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 14 Monday.
   • Exercises from §5 Practice Exercises (pages 342–344): 9, 45, 51, 55, 65, 73, 75, 77, 85, 87, 95, 105, 121, 125.
   • Exercises from §6.5 (pages 380–382): 5, 8, 9, 11, 13, 15, 16, 21, 23, 25.
   • Exercises from §6.6 (pages 389&390): 3, 6, 7, 11, 13, 15, 25.
   • Exercises 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 − 2x f(x) with f(2) = 0 (based on Exercise 6);
     • Check that f(x) = (cos x)/x is a solution of the initial-value problem x f′(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.
   • Additional extra-credit exercise: Exponential growth forever is unrealistic. Instead of assuming that dy/dt is proportional to y (where t is time and y is a growing population), assume that there is a carrying capacity c giving the maximum sustainable value of y and that dy/dt is proportional to y(c − y). (So the population grows most when there is both a large population y to reproduce and a large amount c − y of remaining room to grow.) To be explicit, write dy/dt = ky(c − y), where k and c are positive constants, and use the initial value that y = y₀ when t = 0 (where y₀ is a positive constant satsifying y₀ < c). Solve this initial-value problem (showing at least what integrals you evaluate and at least one step each for determining any constants of integration, as well as the final solution); the solution is called logistic growth. (Hint: There is a tricky integral here; use the algebraic identity that 1/(A(B − A)) = 1/(AB) + 1/(B(B − A)) to help.)
2. Integration techniques:
   • Date due: January 23 Wednesday.
   • Exercises 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.
   • Exercises from §8.2 (pages 434&435): 7, 11, 16, 19, 26, 35, 41, 46, 51, 69.
   • Exercises from §8.3 (pages 439&440): 4, 5, 11, 14, 17, 29, 57.
   • Exercises from §8.4 (pages 445&446): 4, 7, 11, 17, 23, 27, 38, 39.
   • Additional extra-credit exercise: 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.)
3. Transitional topics:
   • Date due: January 30 Wednesday.
   • Note: In this quiz, you'll be able to use pages T-1 through T-6 from the back of the textbook; I'll have copies on hand, so you don't need to remember to bring them.
   • Exercises from §8.5 (pages 451&452): 9, 13, 15, 19, 29, 33.
   • Exercises from §8.6 (pages 459–462): 5, 7, 15, 17, 23, 27.
   • Exercises from §8.7 (page 471): 3, 9, 14, 17, 19, 31.
   • Exercises 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.
   • Exercises from §9.2 (pages 497–499): 27, 31, 32, 88.
   • Additional extra-credit exercise: Use a Wolfram product (Mathematica or Wolfram Alpha) to integrate ∫ x⁻¹(x − x²)^1/2 dx. Then use Sage or 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.)
4. Convergence tests:
   • Date due: February 6 Wednesday.
   • Exercises 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.
   • Exercises from §9.3 (pages 504&505): 3, 9, 13, 16, 19, 21, 25, 28, 29, 31, 33, 35, 37, 49, 51.
   • Exercises from §8.7 (pages 471–473): 41, 43, 45, 53, 59.
   • Exercises from §9.4 (pages 509&510): 5, 6, 10, 14, 19, 21, 22, 25, 27, 29, 32, 33, 34, 37, 45, 47, 51.
   • Additional extra-credit exercise: 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 9.3.6 on page 503 of the textbook does this estimate in the case of p = 2 and m = 10, finding an error of less than 0.005, which you can use as a guide and to check your answer.)
5. Tests for power series:
   • Date due: February 13 Wednesday.
   • Exercises 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.
   • Exercises from §9.5 (pages 515&516): 1, 9, 17, 21, 23, 24, 25, 29, 33, 34, 35, 41, 47, 51, 56, 59, 61, 63.
   • Exercises from §9.7 (pages 530–532): 5, 7, 8, 15, 17, 18, 23, 31, 39, 53.
   • Additional extra-credit exercise: 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.)
6. Taylor series:
   • Date due: February 21 Thursday.
   • Exercises from §9.9 (pages 542&543): 1, 5, 11, 12, 17, 21, 22, 37, 39, 41, 48.
   • Exercises from §9.8 (pages 536&537): 1, 3, 5, 7, 9, 11, 13, 14, 15, 17, 23, 27.
   • Exercises 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 Thursday.
   • Exercises 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.
   • Exercises from §11.2 (pages 608–610): 3, 7, 8, 11, 13, 15, 18, 19, 23, 31, 34, 35, 42, 47.
   • Exercises from §11.3 (pages 616–618): 1, 5, 7, 8, 10, 11, 23, 31, 32, 35, 37.
   • Exercises 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 8 Friday.
   • Exercises from §10.1 (pages 562–564): 5, 7, 13, 23, 33.
   • Exercises 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.
   • Exercises from §10.2 (pages 572&573): 1, 13, 19, 23, 25, 29, 33.
   • Additional extra-credit exercise: Look at equation (4) from Section 10.2 on page 570 of the textbook, 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| (which is not simply a). Treating equation (5) as correct (as motivated by Figure 6.27 on page 370 of the textbook 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 Thursday.
   • Exercises from §10.3 (pages 577&578): 1, 2, 3, 4, 5, 6, 9, 27, 31, 45, 50, 53, 59, 63.
   • Exercises from §10.4 (page 581):
     • Ignore the symmetries: 3, 7, 8, 9, 13, 16;
     • 17, 19, 21.A, 22.A, 23.A, 24.A, 25, 26.
   • Exercises from §10.5 (pages 584&585): 3, 5, 7, 9, 11, 12, 13, 21, 25, 27.
   • Additional extra-credit exercise: Find the area of a rectangle with width l and height h using polar coordinates. Check that the answer is correct. (Show at least what integral or integrals you evaluate. Hint: I would put one corner of the rectangle at the origin and make its sides parallel to the coordinate axes.)

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