Readings and homework

Here are the assigned readings and exercises:

1. Work:
   • Date assigned: January 6 Wednesday.
   • Date due: January 7 Thursday.
   • Textbook: Pages 383–386 (§6.5).
   • Problems due: none (this time only).
2. Moments:
   • Date assigned: January 7 Thursday.
   • Date due: January 11 Monday.
   • Textbook: Pages 389–396 (§6.6).
   • Problems due:
     1. If you wish to find the total mass of the plate described in Exercise 6.6.15, will you do an integral with respect to x or an integral with respect to y?
     2. Give the formula for the centre of mass (x̄, ȳ) in terms of the total mass M and the moments M_x and M_y.
3. Differential equations:
   • Date assigned: January 11 Monday.
   • Date due: January 12 Tuesday.
   • Textbook: Pages 412–417 (§7.2).
   • Handout: Differential equations.
   • Problems due (each just needs one vocabulary word, an adjective or adverb):
     1. What kind of equation has differentials or derivatives in it?
     2. If a quantity is growing over time in such a way that its rate of growth is proportional to its size, then how is it growing?
4. Integration by parts:
   • Date assigned: January 12 Tuesday.
   • Date due: January 15 Friday.
   • Textbook: Pages 432–436 (§8.1).
   • Online notes: Integration by parts.
   • Problems due:
     1. To integrate ∫ x e^x dx by parts, what should be u and what should be dv?
     2. To integrate ∫ x ln x dx by parts, what should be u and what should be dv?
5. Advanced integration techniques:
   • Date assigned: January 14 Thursday.
   • Date due: January 19 Tuesday.
   • Textbook:
     • Pages 439–443 (§8.2);
     • Pages 444–447 (§8.3);
     • Pages 448–454 (§8.4).
   • Problems due:
     1. To integrate sin²x, what trigonemetric identity would you use? That is, sin²x = …?
     2. Give a trigonometric substitution that will help to integrate 1/(x² + 25).
     3. If a rational expression has the denominator x² + 7x + 12, then what will be the denominators of its partial fractions?
6. Integration using computers and tables:
   • Date assigned: January 19 Tuesday.
   • Date due: January 20 Wednesday.
   • Textbook:
     • Read pages 456–459 (§8.5);
     • Glance over pages T1–T6 (the table of integrals).
   • Here are some links to computer algebra systems that will do integrals (as well as much more):
     • Sage (free to download, free to use online, $7.00 per month or more for advanced fatures);
     • Maxima (free to download);
     • Maple ($99.00 to download for students);
     • Wolfram Mathematica ($45.00 per semester to download for students);
     • Wolfram Mathematica Online Integrator (free to use online, indefinite integrals only);
     • Wolfram Alpha (free for some uses online, $5.49 per month for advanced features, or $2.99 for a smartphone app).
   • Problem due: Which entry in the table of integrals in the back of the book (pages T1–T6) tells you how to integrate ∫ (x² + 3)^−1/2 dx? (Hints: That table doesn't use fractional exponents; it uses roots instead. And it doesn't use negative exponents; it uses fractions instead.)
7. Numerical integration:
   • Date assigned: January 20 Wednesday.
   • Date due: January 21 Thursday.
   • Textbook: Pages 461–468 (§8.6).
   • Problems due:
     1. Which numerical method of integration approximates a function with a piecewise linear function?
     2. Which numerical method of integration approximates a function with a piecewise quadratic function?
8. Improper integrals:
   • Date assigned: January 22 Friday.
   • Date due: January 25 Monday.
   • Textbook: Pages 471–475 (the first half of §8.7).
   • Problems due: Consider the integral ∫_−∞^∞ (x² + |x|^1/2)⁻¹ dx, the integral of (x² + |x|^1/2)⁻¹ dx as x runs from −∞ to ∞.
     1. Explain why this integral is improper. (There are several possible answers for this, and it's best if you know them all, but one is enough for full credit.)
     2. Write this integral as a sum of limits of proper Riemann integrals. (Don't worry about whether these limits exist.)
9. Infinite sequences:
   • Date assigned: January 25 Monday.
   • Date due: January 26 Tuesday.
   • Textbook: Pages 486–495 (§9.1).
   • Handout: Sequences and series, page 1 and the top of page 2.
   • Problems due: Problems 9.1.2 and 9.1.4 from page 495.
10. Infinite series:
    • Date assigned: January 26 Tuesday.
    • Date due: January 27 Wednesday.
    • Textbook: Pages 498&499, pages 502–504 (most of §9.2).
    • Handout: Sequences and series, pages 2&3.
    • Problems due: Fill in the vocabulary words:
      1. lim_n→∞ a_n, the limit of a_n as n goes to infinity, is the limit of a ___.
      2. Σ^∞_n=0 a_n, the sum of a_n as n runs from zero to infinity, is the sum of a ___.
11. Evaluating special series:
    • Date assigned: January 27 Wednesday.
    • Date due: January 28 Thursday.
    • Textbook: Pages 500&501 (§9.2, Geometric series).
    • Problems due: Finish these formulas and attach any conditions necessary for them to be true:
      1. The sum of rⁿ as n runs from zero to infinity: Σ^∞_n=0 rⁿ = ⋯.
      2. The sum of b_n+1 − b_n as n runs from zero to infinity: Σ^∞_n=0 (b_n+1 − b_n) = ⋯.
12. The Integral Test:
    • Date assigned: January 29 Friday.
    • Date due: February 1 Monday.
    • Textbook: Pages 507–510 (§9.3).
    • Handout: Convergence tests, pages 1&2 except for the bottom section of page 2.
    • Problems due:
      1. Does the Integral Test apply to the function f(x) = sin²(πx)? Why or why not?
      2. For which values of p does Σ^∞_n=1 (1/n^p), the sum of 1/n^p as n runs from 1 to infinity, converge?
13. Comparison tests for integrals:
    • Date assigned: February 1 Monday.
    • Date due: February 4 Thursday.
    • Textbook: Pages 476–478 (the second half of §8.7).
    • Handout: Convergence tests, skim the bottom section of page 2 and the top section of page 3.
    • Problems due: Suppose that you want to know whether ∫^∞_x=1 (x − 1)/x² dx, the infinite integral of (x − 1)/x², converges.
      1. Knowing that the infinite integral of 1/x² converges, can you use the Direct Comparison Test to decide?
      2. Knowing that the infinite integral of 1/x² converges, can you use the Limit Comparison Test to decide?
      3. Knowing that the infinite integral of 1/x diverges, can you use the Direct Comparison Test to decide?
      4. Knowing that the infinite integral of 1/x diverges, can you use the Limit Comparison Test to decide?
14. Comparison tests for series:
    • Date assigned: February 4 Thursday.
    • Date due: February 5 Friday.
    • Textbook: Pages 512–515 (§9.4).
    • Handout: Convergence tests, read the bottom section of page 2 and the top two sections of page 3.
    • Problems due (optional): Suppose that you want to know whether Σ^∞_n=1 (n − 1)/n², the infinite series of (n − 1)/n², converges.
      1. Knowing that the infinite series of 1/n² converges, can you use the Direct Comparison Test to decide?
      2. Knowing that the infinite series of 1/n² converges, can you use the Limit Comparison Test to decide?
      3. Knowing that the infinite series of 1/n diverges, can you use the Direct Comparison Test to decide?
      4. Knowing that the infinite series of 1/n diverges, can you use the Limit Comparison Test to decide?
15. Alternating series:
    • Date assigned: February 5 Friday.
    • Date due: February 8 Monday.
    • Textbook: From page 522 to the top half of page 524 (the first half of §9.6).
    • Handout: Convergence tests, the middle of page 3.
    • Problems due: Identify which of these series are alternating:
      1. The sum of (−1)ⁿ/(1 + n);
      2. The sum of (1 + (−1)ⁿ)/n;
      3. The sum of cos n.
16. The Ratio and Root Tests:
    • Date assigned: February 8 Monday.
    • Date due: February 9 Tuesday.
    • Textbook: Pages 517–520 (§9.5).
    • Handout: Convergence tests, the bottom section of page 3 and the top section of page 4.
    • Problems due:
      1. Under what circmstances does the Ratio Test not tell you whether a series converges?
      2. Under what circmstances does the Root Test not tell you whether a series converges?
      3. If the Ratio Test doesn't tell you, is it possible that the Root Test will?
17. Absolute convergence:
    • Date assigned: February 9 Tuesday.
    • Date due: February 10 Wednesday.
    • Textbook: From the bottom half of page 524 to page 527 (the second half of §9.6).
    • Handout: Convergence tests, the rest of page 4.
    • Problems due:
      1. If a series converges, is it necessarily true that its series of absolute values also converges?
      2. If the series of absolute values also converges, is it necessarily true that the original series converges?
18. Power series:
    • Date assigned: February 10 Wednesday.
    • Date due: February 11 Thursday.
    • Textbook: Pages 529–536 (§9.7).
    • Problems due: Which of the following are power series (in the variable x)?
      1. Σ^∞_n=0 n²(x − 3)ⁿ
      2. Σ^∞_n=5 (2x − 3)ⁿ
      3. Σ^∞_n=0 (√x − 3)ⁿ
      4. 5 + 7x − 3x³
19. Taylor series:
    • Date assigned: February 12 Friday.
    • Date due: February 15 Monday.
    • Textbook: Pages 538 to 542 (§9.8).
    • Problems due:
      1. Fill in the blank: If f is a function whose derivatives of all orders exist everywhere, then the Maclaurin series generated by f if the Taylor series generated by f at ___.
      2. True or false: Whenever a is a constant, f is a function whose derivatives of all orders exist at a, and the Taylor series generated by f at a converges, then it must converge to f.
20. Taylor polynomials:
    • Date assigned: February 15 Monday.
    • Date due: Feburary 16 Tuesday.
    • Textbook: Page 543 through the top of half of page 546 (the first half of §9.9).
    • Handout: Taylor's Theorem.
    • Problems due:
      1. If a function f is to have a good approximation on an interval by a polynomial of degree at most k, then it's best if its derivative of what order is close to zero on that interval? (Its first derivative, its second derivative, its kth derivative, or what?)
      2. Given a function f and a number a, let R_k be the Taylor remainder of f at a of order k. True or false: If for every number x, the limit, as k → ∞, of R_k(x) is zero, then the Taylor series of f at a must converge everywhere.
21. The Binomial Theorem:
    • Date assigned: February 16 Tuesday.
    • Date due: February 17 Wednesday.
    • Textbook:
      • Page 550 through the middle of page 552 (§9.10, The Binomial Series for Powers and Roots);
      • The bottom of page 553 and the top of page 554 (§9.10, Arctangents).
    • Problems due:
      1. Using the Binomial Theorem, expand (x + 1)⁶.
      2. Using the Binomial Theorem, write (1 + x²)⁻¹ as an infinite series (assuming that x² < 1).
22. More with Taylor series:
    • Date assigned: February 17 Wednesday.
    • Date due: February 18 Thursday.
    • Textbook:
      • The bottom of page 546 and page 547 (§9.9, Using Taylor Series);
      • The bottom of page 552 and the top of page 553 (§9.10, Evaluating Nonelementary Integrals);
      • From the bottom of page 554 to page 556 (the rest of §9.10).
    • Problems due:
      1. Following Exercise 2 from the previous assignment, integrate your answer to get an infinite series for atan x = tan⁻¹ x.
      2. Can any limit using L'Hôpital's Rule be done using Taylor series instead? Explain why or why not.
23. Parametrized curves:
    • Date assigned: February 19 Friday.
    • Date due: February 22 Monday.
    • Textbook: Pages 563–567 (§10.1).
    • Problems due: Define a parametrized curve by (x, y) = (2t², 3t³) for 0 ≤ t ≤ 2.
      1. Which variable(s) is/are the parameter(s)?
      2. What are the beginning/initial point and the ending/final/terminal point of the curve?
24. Calculus with parametrized curves:
    • Date assigned: February 22 Monday.
    • Date due: February 23 Tuesday.
    • Textbook: From page 570 to the top of page 572 (§10.2, Tangents and areas).
    • Online notes: Differentiation of parametrized curves.
    • Problems due: If x and y are each functions of t:
      1. Give a formula for the derivative of y with respect to x in terms of the derivatives of x and y with respect to t. (There is basically only one possible correct answer to this.)
      2. Give a formula for the second derivative of y with respect to x in terms of derivatives of x and y with respect to t. (There is more than one possible correct answer to this, and you only need to give one of them, but make sure that all of the derivatives appearing are with respect to t as required!)
25. Arclength:
    • Date assigned: February 23 Tuesday.
    • Date due: February 24 Wednesday.
    • Textbook: From middle of page 572 to page 577 (the rest of §10.2).
    • Problems due:
      1. If a curve is parametrized by x = f(t) and y = g(t) for a ≤ t ≤ b, then what integral gives the length of this curve?
      2. How does changing the parametrization of a curve affect its arclength?
26. Polar coordinates:
    • Date assigned: February 24 Wednesday.
    • Date due: February 25 Thursday.
    • Textbook: Pages 579–582.
    • Problems due: True or false
      1. When using polar coordinates (r, θ), r must be nonnegative (r ≥ 0) and similarly it must be that 0 ≤ θ < 2π.
      2. Every point in the coordinate plane can be given by polar coordinates (r, θ) such that r ≥ 0 and 0 ≤ θ < 2π.
27. Graphs in polar coordinates:
    • Date assigned: February 26 Friday.
    • Date due: February 29 Monday.
    • Textbook: Pages 584–586 (most of §10.4).
    • Problems due:
      1. If a curve is parametrized in polar coordinates by r = f(θ) for some differentiable function f, then what is the slope of the curve at a given value of θ? (Give a formula that uses only the variable θ, although you can also apply f and its derivatives.)
      2. Under what circumstances is this slope undefined? (Again, express this condition using only the variable θ.)
28. Area in polar coordinates:
    • Date assigned: February 29 Monday.
    • Date due: March 1 Tuesday.
    • Textbook: From page 587 to the top of page 589 (§10.5, Area in the plane).
    • Problems due:
      1. Can the area of a region in the plane ever be negative?
      2. What is the formula for the area of the region satisfying f(θ) ≤ r ≤ g(θ) and α ≤ θ ≤ β in polar coordinates? (Assume that α and β are real numbers with α ≤ β and β − α ≤ 2π, and that f and g are continuous functions defined at least on [α, β] with 0 ≤ f(θ) ≤ g(θ) whenever α ≤ θ ≤ β.)
29. Length in polar coordinates:
    • Date assigned: March 1 Tuesday.
    • Date due: March 2 Wednesday.
    • Textbook: Pages 589&590 (§10.5, Length of a polar curve).
    • Problems due:
      1. Can the length of a curve ever be negative?
      2. What is the formula for the length of the curve given by r = f(θ) and α ≤ θ ≤ β in polar coordinates? (Assume that α and β are real numbers with α ≤ β, and that f is a continuously differentiable function defined at least on [α, β] with (f(θ₁), θ₁) always defining a different point in polar coordinates than (f(θ₂), θ₂) whenever α ≤ θ₁, θ₂ ≤ β and θ₁ ≠ θ₂.)
30. Graphs in three dimensions:
    • Date assigned: March 2 Wednesday.
    • Date due: March 3 Thursday.
    • Textbook: Pages 602–605 (§11.1).
    • Problems due:
      1. In a right-handed rectangular coordinate system using the variables x, y, and z, if you curl the fingers of your right hand from the direction of the positive x-axis to the direction of the positive y-axis and stick out your thumb, then in what direction approximately should your thumb point?
      2. What is the name of a shape whose equation in a three-dimensional rectangular coordinate system is linear and contingent? (For example, 2x + 3y + 5z = 8.)
      3. What is the equation in the rectangular (x, y, z)-coordinate system of a sphere whose radius is r and whose centre is (h, k, l)?
31. Vectors:
    • Date assigned: March 4 Friday.
    • Date due: March 7 Monday.
    • Textbook: Pages 607–612 (most of §11.2).
    • Handout: Vectors, from page 1 to the top of page 5.
    • Problems due:
      1. Give a formula for the vector from the point (x₁, y₁) to the point (x₂, y₂).
      2. Give a formula for the magnitude (or norm, or length) of the vector ⟨a, b, c⟩ = ai + bj + ck.
32. Length and angle:
    • Date assigned: March 7 Monday.
    • Date due: March 8 Tuesday.
    • Textbook: Pages 613&614 (§11.2, Applications).
    • Handout: Vectors, pages 5&6.
    • Problems due:
      1. If u, v, and w are vectors, simplify the expression 2(u + 3v) − 6(v − 3w) − 18(w + u/9).
      2. If |u + v|² = |u|² + |v|², then what is the angle between u and v? (You might want to draw a picture.)
33. The dot product:
    • Date assigned: March 8 Tuesday.
    • Date due: March 9 Wednesday.
    • Textbook: Pages 616–621 (§11.3).
    • Handout: Multiplying vectors, from page 1 to the top of page 4.
    • Problems due:
      1. State a formula for the dot product u ⋅ v of two vectors using only their lengths |u| and |v|, the angle ∠(u, v) between them, and real-number operations.
      2. State a formula for the projection of u onto v using only dot products and real-number operations (so no lengths or angles unless expressed using dot products).
34. The cross product:
    • Date assigned: March 9 Wednesday.
    • Date due: March 10 Thursday.
    • Textbook: Pages 624–628 (§11.4).
    • Handout: Multiplying vectors, pages 4–7.
    • Problems due:
      1. State a formula for the magnitude |u × v| of the cross product of two vectors u and v, using only their lengths |u| and |v|, the angle ∠(u, v) between them, and real-number operations.
      2. If u and v are vectors in 2 dimensions, then is u × v a scalar or a vector?
      3. If u and v are vectors in 3 dimensions, then is u × v a scalar or a vector?
35. Geometry with vectors:
    • Date assigned: March 11 Friday.
    • Date due: March 14 Monday.
    • Textbook: Pages 630–636 (§11.5).
    • Online notes: Linear geometry with vectors.
    • Problems due:
      1. Give a parametrization for the line through the point (x₀, y₀, z₀) and parallel to the vector ⟨a, b, c⟩ = ai + bj + ck.
      2. Give an equation for the plane through the point (x₀, y₀, z₀) and perpendicular to the vector ⟨a, b, c⟩ = ai + bj + ck.

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