 11, 16, 19, 26, 35, 41, 46, 51, 69.
   • Problems from §8.3 (pages 439&440): 4, 5, 11, 14, 17, 29, 57.
   • Problems from §8.4 (pages 445&446): 4, 7, 11, 17, 23, 27, 38, 39.
3. Transitional topics:
   • Date due: January 30 Monday.
   • Problems from §8.5 (pages 451&452): 9, 10, 13, 14, 15, 16, 19, 20, 29, 30, 33, 34.
   • Additional extra-credit problem: Use a Wolfram product (Mathematica or Wolfram Alpha) to integrate ∫ x⁻¹(x − x²)^1/2 dx. Then use the tables in the back of the textbook (try the middle of page T-5) to do the same integral. Do these answers agree? Do they have any problems? Can you reconcile them? Explain. (For a thorough understanding, you may need to look up Euler's Formula relating exponential and trigonometric functions.)
   • Problems from §8.6 (pages 459–462): 5, 7, 15, 17, 23, 27.
   • Problems from §8.7 (page 471): 3, 9, 14, 17, 19, 31.
   • Problems from §9.1 (pages 487–490): 9, 12, 15, 21, 22, 23, 27, 35, 37, 38, 41, 43, 45, 48, 51, 53, 55, 60, 66, 73, 83, 99.
   • Problems from §9.2 (pages 497–499): 27, 31, 32, 88.
4. Convergence tests:
   • Date due: February 6 Monday.
   • Problems from §9.2 (pages 497–499): 3, 5, 9, 12, 13, 15, 17, 18, 35, 37, 40, 43, 46, 51, 55, 61, 62, 65, 69, 73, 81, 82, 90.
   • Problems from §9.3 (pages 504&505): 3, 9, 13, 16, 19, 21, 25, 28, 29, 31, 33, 35, 37, 49, 51.
   • Additional extra-credit problem: Based on the error estimation given by the Integral Test, if you can work out integrals of f (where f is a decreasing function whose infinite integral converges), then you can estimate the infinite sum of a_n = f(n) using m terms and a couple of integrals with an error of at most ½∫_m^m+1 f(x) dx, half of the integral of f from m to m + 1. Using this, and if p > 1, what is the maximum possible error if you estimate the convergent p-series Σ^∞_n=1 (1/n^p), the infinite series of 1/n^p starting with n = 1, using m terms? (Example 9.3.5 on page 510 of the textbook does this estimate in the case of p = 2 and m = 10, finding an error of at most ½∫₁₀¹¹(1/x²) dx = ½(1/10 − 1/11) = 1/220 < 0.005.) When you have this expression, consider these limits:
     1. As m → ∞ (so what happens to the error as you add more and more terms?)
     2. As p → 1⁺ (so what happens to the error as you approach the divergent harmonic series?)
   • Problems from §8.7 (pages 471–473): 41, 43, 45, 53, 59.
   • Problems from §9.4 (pages 509&510): 5, 6, 10, 14, 19, 21, 22, 25, 27, 29, 32, 33, 34, 37, 45, 47, 51.
5. Tests for power series:
   • Date due: February 13 Monday.
   • Problems from §9.6 (pages 521&522): 1, 9, 10, 12, 13, 17, 21, 22, 27, 29, 30, 33, 35, 37, 39, 40, 43, 49, 51, 57.
   • Problems from §9.5 (pages 515&516): 1, 9, 17, 21, 23, 24, 25, 29, 33, 34, 35, 41, 47, 51, 56, 59, 61, 63.
   • Additional extra-credit problem: Consider the ‘alternating p-series’ Σ^∞_n=1 (−1)ⁿ/n^p, the infinite sum of (−1)ⁿ/n^p.
     1. For which values of p does this converge?
     2. If you approximate this series with m terms, then what is the maximum possible absolute error (the maximum possible absolute value of the difference between the approximation and the true value) using the error estimate that goes with the Alternating Series Test? (Your answer will depend on both p and m.)
     3. What is the limit of this expression as m → ∞?
     4. What is the limit of this expression as p approaches the first value of p for which the alternating p-series diverges?
   • Problems from §9.7 (pages 530–532): 5, 7, 8, 15, 17, 18, 23, 31, 39, 53.
6. Taylor series:
   • Date due: February 20 Monday.
   • Problems from §9.9 (pages 542&543): 1, 5, 11, 12, 17, 21, 22, 37, 39, 41, 48.
   • Problems from §9.8 (pages 536&537): 1–9 odd, 11, 13, 14, 15, 17, 23, 27.
   • Problems from §9.10 (pages 549–551):
     • 3, 5, 7, 12, 13, 25, 27, 29, 31, 33, 59, 61.
     • Extra credit: 68, 69.
7. Vectors:
   • Date due: February 28 Tuesday.
   • Problems from §11.1 (pages 599&600): 1, 3, 7, 8, 11, 19, 25, 29, 30, 35, 37, 39, 41, 45, 49, 51, 52, 57, 59, 62.
   • Problems from §11.2 (pages 608–610): 3, 7, 8, 11, 13, 15, 18, 19, 23, 31, 34, 35, 42, 47.
   • Problems from §11.3 (pages 616–618): 1, 5, 7, 8, 10, 11, 23, 31, 32, 35, 37.
   • Problems from §11.4 (pages 622–624):
     • 3, 6, 11, 12, 17, 20, 21, 23, 27, 28, 29, 31;
     • Extra credit: 34;
     • 35, 38, 39, 43.
8. Analytic geometry:
   • Date due: March 7 Tuesday.
   • Problems from §11.5 (pages 630–632): 1, 6, 7, 9, 17, 21, 23, 27, 28, 31, 33, 36, 37, 41, 43, 45, 47, 53, 59, 61, 67.
   • Problems from §10.1 (pages 562–564): 5, 7, 13, 23, 33.
   • Problems from §10.2 (pages 572&573): 1, 13, 19, 23, 25, 29, 33.
   • Additional extra-credit problem: Look at equation (10.2.4) on page 570, and compare it to the equation that follows it (which I'll call equation (5), although the textbook doesn't number it). Treating ds, dx, dy, and dt according to the ordinary rules of algebra for real numbers, and assuming dt ≠ 0, simplify equation (4) as much as you can to get something like equation (5); pay especial attention to the algebraic rule √(a²) = |a| (not simply a). Treating equation (5) as correct (as motivated by Figure 6.27 on page 370 and noting that ds is supposed to represent a length), what should equation (4) be so that it will simplify to equation (5) exactly? Can you think of a situation in which integrating this modified version of equation (4) would give the correct value for the arclength of a parametrized curve while integrating the textbook's version of equation (4) would give a wrong value? (This would have to be a situation that violates some of the fine print in the definition on page 567 of the textbook, because there is nothing wrong with that definition as far as it goes.)
9. Polar coordinates:
   • Date due: March 14 Tuesday.
   • Problems from §10.3 (pages 577&578): 1–4, 5, 6, 9, 27, 31, 45, 50, 53, 59, 63.
   • Problems from §10.4 (page 581):
     • Ignore the symmetries: 3, 7, 8, 9, 13, 16;
     • 17, 19, 21–24.A, 25, 26.
   • Problems from §10.5 (pages 584&585):
     • 3, 5, 7, 9, 11, 12, 13, 21, 25, 27;
     • Extra credit: Show at least the derivatives that you substitute: 29.

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