# Homework

In case you miss a homework assignment in class, you can find it below. Unless otherwise specified, all problems are from the 2nd Edition of University Calculus: Early Transcendentals by Hass et al published by Addison Wesley (Pearson). When I return graded homework, I may post some solutions here too; see the downloading help if you have trouble reading them. See the grading policies for general instructions on doing homework and how it will be graded.

You can always do more homework problems! You may need to practise the material if you want to remember it for the final exam, a subsequent course, or the rest of your life. If you bought MyMathLab access with your course textbook (or separately), then you can find supplementary problems through the Moodle page. (However, MyMathLab is not required for this section.)

Here is the assigned homework:

1. Vectors:
• Date assigned: April 1 Wednesday.
• Date due: April 6 Monday.
• Pages in the textbook: §§11.1–5, my notes on vector operations.
• Problems from the Chapter 11 Practice Exercises (pages 644&645):
• Show what numerical calculations (if any) you make: 17, 19, 25, 29, 31, 35, 37, 43, 50;
• No additional work needed: 67, 69.
• Extra-credit essay question: Explain your background in mathematics and what you are going to use this course for.
2. Parametrized curves:
• Date due: April 6 Monday.
• Date due: April 8 Wednesday.
• Pages in the textbook: 649–655, 657–661.
• Problems from §12.1 (pages 655–657):
• Show at least one intermediate step for each calculation: 1, 4, 6, 7;
• Show also the velocity vector at the given value of t: 11, 14;
• Show also the velocity and acceleration vectors at t = 0: 15, 17;
• Show at least one intermediate step for each: 19, 20;
• No additional work needed: 23.
• Problems from §12.2 (pages 661–663):
• Show at least one intermediate step for each: 1, 4, 6, 12, 15, 17;
• Show what calculations you make or what equations you solve: 21, 22.
• Problems from §15.1 (pages 832–834): No additional work needed: 1–8.
3. Functions of several variables:
• Date assigned: April 8 Wednesday.
• Date due: April 13 Monday.
• Pages in the textbook: 686–691, my notes on functions of several variables.
• Problems from §13.1 (pages 692–694):
• Show what numerical calculations you make: 3, 4;
• Show what equations or inequalities you solve: 7, 9, 10;
• Give an equation for each level curve: 15;
• No additional work needed: 18, 24, 30, 31–36, 40, 42;
• State the value of c used: 52, 54, 59, 62.
4. Limits and continuity in several variables:
• Date assigned: April 13 Monday.
• Date due: April 15 Wednesday.
• Pages in the textbook: 694–700.
• Problems from §13.2 (pages 700–703):
• Show what numerical calculations you make: 2, 6, 11;
• Show the rewritten expressions: 18, 23;
• Show what numerical calculations you make: 28;
• Show what equations or inequalities (if any) you solve: 31, 32, 36, 39;
• State which paths you use: 43, 46;
• Give a reason: 55.
5. Vector fields and differential forms:
• Date assigned: April 15 Wednesday.
• Date due: April 20 Monday.
• Pages in the textbook: 835&836, my handout on differential 1-forms.
• Problems from §15.2 (pages 844–847):
• No additional work needed: 5, 6, 39–44.
1. No additional work needed: Given u = ⟨2x, −3x, 4xy⟩, express u ⋅ dr as a differential form (using r = ⟨x, y, z⟩).
2. No additional work needed: Given v = ⟨x2, 3⟩, express v ⋅ dr and v × dr as differential forms (using r = ⟨x, y⟩).
3. Show what numerical calculation you make: Given α = 3x dx + 4x2y dy, evaluate α at (x, y) = (2, 6) along ⟨dx, dy⟩ = ⟨0.003, 0.005⟩.
6. Partial differentiation:
• Date assigned: April 20 Monday.
• Date due: April 22 Wednesday.
• Pages in the textbook: 703–710, 723–726, 728&729, my handout on differentials.
• Problems from §13.3 (pages 711–714):
• Show at least one intermediate step for each: 3, 4, 10, 12, 24, 26, 30, 39;
• Show the first-order partial derivatives along the way: 43, 46;
• No additional work needed: 55;
• Show what limits you evaluate: 57;
• Show what algebraic equations you verify: 75, 82.
• Problems from §13.5 (pages 729&730):
• Show at least one intermediate step: 2, 3, 7, 8;
• Show the gradient, the differential, or the partial derivatives; and show either the direction of u or a result before adjusting for the magnitude of u: 14, 15, 16;
• Show the gradient as an intermediate step: 20, 23.
7. More with partial derivatives:
• Date assigned: April 22 Wednesday.
• Date due: April 27 Monday.
• Pages in the textbook: 714–720, 727, 730–733.
• Problems from §13.4 (pages 721&722):
• Use any method (including differentials or gradients), but show at least one intermediate step for each derivative: 2, 4, 7, 10;
• No additional work needed: 19, 20;
• Show at least one intermediate step: 27, 28, 33, 41.
• Problems from §13.5 (pages 729&730): Show the gradient, the differential, or the partial derivatives: 28.
• Problems from §13.6 (pages 737–739):
• Show the gradient, the differential, or the partial derivatives: 3, 6, 10, 13, 14.
• Problems from §15.2 (pages 844–847): Show at least one intermediate step: 1, 4.
8. Linearization and Taylor polynomials:
• Date assigned: April 27 Monday.
• Date due: April 30 Thursday.
• Pages in the textbook: 710&711, 733–737, my handout on Taylor polynomials.
• Problems from §13.6 (pages 737–739):
• For each problem, show what numerical calculation you make: 19, 21;
• For each problem, show the gradient, the differential, or the partial derivatives: 29, 30;
• For each problem, show what calculations you make or what inequalities you solve to estimate the error: 33, 35;
• For each problem, show the gradient, the differential, or the partial derivatives: 39, 50, 54.
• Additional extra-credit problem: Let f be the function of two variables given by f(x, y) = 3 sin(x + y) + 4 cos(x − y). Evaluate f, both of its partial derivatives, and all four of its second partial derivatives at (0, 0). Then use these results to approximate f near (0, 0) with a second-degree polynomial.
9. Optimization:
• Date assigned: May 4 Monday.
• Date due: May 7 Thursday.
• Pages in the textbook: 740–745, 748–755.
• Problems from §13.7 (pages 745–748): Show what equations you solve and what numerical calculations you make: 2, 7, 9, 15, 27, 32, 34, 37, 43, 52, 57.
• Problems from §13.8 (pages 755–757): Show what equations you solve and what numerical calculations you make: 1, 5, 10, 11, 16, 23, 29.
10. Integration on curves:
• Date assigned: May 7 Thursday.
• Date due: May 13 Wednesday.
• Pages in the textbook: 664–667, 828–832, 836–844.
• Problems from §15.2 (pages 844–847):
• Show what one-variable integrals you evaluate: 14, 16, 17.A&B, 23, 24;
• Show what one-variable integrals you evaluate: 10, 11, 19, 22, 29.
• Problems from §15.1 (pages 832–834): Show what one-variable integrals you evaluate: 10, 13, 16, 22, 30, 35.
• Problems from §12.3 (pages 667&668): Show what one-variable integrals you evaluate: 1, 5, 8, 9, 11, 15, 18.
• Additional extra-credit problem: Show at least one intermediate step: Show that the arclength element along the graph of y = f(x) is ds = √(1 + f′(x)²) |dx| (whenever f is differentiable).
11. Multiple integration:
• Date assigned: May 13 Wednesday.
• Date due: May 19 Tuesday.
• Pages in the textbook: 763–767, 768–774, 786–791.
• Problems from §14.1 (pages 767&768):
• Show at least the intermediate one-variable integral: 3, 7, 10;
• Show at least an iterated integral and an intermediate one-variable integral: 15, 20;
• Show a two-variable iterated integral: 25.
• Problems from §14.2 (pages 774–777):
• No additional work needed: 1, 2, 7, 9, 12, 14, 17;
• Show also the intermediate one-variable integral: 19, 23;
• No additional work needed: 35, 41;
• Show also the intermediate one-variable integral: 47, 51;
• Show a two-variable iterated integral: 57, 61.
• Problems from §14.5 (pages 792–795):
• Show also the two intermediate integrals: 3;
• No additional work needed: 6;
• Show at least the two intermediate integrals: 9, 15;
• No additional work needed: 21;
• Show a three-variable iterated integral: 25, 29, 34.
12. Applications of multiple integration:
• Date assigned: May 19 Tuesday.
• Date due: May 26 Tuesday.
• Pages in the textbook: 777–779, 791&792, 795–800.
• Problems from §14.3 (page 779):
• Show what integrals you evaluate: 1, 4, 7, 12;
• No additional work necessary: 13, 14, 17;
• Show what integrals you evaluate: 20, 21.
• Problems from §14.5 (pages 792–795): Show what integrals you evaluate: 37.
• Problems from §14.6 (pages 800–802): Show what integrals you evaluate: 3, 14, 19, 25, 29.
13. Change of variables in multiple integration:
• Date assigned: May 26 Tuesday.
• Date due: June 1 Monday.
• Pages in the textbook: 814–821, 780–784, 802–810.
• Problems from §14.4 (pages 784–786):
• No additional work needed: 1, 3, 5, 7;
• Show the iterated integrals in polar form: 9, 17, 20;
• No additional work needed: 23, 24;
• Show what iterated integrals you evaluate: 28, 29, 34;
• Show the iterated integral in polar form: 37.
• Problems from §14.7 (pages 810–813):
• Show also the two intermediate integrals for each: 1, 2, 8;
• Show the iterated integrals: 12;
• Show the iterated integral in cylindrical coordinates: 14;
• Show also the two intermediate integrals for each: 23;
• Show the iterated integral in spherical coordinates: 37;
• Show what iterated integrals you evaluate: 43, 46, 57, 77.
• Extra-credit problem from §14.8 (pages 821–823): Show the integral in u and v that you evaluate: 16.
14. Integrals on surfaces:
• Date assigned: June 1 Monday.
• Date due: June 4 Thursday.
• Pages in the textbook: 870–877, 880–887, my handout on the wedge product.
• Problems from §15.5 (pages 878–880):
• No additional work needed: 2, 3, 6, 9, 13;
• Show what integrals you evaluate: 20, 23.
• Problems from §15.6 (pages 877–889):
• Show what parmetrisations you use and what iterated integrals you evaluate: 1, 5, 8, 11, 16, 17, 19, 23, 25, 34, 35, 37, 41;
• Show what integrals you evaluate: 45.
• Additional extra-credit problem: Show at least one intermediate step: Show that the surface-area element of the surface of revolution given by r = f(z) in cylindrical coordinates is dσ = f(z) √(1 + f′(z)²) dz dθ.
15. Conservative vector fields and exact differential forms:
• Date assigned: June 4 Thursday.
• Date due: June 11 Thursday.
• Pages in the textbook: 847–856.
• Problems from §15.3 (pages 856–858):
• Show what calculations you make to check: 1, 3, 6;
• Show what integrals you take: 7, 8, 11;
• Show what numerical calculations you make: 14, 17, 21;
• Explain: 25.
16. Stokes Theorems:
• Date assigned: June 11 Thursday.
• Date due: Never.
• Pages in the textbook: 858–867, 889–898, 900–909, my handout on exterior differential forms.
• Problems from §15.4 (pages 867–869): Show what integrals you evaluate: 1, 4, 7, 9, 12, 15, 21, 24, 26, 33.
• Problems from §15.7 (pages 898–900):
• Show what integrals you evaluate: 1, 3, 5, 6, 9, 14, 17;
• Show what calculations you make: 19, 26.
• Problems from §15.8 (pages 909–911):
• Show what calculations you make: 1, 2;
• Show what integrals you evaluate: 6, 7, 8, 13;
• Explain: 17.
That's it!
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