# Problem sets and quizzes

Almost every week, there will be a quiz during the last hour of the class period, closely based on an associated problem set. (The day of the week will vary, so check the dates below carefully. Also, there is a final exam on the last day of the term.) Unless otherwise specified, all exercises in the problem sets are from the 3rd Edition of University Calculus: Early Transcendentals by Hass et al published by Addison–Wesley (Pearson).

Here are the quizzes and their associated problem sets (Quiz 1, Quiz 2, Quiz 3, Quiz 4, Quiz 5, Quiz 6, Quiz 7); but anything whose assigned date is in the future is subject to change!

1. Vectors and curves:
• Date taken: January 15 Tuesday.
• Exercises from the Chapter 11 Practice Exercises (pages 638&639): 19, 25, 29, 31, 35, 37, 43, 50.
• Exercises from §15.1 (page 826): 1–8.
• Exercises from §12.1 (pages 648–650): 1, 4, 6, 7, 11, 14, 15, 17, 19, 20, 23.
• Exercises from §12.2 (pages 654–656): 1, 4, 6, 12, 15, 17, 21, 22.
• Exercises from §12.3 (page 660): 1, 5, 8, 9, 11, 15, 18.
• Extra-credit essay question: Explain your background in mathematics and what you are going to use this course for.
2. Functions of several variables:
• Date taken: January 29 Tuesday.
• Exercises from §13.1 (pages 682–684): 3, 4, 7, 9, 10, 15, 17, 18, 19, 23, 24, 30, 31–36, 39, 40, 42, 52, 54, 59, 62.
• Exercises from §13.2 (pages 690–693): 2, 6, 11, 18, 23, 28, 31, 32, 36, 39, 43, 46, 55.
• Exercises from §15.2 (pages 838–840): 5, 6, 39, 41, 43.
• Additional exercises (you must attempt these to get full credit for the problem set):
1. Given α = 3x dx + 4x2y dy, evaluate α at (x, y) = (2, 6) along ⟨dx, dy⟩ = ⟨0.003, 0.005⟩. (Answer.)
2. Given α = 2xy dx + 2yz dy + 2xz dz, evaluate α at (x, y, z) = (−1, 3, 2) along ⟨dx, dy, dz⟩ = ⟨0.01, 0.02, −0.01⟩.
3. Given β = x2 dx + xy dy + xz dz, evaluate β at (x, y, z) = (4, 3, −2). (Answer.)
4. Given β = 5x2 dx − 3xy dy, evaluate β at (x, y) = (1, 2).
• Exercises from §13.3 (pages 702–704): 3, 4, 10, 12, 24, 26, 30, 39, 43, 46, 55, 57, 75, 82, 91.
• Additional extra-credit exercise: Prove that the two definitions of continuity on page 24 of my notes are equivalent. Actually, just do the direction that is not incredibly difficult: Given a function f of several variables and a point P0, show that if the conditions in the second definition of the continuity of f at P0 (the one in terms of ε and δ in the third paragraph on page 24) are met, then the conditions in the first definition (the one in terms of the continuity of composite functions in the second paragraph on page 24) must also be met (but don't try to prove it the other way around). To provide the link between these definitions, you will need to refer to the ε-δ definition of continuity for an ordinary function of one variable; see the definition about two-thirds of the way down page 8 in my notes from last term if you don't know precisely how that goes.
3. Applications of differentiation:
• Date taken: February 5 Tuesday.
• Exercises from §13.5 (pages 720&721): 2, 3, 7, 8, 14, 15, 16, 20, 23, 28.
• Exercises from §15.2 (pages 838–840): 1, 4.
• Exercises from §13.6 (pages 727–730): 3, 6, 10, 13, 14.
• Exercises from §13.7 (pages 737–739): 2, 7, 9, 15, 27, 32, 34, 37, 43, 52, 57.
• Exercises from §13.8 (pages 746–748): 1, 5, 10, 11, 16, 23, 29.
• Additional extra-credit exercise: Suppose that x = 4r cos θ, y = 4r sin θ, z = 3r, u = x2 + y2 + z2, and v = xy + xz + yz.
1. Find the matrix d(x, y, z)/d(r, θ) as a function of r and θ.
2. Find the matrix d(u, v)/d(x, y, z) as a function of x, y, and z, then express this as a function of r and θ.
3. Express u and v as functions of r and θ, then find the matrix d(u, v)/d(r, θ) as a function of r and θ.
4. Transitional topics:
• Date taken: February 13 Wednesday.
• Exercises from §13.6 (pages 727–730): 19, 21, 29, 30, 33, 35, 39, 50, 54.
• Exercises from §15.2 (pages 838–840): 10, 11, 14, 16, 17.A&B, 19, 22, 23, 24, 29.
• Exercises from §15.1 (pages 826–828): 10, 13, 16, 22, 30, 35.
• Exercises from §14.1 (pages 759&760): 3, 7, 10, 17, 22, 27.
• Exercises from §14.2 (pages 767–769): 19, 23.
• Additional extra-credit exercise: 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 quadratic polynomial (that is one whose degree is at most 2).
5. Multiple integrals:
• Date taken: February 25 Monday.
• Exercises from §14.2 (pages 767–769):
• 1, 2, 7, 9, 12, 14, 17, 35, 41, 47, 51, 57, 61, 82;
• Extra credit: 80.
• Exercises from §14.5 (pages 785–788): 3, 6, 9, 15, 21, 25, 29, 34, 37.
• Exercises from §14.3 (page 772): 1, 4, 7, 12, 13, 14, 17, 20, 21.
• Exercises from §14.6 (pages 793–795): 3, 14, 19, 25, 29.
6. General higher-dimensional integration:
• Date taken: March 5 Tuesday.
• Exercises from §14.8 (pages 814–816): 1, 3, 7, 9, 15.
• Exercises from §14.4 (pages 777–786): 1, 3, 5, 7, 9, 17, 20, 23, 24, 28, 29, 34, 37.
• Exercises from §14.7 (pages 803–806): 1, 2, 8, 12, 14, 23, 37, 43, 46, 57, 77.
• Exercises from §15.5 (pages 872–874): 2, 3, 6, 9, 13, 20, 23.
• Exercises from §15.6 (pages 883–884): 1, 5, 8, 11, 16, 17, 19, 23, 25, 34, 35, 37, 41, 45.
• Additional extra-credit exercise: Consider the surface given by r = f(z) in cylindrical coordinates, where f is a differentiable function defined on the interval [a, b]. Use the methods of §6.6 of my notes (or §15.5 of the textbook) to show that the area of this surface is 2π ∫abf(z) √(f′(z)2 + 1) dz.
7. The Stokes theorems:
• Date taken: March 14 Thursday.
• Exercises from §15.3 (pages 849–851): 1, 3, 6, 7, 8, 11, 14, 17, 21, 25.
• Exercises from §15.4 (pages 861–863): 1, 4, 7, 9, 12, 15, 21, 24, 26, 33.
• Exercises from §15.7 (pages 895–897): 1, 3, 5, 6, 9, 14, 17, 21, 28.
• Exercises from §15.8 (pages 906–908): 1, 2, 6, 7, 8, 13, 17.
• Additional extra-credit exercise: Suppose that F is a conservative vector field defined on all of 3-dimensional space; then there exists a scalar field f such that F = ∇f. Let U = −f. In physics, if F is a force field, then we call U a potential energy field for F. Recall that, if an object travels along a curve C in the force field F, then the work done on that object by that force field, or in other words the energy transferred to that object by that force field, is the integral ∫PCF(P) ⋅ dP (or ∫CF ⋅ dr for short). If the curve C begins at the point P1 and ends at the point P2, then use that F = −∇U to express the value of this work using values of the scalar field U at those points. If you imagine that U(P) is the amount of ‘potential’ energy held by an object at P by virtue of its position within this force field, then check that the amount of energy transferred to the object by the field (the work) is the opposite of the change in the object's potential energy. (In other words, we have conservation of energy: the total change in energy is zero. This conservation is why conservative vector fields are called ‘conservative’.)
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