Problem sets and quizzes

Here are the quizzes and their associated problem sets:

1. Quiz 1:
   • Date taken: April 7 Thursday.
   • 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.
   • Extra-credit essay question: Explain your background in mathematics and what you are going to use this course for.
   • Problems from §15.1 (pages 832–834): No additional work needed: 1–8.
   • 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 §12.3 (pages 667&668): Show what one-variable integrals you evaluate: 1, 5, 8, 9, 11, 15, 18.
2. Quiz 2:
   • Date taken: April 14 Thursday.
   • 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: 17, 18, 19, 23, 24, 30, 31–36, 39, 40, 42;
     • State the value of c used: 52, 54, 59, 62.
   • 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.
   • Additional extra-credit problem: Prove that the two definitions of continuity on page 1 of my handout on definitions (one in terms of the continuity of composite functions and the other in terms of ε and δ) are equivalent. (In other words: Given a function f of several variables and a point P₀, show that if the conditions in the first definition of the continuity of f at P₀ are met, then the conditions in the second definition must also be met, and the other way around.)
   • Problems from §15.2 (pages 844–847):
     • No additional work needed: 5, 6, 39–44.
   • Additional problems (you must attempt these to get full credit for the problem set): Show what numerical calculations you make:
     1. Given α = 3x dx + 4x²y dy, evaluate α at (x, y) = (2, 6) along ⟨dx, dy⟩ = ⟨0.003, 0.005⟩, that is along dr = 0.003i + 0.005j. (Hint: Your final answer will simply be a number.)
     2. Given β = x² dx + xy dy + xz dz, evaluate β at (x, y, z) = (4, 3, −2). (Hint: Your final answer will not simply be a number.)
3. Quiz 3:
   • Date taken: April 21 Thursday.
   • 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;
     • Show what limits you take to find the partial derivatives, and state why the function is not continuous: 91.
   • 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.
   • Additional extra-credit problem: If you don't know about matrices (at least how to multiply them), read http://www.mathsisfun.com/algebra/matrix-multiplying.html first. Now, if you have m functions of n variables each, then you can put their partial derivatives into an m-by-n matrix; for example, if you have 2 functions of 3 variables each, say u = f(x, y, z) and v = g(x, y, z), then the partial derivatives fit into a 2-by-3 matrix:

     ∂u/∂x	∂u/∂y	∂u/∂z
     ∂v/∂x	∂v/∂y	∂v/∂z

     We may call this matrix d(u, v)/d(x, y, z).
     1. If you have an ordinary function y = f(x), think of this as a group of only 1 function of only 1 variable each, so that d(y)/d(x) in the notation above is a 1-by-1 matrix, consisting of a single entry. Check that the usual derivative dy/dx is this entry. (You don't have to even write anything down for this if you don't want to, but make sure that it makes sense before you go on.)
     2. If you have a parametrized curve in 3 dimensions, say P = (x, y, z) = (f(t), g(t), h(t)), think of this as a group of 3 functions of only 1 variable each, so that d(x, y, z)/d(t) is a 3-by-1 matrix, consisting of a single column with 3 entries. Check that the components of the velocity vector dP/dt = dr/dt are the same as the entries of this matrix. (For this reason, ordinary vectors that represent change of position are sometimes called column vectors.)
     3. If you have a function of 3 variables, say u = F(x, y, z), think of this as a group of only 1 function of 3 variables each, so that d(u)/d(x, y, z) is a 1-by-3 matrix, consisting of a single row with 3 entries. Check that the components of the gradient vector ∇F(x, y, z) are the same as the entries of this matrix. (For this reason, vectors such as gradients that represent change with respect to position are sometimes called row vectors.)
     4. If you have both (x, y, z) = (f(t), g(t), h(t)) and u = F(x, y, z), then composition makes u an ordinary function of t. Show that d(u)/d(t) = d(u)/d(x, y, z) d(x, y, z)/d(t), using matrix multiplication, and check that this matches the defining property of the gradient from page 3 of my handout on definitions.
     5. If you have both r = ⟨f(u), g(u), h(u)⟩ and u = F(x, y, z), then composition in the other order makes a vector field. Use matrix multiplication to get a 3-by-3 matrix; this matrix is called the total derivative of the vector field. (By the way, even if a vector field is not obtained as a composite in this way, it still has a matrix-valued function like this as its total derivative.)
   • 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;
     • Show the gradient, the differential, or the partial derivatives: 28.
   • Problems from §15.2 (pages 844–847): Show at least one intermediate step: 1, 4.
   • Problems from §13.6 (pages 737–739):
     • Show the gradient, the differential, or the partial derivatives: 3, 6, 10, 13, 14.
4. Quiz 4:
   • Date taken: April 28 Thursday.
   • 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 quadratic polynomial (that is one whose degree is at most 2).
   • 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.
5. Quiz 5:
   • Date taken: May 5 Thursday.
   • 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 §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.
   • Additional extra-credit problem: Suppose that F is a conservative vector field; in other words, 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 ∫_C F(P) ⋅ dr. If the curve C begins at the point P₁ and ends at the point P₂, then express the value of this work using values of the scalar field U. 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 vectors fields are called ‘conservative’.)
6. Quiz 6:
   • Date taken: May 12 Thursday.
   • 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;
     • Extra credit: 80.
   • 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 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;
     • Show what integrals you evaluate: 37.
7. Quiz 7:
   • Date taken: May 19 Thursday.
   • Problems from §14.6 (pages 800–802): Show what integrals you evaluate: 3, 14, 19, 25, 29.
   • Extra-credit problem from §14.8 (pages 821–823): Show the integral in u and v that you evaluate: 16.
   • 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.
8. Quiz 8:
   • Date taken: May 26 Thursday.
   • Problems from §15.5 (pages 878–880):
     • No additional work needed: 2, 3, 6, 9, 13;
     • Show what integrals you evaluate: 20, 23.
   • Additional extra-credit problem: 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 §15.5 (or the handout) to show that the area of this surface is 2π ∫_a^b f(z) (f′(z)² + 1)^1/2 dz.
   • Problems from §15.6 (pages 877–889):
     • Show what parametrizations you use and what iterated integrals you evaluate: 19, 23, 25, 34, 35, 37, 41;
     • Show what parametrizations you use and what iterated integrals you evaluate: 1, 5, 8, 11, 16, 17;
     • Show what integrals you evaluate: 45.
9. Quiz 9:
   • Date taken: June 6 Monday.
   • 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.
     • Extra-credit: 31.

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