Readings and homework

Here are the assigned readings and problems (Reading 1, Reading 2, Reading 3, Reading 4, Reading 5, Reading 6, Reading 7, Reading 8, Reading 9, Reading 10, Reading 11, Reading 12, Reading 13, Reading 14, Reading 15, Reading 16, Reading 17, Reading 18, Reading 19, Reading 20, Reading 21, Reading 22, Reading 23, Reading 24, Reading 25, Reading 26, Reading 27, Reading 28, Reading 29, Reading 30, Reading 31, Reading 32, Reading 33, Reading 34, Reading 35, Reading 36, Reading 37, Reading 38).

1. Review:
   • Date assigned: January 5 Thursday.
   • Date due: January 6 Friday.
   • Reading from the textbook: Review at least Sections 5.4 and 5.5.
   • Problems due:
     1. If F′(x) = f(x) for all x, then what is ∫ f(x) dx, that is the indefinite integral of f(x) with respect to x?
     2. If ∫ f(x) dx = F(x) + C, then what is ∫^b_x=a f(x) dx, that is the definite integral of f(x) with respect to x as x runs from a to b?
2. Work:
   • Date assigned: January 6 Friday.
   • Date due: January 9 Monday.
   • Reading from the textbook: Pages 376–380 (§6.5).
   • Problems due:
     1. Suppose that a variable force is applied on object moving from a to b along the x-axis, such that the amount of the force in the direction of the positive x-axis is a function F of the object's position along the x-axis. Write down an integral for the work done on the object by that force.
     2. Suppose that a spring with a spring constant k is stretched or compressed a distance x from its equilibrium length. Write down a formula for the force of restoration on the spring.
3. Moments:
   • Date assigned: January 9 Monday.
   • Date due: January 10 Tuesday.
   • Reading from the textbook: Pages 382–389 (§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.
4. Differential equations:
   • Date assigned: January 10 Tuesday.
   • Date due: January 11 Wednesday.
   • Reading from the textbook: Pages 403–409 (§7.2).
   • Handout: Differential equations.
   • Problems due: Fill in the blanks with vocabulary words:
     1. An equation with differentials or derivatives in it is a(n) _____ equation.
     2. A quantity is growing _____ if it is growing in such a way that its rate of growth is proportional to its size.
5. Integration by parts:
   • Date assigned: January 12 Thursday.
   • Date due: January 13 Friday.
   • Reading from the textbook: Pages 423–427 (§8.1).
   • Handout: 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?
6. Trigonometric integration:
   • Date assigned: January 13 Friday.
   • Date due: January 18 Wednesday.
   • Reading from the textbook: Pages 429–434 (§8.2).
   • Problem due: To integrate sin²x (with respect to x), what trigonemetric identity would you use? That is, sin²x = _____?
7. Trigonometric substitution:
   • Date assigned: January 18 Wednesday.
   • Date due: January 19 Thursday.
   • Reading from the textbook: Pages 435–438 (§8.3).
   • Problem due: Give a trigonometric substitution that will help to integrate (x² + 25)^1/2 with respect to x.
8. Partial fractions:
   • Date assigned: January 18 Wednesday.
   • Date due: January 19 Thursday.
   • Reading from the textbook: Pages 440–445 (§8.4).
   • Problem due: If a rational expression has the denominator (x + 3)(2x − 1)², then what will be the denominators of its partial fractions?
9. Integration using computers and tables:
   • Date assigned: January 20 Friday.
   • Date due: January 23 Monday.
   • Reading from the textbook:
     • Read pages 447–451 (§8.5);
     • Glance over pages T1–T6 (the table of integrals).
   • Optional: Look at these some or all of these computer algebra systems that will do integrals (as well as much more):
     • Sage (free to download, free for most uses online, $7.00 per month to make intensive computations online);
     • Maxima (free to download);
     • Maple ($99.00 to download);
     • Wolfram Mathematica ($150.00 to download, $75.00 per year online);
     • Wolfram Alpha (free for some uses online, $4.75 per month for advanced features, $2.99 for a smartphone app with intermediate features).
   • 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.)
10. Numerical integration:
    • Date assigned: January 23 Monday.
    • Date due: January 24 Tuesday.
    • Reading from the textbook: Pages 452–459 (§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?
11. Improper integrals:
    • Date assigned: January 24 Tuesday.
    • Date due: January 25 Wednesday.
    • Reading from the textbook: Pages 462–467 (all of §8.7 except for Tests for Convergence and Divergence).
    • Problem due: Consider the integral ∫^∞_x=−∞ (x² + |x|^1/2)⁻¹ dx, that is the integral of (x² + |x|^1/2)⁻¹ dx as x runs from −∞ to ∞. List all of the reasons why this integral is improper.
12. Infinite sequences:
    • Date assigned: January 25 Wednesday.
    • Date due: January 26 Thursday.
    • Reading from the textbook: Pages 478–487 (§9.1).
    • Handout: Sequences and series, page 1 and the top of page 2.
    • Problems due: Exercises 2 and 4 from §9.1 (page 487).
13. Infinite series:
    • Date assigned: January 26 Thursday.
    • Date due: January 27 Friday.
    • Reading from the textbook:
      • From page 490 through the top of page 492 (§9.2, introduction);
      • From Example 6 on page 494 through page 497 (all of the rest of §9.2 except for Geometric Series).
    • Handout: Sequences and series the rest of page 2 through the top of page 4.
    • 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(n) ___.
      2. Σ^∞_n=0 a_n, the sum of a_n as n runs from zero to infinity, is the sum of a(n) ___.
14. Evaluating special series:
    • Date assigned: January 30 Monday.
    • Date due: January 31 Tuesday.
    • Handout: Sequences and series, the rest of page 4 and the first half of page 5.
    • Reading from the textbook: The rest of page 492 and through the end of Example 5 on page 494 (§9.2, Geometric series).
    • Problems due: Finish these formulas and attach any conditions necessary for them to be true:
      1. The sum of b_n+1 − b_n as n runs from i to infinity (where i is a natural number and b is an infinite sequence of real numbers): Σ^∞_n=i (b_n+1 − b_n) = _____.
      2. The sum of rⁿ as n runs from i to infinity (where i is a natural number and r is a real number): Σ^∞_n=i rⁿ = _____.
15. The Integral Test:
    • Date assigned: January 31 Tuesday.
    • Date due: February 1 Wednesday.
    • Reading from the textbook: Pages 499–504 (§9.3).
    • Handout: Sequences and series, the rest of page 5 through the top third of page 7.
    • 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?
16. Comparison tests for integrals:
    • Date assigned: February 1 Wednesday.
    • Date due: February 2 Thursday.
    • Reading from the textbook: The rest of page 468 and through page 470 (§8.7: Tests for Convergence and Divergence).
    • 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?
17. Comparison tests for series:
    • Date assigned: February 2 Thursday.
    • Date due: February 3 Friday.
    • Reading from the textbook: Pages 506–509 (§9.4).
    • Handout: Sequences and series, the middle half of page 7.
    • Problems due: 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?
18. Absolute convergence:
    • Date assigned: February 6 Monday.
    • Date due: February 7 Tuesday.
    • Reading from the textbook:
      • Pages 510&511 (§9.5: introduction).
      • From page 519 through the paragraph following Example 5 on page 520 (§9.6: Conditional Convergence; Rearranging Series).
    • Handout: Sequences and series, the rest of page 7.
    • 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 converges, is it necessarily true that the original series converges?
19. The Ratio and Root Tests:
    • Date assigned: February 7 Tuesday.
    • Date due: February 8 Wednesday.
    • Reading from the textbook: Pages 512–515 (§9.5: The Ratio Test; The Root Test).
    • Handout: Sequences and series, the top half of page 8.
    • Problems due:
      1. Under what circumstances does the Ratio Test not tell you whether a series converges?
      2. Under what circumstances 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?
20. Alternating series:
    • Date assigned: February 8 Wednesday.
    • Date due: February 9 Thursday.
    • Reading from the textbook:
      • Pages 516–518 (§9.6: introduction);
      • The rest of page 520 (§9.6: Summary of Tests).
    • Handout: Sequences and series, the rest of page 8.
    • Problems due: Identify which of these series are alternating:
      1. The sum of (−1)ⁿ/(2 + n);
      2. The sum of (2 + (−1)ⁿ)/n;
      3. The sum of (cos n)/n.
21. Power series:
    • Date assigned: February 9 Thursday.
    • Date due: February 10 Friday.
    • Reading from the textbook: Pages 522–530 (§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
22. Taylor polynomials:
    • Date assigned: February 13 Monday.
    • Date due: Feburary 14 Tuesday.
    • Handout: Taylor series, from page 1 through the first half of page 3.
    • Reading from the textbook:
      • Pages 534–536 (§9.8: Taylor Polynomials);
      • From page 537 through Example 3 on page 540 (§9.9: introduction, Estimating the Remainder).
    • 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.
23. Taylor series:
    • Date assigned: Feburary 14 Tuesday.
    • Date due: February 15 Wednesday.
    • Handout: Taylor series, the rest of page 3.
    • Reading from the textbook: Pages 532 to 534 (the rest of §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 is 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.
24. The Binomial Theorem:
    • Date assigned: February 15 Wednesday.
    • Date due: February 16 Thursday.
    • Reading from the textbook: From page 543 through Example 2 on page 545 (§9.10: introduction; The Binomial Series for Powers and Roots).
    • Handout: Taylor series, the first line in the list on page 4 and the paragraph about it in the middle of page 4.
    • 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).
25. More common Taylor series:
    • Date assigned: February 16 Thursday.
    • Date due: February 17 Friday.
    • Reading from the textbook:
      • Pages 540&541 (§9.9: Using Taylor Series);
      • Pages 545–548 (the rest of §9.10).
    • Handout: the rest of Taylor series.
    • 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.
26. Graphs in three dimensions:
    • Date assigned: February 20 Monday.
    • Date due: February 21 Tuesday.
    • Reading from the textbook: Pages 596–599 (§11.1).
    • Handout: Vectors, the first half of page 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)?
27. Vectors:
    • Date assigned: February 21 Tuesday.
    • Date due: February 22 Wednesday.
    • Handout: Vectors, the rest of page 1 and through the first half of page 5.
    • Reading from the textbook:
      • From page 601 to the very top of page 605 (§11.2: introduction; Component Form; Vector Algebra Operations);
      • The bottom half of page 606 (§11.2: Midpoint of a Line Segment).
    • Problems due:
      1. Give a formula for the vector from the point (x₁, y₁) to the point (x₂, y₂).
      2. If u, v, and w are vectors, simplify the expression 2(u + 3v) − 6(v − 3w) − 18(w + u/9).
28. Length and angle:
    • Date assigned: February 22 Wednesday.
    • Date due: February 23 Thursday.
    • Handout: Vectors, the rest of page 5 and through the top of page 7.
    • Reading from the textbook:
      • Pages 605&606 (§11.2: Unit Vectors);
      • Page 607&608 (§11.2: Applications).
    • Problems due:
      1. Give a formula for the magnitude (or norm, or length) of the vector ⟨a, b, c⟩.
      2. If |u + v|² = |u|² + |v|², then what is the angle between u and v? (Hint: Use the Pythagorean Theorem, the Law of Cosines, or the formula for angles in terms of lengths in the handout; any one of these could do the job! You might also want to draw a picture.)
29. The dot product:
    • Date assigned: February 23 Thursday.
    • Date due: February 24 Friday.
    • Handout: Vectors, the rest of page 7 and through the top half of page 10.
    • Reading from the textbook: Pages 610–615 (§11.3).
    • 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).
30. The cross product:
    • Date assigned: February 24 Friday.
    • Date due: February 27 Monday.
    • Handout: Vectors, the rest of page 10 and through the top of page 14.
    • Reading from the textbook: Pages 618–622 (§11.4).
    • 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?
31. Geometry with vectors:
    • Date assigned: February 28 Tuesday.
    • Date due: March 1 Wednesday.
    • Reading from the textbook: Pages 624–630 (§11.5).
    • Handout: Vectors, the rest of page 14.
    • Problems due:
      1. Give a parametrization for the line through the point (x₀, y₀, z₀) and parallel to the vector ⟨a, b, c⟩.
      2. Give an equation for the plane through the point (x₀, y₀, z₀) and perpendicular to the vector ⟨a, b, c⟩.
32. Parametrized curves:
    • Date assigned: March 1 Wednesday.
    • Date due: March 2 Thursday.
    • Reading from the textbook: Pages 557–562 (§10.1).
    • Handout: Parametrized curves:
      • Page 1 and the very top of page 2;
      • Optional: the rest of page 2.
    • 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?
33. Calculus with parametrized curves:
    • Date assigned: March 2 Thursday.
    • Date due: March 3 Friday.
    • Reading from the textbook: From page 564 through end of Example 3 on page 566 (§10.2: introduction; Tangents and areas).
    • Handout: Parametrized curves, pages 3 and 4.
    • 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!)
34. Arclength of parametrized curves:
    • Date assigned: March 3 Friday.
    • Date due: March 6 Monday.
    • Reading from the textbook: Pages 566 to 572 (the rest of §10.2).
    • Handout: Parametrized curves, page 5.
    • 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?
35. Polar coordinates:
    • Date assigned: March 7 Tuesday.
    • Date due: March 8 Wednesday.
    • Reading from the textbook: Pages 574–577 (§10.3).
    • Problems due: True or false:
      1. If a point in the coordinate plane is given by polar coordinates (r, θ), then r ≥ 0 and 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.
36. Graphs in polar coordinates:
    • Date assigned: March 8 Wednesday.
    • Date due: March 9 Thursday.
    • Reading from the textbook: Pages 578–580 (§10.4).
    • Problems due: Suppose that a curve is parametrized in polar coordinates by r = f(θ) for some differentiable function f. (In the following answers, refer directly to only f, its derivatives, and θ.)
      1. What is the slope of the curve at a given value of θ?
      2. Under what circumstances is this slope undefined?
37. Area in polar coordinates:
    • Date assigned: March 9 Thursday.
    • Date due: March 10 Friday.
    • Reading from the textbook: Pages 581&582 and most of page 583 (§10.5: introduction; 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.)
38. Length in polar coordinates:
    • Date assigned: March 10 Friday.
    • Date due: March 13 Monday.
    • Reading from the textbook: The rest of page 583 and page 584 (§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).)

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