Problem sets and quizzes

Here are the quizzes and their associated problem sets:

1. Quiz 1:
   • Date: January 14 Thursday.
   • Problems from §5 Practice Exercises (pages 345–348):
     • Show at least one intermediate step for each part: 9;
     • Show at least one step before and after performing a substitution: 45, 51, 55, 65;
     • Show at least what numerical calculation you make: 73, 75, 77, 85, 87, 95, 105;
     • No additional work needed: 121;
     • Show at least one intermediate step: 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 386–389):
     • For each result, show what numerical calculation you make or what definite integral you evaluate: 5, 8, 9, 11, 13, 15, 16, 21;
     • Show at least two intermediate steps: 23;
     • Show what numerical calculation you make using kinetic energy: 25.
   • Problems from §6.6 (pages 396&397): Show what definite integrals you evaluate: 3, 6, 7, 11, 13, 15, 25.
   • Problems from §7.2 (pages 418–420):
     • Show y′ (meaning dy/dx) as an intermediate step: 3, 6, 7;
     • Show what integrals you take after separating variables, and be sure to solve for y (which the answers in the back do not finish, except for #17): 11, 13, 17, 18, 21;
     • Show what equation (differential or otherwise) you solve or what numerical calculation you make: 33, 35, 36, 39.
2. Quiz 2:
   • Date: January 22 Friday.
   • Problems from §8.1 (pages 436–438):
     • Show at least u, du, v, and dv for each integration by parts: 3, 6, 11, 17, 18, 25, 41, 47, 49;
     • Assume that n is a constant n ≥ 1, and show at least u, du, v, and dv for each integration by parts: 61, 64;
     • Show at least u, du, v, and dv for each integration by parts: 67.
   • Problems from §8.2 (pages 443&444): Show at least one intermediate step showing the integral after all trigonometric identities have been applied: 11, 19, 35.
   • Problems from §8.3 (pages 447&448):
     • Show at least one intermediate step showing the integral after all trigonometric substitutions have been applied: 5, 11, 17;
     • For each part, show at least one intermediate step showing the integral after all substitutions have been applied: 57.
   • Problems from §8.4 (pages 454&455): Show at least one intermediate step showing the integral after all fractions have been parted: 11, 17, 27.
   • Problems from §8.5 (pages 460&461): No additional work needed: 10, 14, 16, 20, 30, 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 book (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 468–470): Show what numerical calculations you make: 5, 7, 15, 17, 23, 27.
3. Quiz 3:
   • Date: January 29 Friday.
   • Problems from §8.7 (pages 479–481): Show at least what limits you evaluate: 3, 9, 14, 17, 19, 31.
   • Problems from §9.1 (pages 495–498):
     • Show what numerical calculation you make to find each term: 9, 12;
     • No additional work needed: 15, 21, 22, 23;
     • Show at least one intermediate step for each limit, or give a reason for divergence: 27, 35, 37, 38, 41, 43, 45, 48, 51, 53, 55, 60, 66, 73, 83;
     • Show what numerical calculation you make to find each term: 99.
   • Problems from §9.2 (pages 505&506):
     • No additional work needed: 3, 5, 9, 12, 13;
     • Show a formula for the partial sums or show a calculation using a formula for the sum of a geometric series: 15, 17, 18;
     • Show the limit of the terms: 27, 31, 32;
     • No additional work needed: 35, 37, 40;
     • Show a formula for the partial sums or show a calculation using a formula for the sum of a telescoping series: 43, 46;
     • No additional work needed (but be sure to do everything in the instructions): 51, 55, 61, 62, 65, 69;
     • Show a formula for the partial sums or show a calculation using a formula for the sum of a geometric series: 73;
     • No additional work necessary: 81, 82;
     • Extra credit: 88;
     • Show what finite equation you solve: 90.
4. Quiz 4:
   • Date: February 5 Friday.
   • Problems from §9.3 (pages 511&512):
     • Write down what integral you check; also note at what point (if not immediately) the Integral Test applies: 3, 9;
     • State at least which test you are using: 13, 16, 19, 21, 25, 28, 29, 31, 33, 35, 37;
     • Show what numerical calculations you make: 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 479–481): Show what integrals you compare to, and indicate whether the comparison is direct or in the limit: 41, 43, 45, 53, 59.
   • Problems from §9.4 (pages 516&517):
     • Show what series you compare to, and show which inequality holds between the two series (and for which terms, if not all of them): 5, 6;
     • Show what series you compare to, and show the limit of the ratio of the two series: 10, 14;
     • State at least which test you are using; in the case of a comparison test, show what series you compare to: 19, 21, 22, 25, 27, 29, 32, 33, 34, 37, 45, 47, 51.
5. Quiz 5:
   • Date: February 12 Friday.
   • Problems from §9.6 (pages 527&528):
     • Show the limit of the terms (or of their absolute values); if the series converges, state when its terms' absolute values begin decreasing (if not immediately): 1, 9, 10, 12, 13;
     • Show your numerical calculation: 49, 51;
     • State at least the number of terms that you use: 57.
   • 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 §9.5 (pages 521&522):
     • Show the limit of the ratios: 1;
     • Show the limit of the roots: 9;
     • State at least which test you are using (and state what series you compare to if you use a comparison test): 17, 21, 23, 24, 25, 29, 33, 34, 35, 41;
     • Use the Ratio Test, and show the limit of the ratios: 47, 51;
     • State at least which test you are using (and state what series you compare to if you use a comparison test): 56, 59, 61;
     • Show the limit of the ratios and the limit of the roots: 63.
   • Problems from §9.6 (pages 527&528):
     • State at least which test or tests you use (you may need two for some of these): 17, 21, 22, 27, 29, 30, 33, 35, 37, 39, 40, 43.
   • Problems from §9.7 (pages 536–538):
     • Show what limit you take to find the radius of convergence, and state at least what tests you use to treat the endpoints (if any) of the interval of convergence: 5, 7, 8, 15, 17, 18, 23, 31;
     • Show what limit you take: 39.
     • Show what calculations you make (if any) to find the radii of convergence, show a formula for the sum of a geometric series to calculate the sum of the series, and show at least one intermediate step for the sum of the derivative series: 49.
6. Quiz 6:
   • Date: February 19 Friday.
   • Problems from §9.8 (pages 542&543):
     • Show at least the values of the relevant derivatives at a: 1–9 odd;
     • Either show enough derivatives that the pattern is clear or show substitution into a formula given in the textbook: 11, 13, 14, 15, 17, 23, 27.
   • Problems from §9.9 (pages 549&550):
     • Either show enough derivatives that the pattern is clear or show substitution into a formula given in the textbook: 1, 5, 11, 12, 17, 21, 22.
     • Show what numerical calculations you make or what equations or inequalities you solve: 37, 39, 41, 48.
   • Problems from §9.10 (pages 556–558):
     • Show at least one intermediate step for each: 3, 5, 7, 12, 13;
     • Show at least what equations or inequalities you solve or what numerical calculations you make, to determine the number of terms to use or to check that you have enough terms: 25, 27;
     • Show at least one intermediate step for each: 29, 31, 33, 59, 61.
     • Extra credit: 68, 69.
7. Quiz 7:
   • Date: February 26 Friday.
   • Problems from §10.1 (pages 568–570):
     • Show at least one intermediate step on the way to finding the cartesian equation: 5, 7, 13;
     • Show at least one intermediate step of algebra: 23;
     • Show at least one intermediate step (besides the hint) for each variable (hint: tan θ = y/x): 31.
   • Problems from §10.2 (pages 577–579):
     • Show what differentials or derivatives you find along the way: 1, 13, 19;
     • Show what integrals in t you take: 23, 25, 29, 33.
   • Additional extra-credit problem: Look at equation (10.2.4) on page 575, and compare it to the equation that follows it (which I'll call equation (5), even the book 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 376 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 book'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 573 of the textbook, because there is nothing wrong with that definition as far as it goes.)
   • Problems from §10.3 (pages 582&583):
     • No additional work needed: 1–4;
     • Show what numerical calculations you make: 5, 6, 9;
     • Show at least one intermediate step of algebra for each: 27, 31, 45, 50, 53, 59, 63.
8. Quiz 8:
   • Date: March 4 Friday.
   • Problems from §10.4 (pages 586&587):
     • No additional work needed, and ignore the symmetries if you like: 3, 7, 8, 9, 13, 16;
     • Show what derivatives or differentials you take: 17, 19;
     • No additional work needed: 21–24.a, 25, 26.
   • Problems from §10.5 (pages 590&591):
     • Show what integrals (in one variable) you take: 3, 5, 7, 9, 11, 12, 13, 21, 25, 27;
     • Extra credit: Show at least the derivatives that you substitute: 29.
   • Problems from §11.1 (pages 605&606):
     • No additional work needed: 1, 3, 7, 8, 11, 19, 25, 29, 30, 35, 37, 39;
     • Show what numerical calculation you make or what equation you solve: 41, 45;
     • No additional work needed: 49, 51, 52;
     • Show at least two intermediate steps: 57;
     • No additional work needed: 59;
     • Show what numerical calculations you make or what equations you solve: 62.
9. Quiz 9:
   • Date: March 11 Friday.
   • Problems from §11.2 (pages 614–616):
     • Show at least one intermediate step for each part: 3, 7, 8, 11, 13, 15, 18, 19;
     • Show also u, v, and w as appropriate in each picture: 23;
     • No additional work needed: 31, 34;
     • Show at least one intermediate step for each part: 35;
     • Show what numerical calculations you make or what equations you solve: 42, 47.
   • Problems from §11.3 (pages 622–624):
     • Show at least one intermediate step for each result: 1, 5, 7, 8;
     • Show what numerical calculations you make or what equations you solve; you may leave exact answers involving trigonometric operations: 10, 11, 23;
     • Show at least a formula for each slope: 31, 32;
     • Show what numerical calculations you make or what equations you solve: 35, 37.
   • Problems from §11.4 (pages 628–630):
     • Show what numerical calculations you make: 3, 6, 11, 12, 17, 20, 21, 23;
     • No additional work needed: 27, 28, 29, 31;
     • Extra credit: 34.
     • Show what numerical calculations you make: 35, 38, 39, 43.
10. Additional problems:
    • Date: No quiz, but this material will still be on the final exam!
    • Problems from §11.5 (pages 636–638): 1, 6, 7, 9, 17, 21, 23, 27, 28, 31, 33, 36, 37, 41, 43, 45, 47, 53, 59, 61, 67.

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