MATH-1700-WBP01

Course administration

• Canvas page (where you must log in).
• Help with DjVu (if you have trouble reading the DjVu files on this page).
• Official syllabus (DjVu).
• Course policies (DjVu).
• Class hours: Asynchronous online.
• Final exam: By appointment July 20–24.

Contact information

• Name: Toby Bartels, PhD.
• Canvas messages.
• Email: TBartels@Southeast.edu.
• Voice mail: 1-402-323-3452.
• Text messages: 1-402-805-3021.
• Office hours: By appointment, in room U9C or over Zoom meeting 610-024-2876.

Readings

• The first set, on single-variable Calculus (including material from Calculus 1);
• The second set, on multivariable Calculus (including material for Calculus 3).

Try to read this introduction for the first day of class:

• Objectives:
  • Understand what to expect from this course;
  • Know how to submit assignments.
• Reading: My course policies (DjVu, same as above).
• Reading homework due on May 18 Monday or ASAP thereafter (write this by hand on paper and submit it on Canvas or another way):
  1. If you want to submit something that you've written by hand on paper, how will you send me a picture of it? (submit on Canvas, attach to an email, etc).
  2. How will you get the final exam proctored? (Lincoln Testing Center, ProctorU, etc).
  You can change your mind about these later! (But let me know if you change your mind about #2.)
• Problem set from the textbook due on May 19 Tuesday or ASAP thereafter (submit this through MyLab): O.1.1, O.1.2, O.1.3, O.1.4, O.1.5, O.1.6, O.1.7, O.1.8, O.1.10, O.1.11, O.1.12.

Applications of integration

1. Integration review:
   • Reading from my notes (first set): Review at least Chapter 5 through Section 5.3 (pages 43–46).
   • Reading from the textbook: Review Chapter 5 as needed, at least Sections 5.4 and 5.5.
   • Reading homework due on May 19 Tuesday (write this by hand on paper and submit it on Canvas):
     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?
   • Problem set from the textbook due on May 20 Wednesday (submit this through MyLab): 5.3.9, 5.5.17, 5.5.23, 5.5.55, 5.4.5, 5.4.7, 5.4.9, 5.4.21, 5.6.19, 5.6.29, 5.4.45, 5.4.37.
2. Work:
   • Reading from the textbook: Section 6.5 (pages 386–389).
   • Reading homework due on May 20 Wednesday (write this by hand on paper and submit it on Canvas):
     1. Suppose that a variable force is applied to an 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 (positive) spring constant k is stretched or compressed a (signed) distance x from its equilibrium length. Write down a formula for the force needed to hold the spring in that position.
   • Problem set from the textbook due on May 21 Thursday (submit this through MyLab): 6.5.7, 6.5.9, 6.5.11, 6.5.12, 6.5.15, 6.5.17, 6.5.19, 6.5.23.
3. Moments:
   • Reading from the textbook: Section 6.6 (pages 392–400).
   • Reading homework due on May 21 Thursday (write this by hand on paper and submit it on Canvas):
     1. If you wish to find the total mass of the plate described in Exercise 6.6.24 on page 400 of the textbook, 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.
   • Problem set from the textbook due on May 22 Friday (submit this through MyLab): 6.6.9, 6.6.12, 6.6.13, 6.6.17, 6.6.23, 6.6.33.
4. Differential equations:
   • Reading from my notes (first set):
     • Section 5.4 (page 46);
     • Chapter 6 through Section 6.3 (pages 51–53);
     • Optional: The rest of Chapter 6 (pages 53&54).
   • Reading from the textbook: Section 7.2 (pages 415–422).
   • Reading homework due on May 22 Friday (write this by hand on paper and submit it on Canvas):
     1. Fill in the blanks with vocabulary words:
        1. An equation with differentials or derivatives in it is a(n) _____ equation.
        2. A differential equation of the form dy/dx = g(x) h(y) is called _____.
        3. A quantity is growing _____ if it is growing in such a way that its rate of growth is proportional to its size.
     2. Suppose that a quantity y undergoes exponential growth with a relative growth rate constant of k and an initial value of y₀ at time t = 0. Write down a formula for the value of y as a function of the time t.
   • Problem set from the textbook due on May 26 Tuesday (submit this through MyLab): 7.2.1, 7.2.3, 7.2.5, 7.2.7, 7.2.11, 7.2.13, 7.2.17, 7.2.35, 7.2.37, 7.2.38, 7.2.41.

Advanced integration techniques

5. Integration by parts:
   • Reading from my notes (first set): Section 5.6 (page 48).
   • Reading from the textbook: Chapter 8 through Section 8.1 (pages 436–442).
   • Reading homework due on May 26 Tuesday (write this by hand on paper and submit it on Canvas): Use the formula ∫ u dv = uv − ∫ v du for integration by parts.
     1. To find ∫ x e^x dx, what should be u and what should be dv?
     2. To find ∫ x ln x dx, what should be u and what should be dv?
   • Problem set from the textbook due on May 27 Wednesday (submit this through MyLab): 8.1.3, 8.1.6, 8.1.11, 8.1.17, 8.1.19, 8.1.25, 8.1.41, 8.1.47, 8.1.49, 8.1.67, 8.1.69, 8.1.77.
6. Partial fractions:
   • Reading from the textbook: Section 8.4 (pages 456–461).
   • Reading homework due on May 27 Wednesday (write this by hand on paper and submit it on Canvas): Suppose that a rational expression has the denominator (x² + 3)‍(x + 3)².
     1. What are the denominators of its partial fractions?
     2. For each of these, indicate the maximum degree of the numerator. (If you wish, you may do this by writing a general form for each numerator, such as A or Ax + B.)
   • Problem set from the textbook due on May 28 Thursday (submit this through MyLab): 8.4.4, 8.4.7, 8.4.11, 8.4.17, 8.4.23, 8.4.27, 8.4.37, 8.4.39.
7. Trigonometric integration:
   • Reading from the textbook: Section 8.2 (pages 445–449).
   • Reading homework due on May 28 Thursday (write this by hand on paper and submit it on Canvas):
     1. To find ∫ sin² x dx, what trigonometric identity would you use? That is, sin² x = _____?
     2. For which (if any) of the following other integrals would you also need to use the identity from the previous exercise? (Say Yes or No for each one.)
        1. ∫ sin² x cos² x dx,
        2. ∫ sin² x cos³ x dx,
        3. ∫ sin³ x cos² x dx,
        4. ∫ sin³ x cos³ x dx.
   • Problem set from the textbook due on May 29 Friday (submit this through MyLab): 8.2.10, 8.2.11, 8.2.17, 8.2.19, 8.2.25, 8.2.53, 8.2.35, 8.2.41, 8.2.45, 8.2.67.
8. Trigonometric substitution:
   • Reading from the textbook: Section 8.3 (pages 451–454).
   • Reading homework due on May 29 Friday (write this by hand on paper and submit it on Canvas): For each of the following expressions, give a trigonometric substitution that would tend to help when finding integrals containing that expression:
     1. √(x² + 25), the square root of x² + 25;
     2. √(x² − 25), the square root of x² − 25;
     3. √(25 − x²), the square root of 25 − x².
   • Problem set from the textbook due on June 2 Tuesday (submit this through MyLab): 8.3.5, 8.3.9, 8.3.11, 8.3.13, 8.3.21, 8.3.23, 8.3.33.
9. Integration using computers and tables:
   • Reading from the textbook:
     • Section 8.5 (pages 463–467);
     • Skim “A Brief Table of Integrals” (pages T1–T6).
   • 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, $15 per month to make intensive computations online);
     • Maxima (free to download);
     • Maple ($75 per year for students);
     • Wolfram Mathematica ($75 per year for students);
     • Wolfram Alpha (free for some uses online, $60 per year for advanced features);
     • David Scherfgen's integral calculator (free to use online, not a full CAS, but shows steps).
   • Reading homework due on June 2 Tuesday (write this by hand on paper and submit it on Canvas): Which entry in the table of integrals in the back of the textbook (pages T1–T6) tells you how to integrate ∫ x²(x² + 3)^−½ dx? (Hints: That table doesn't use negative exponents; it uses fractions instead. And it doesn't use fractional exponents; it uses roots instead.)
   • Problem set from the textbook due on June 3 Wednesday (submit this through MyLab): 8.5.9, 8.5.13, 8.5.15, 8.5.19, 8.5.29, 8.5.33.

Numerical analysis

10. Numerical integration:
    • Reading from the textbook: Section 8.6 (pages 469–476).
    • Reading homework due on June 3 Wednesday (write this by hand on paper and submit it on Canvas):
      1. Which numerical method of integration approximates a function with a piecewise-constant function, aka a step function? (Hint: This is not from this lesson's reading; you already know it from Calculus 1.)
      2. Which numerical method of integration approximates a function with a continuous piecewise-linear function?
      3. Which numerical method of integration approximates a function with a continuous piecewise-quadratic function?
      4. Suppose that you are attempting to approximate the integral of a function f on an interval [a, b] and f is at least four times differentiable on [a, b]. Also suppose that M₁ ≥ |‍f′(x)|‍, M₂ ≥ |‍f″(x)|‍, M₃ ≥ |‍f‴(x)|‍, and M₄ ≥ |‍f⁽⁴⁾(x)|‍, whenever a ≤ x ≤ b. Answer the following questions using only a, b, M₁, M₂, M₃, and/or M₄ (and n or ε when one of these appears in the question):
         1. If you approximate the integral with the Trapezoid Rule using n subintervals, then what is the maximum possible absolute error in the approximation?
         2. If you wish to approximate the integral with the Trapezoid Rule and have an absolute error no more than ε, then what is the number of subintervals that you might need?
         3. If you approximate the integral with Simpson's Rule using n subintervals, then what is the maximum possible absolute error in the approximation?
         4. If you wish to approximate the integral with Simpson's Rule and have an absolute error no more than ε, then what is the number of subintervals that you might need?
    • Problem set from the textbook due on June 4 Thursday (submit this through MyLab): 8.6.23, 8.6.24, 8.6.25, 8.6.5, 8.6.7, 8.6.15, 8.6.17, 8.6.27.
11. Improper integrals:
    • Reading from the textbook: Section 8.7 through “Improper Integrals with a CAS” (pages 478–484).
    • Reading homework due on June 4 Thursday (write this by hand on paper and submit it on Canvas): Consider the integral ∫^∞_x=−∞ (x² + |‍x|‍^½)⁻¹ dx, that is the integral of (x² + |‍x|‍^½)⁻¹ dx as x runs from −∞ to ∞. List all of the reasons why this integral is improper.
    • Problem set from the textbook due on June 5 Friday (submit this through MyLab): 8.7.4, 8.7.9, 8.7.13, 8.7.17, 8.7.19, 8.7.31.
12. Comparison tests:
    • Reading from the textbook: The rest of Section 8.7 (pages 484–486).
    • Reading homework due on June 5 Friday (write this by hand on paper and submit it on Canvas): Suppose that you want to know whether ∫^∞_x=1 (x + sin x)/x² dx, the infinite integral of (x + sin x)/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?
    • Problem set from the textbook due on June 9 Tuesday (submit this through MyLab): 8.7.45, 8.7.47, 8.7.49, 8.7.57, 8.7.63.
13. Sequences:
    • Reading from the textbook: Chapter 9 through Section 9.1 (pages 495–504).
    • Reading from my notes (first set): Chapter 7 through Section 7.1 (pages 55&56).
    • Reading homework due on June 9 Tuesday (write this by hand on paper and submit it on Canvas):
      1. For each of the following sequences (each called a), write down the values of a₀ through a₄ (so 5 values in all for each sequence).
         1. a_n = n! (the factorial of n);
         2. a_n = (−1)ⁿ;
         3. a₀ = 0, a₁ = 1, a_n+2 = a_n + a_n+1.
      2. Suppose that f is a function defined everywhere, and a is a sequence given by a_n = f(n) for natural numbers n.
         1. Yes/No: If lim_x→∞ f(x) (the limit of f at infinity) exists, then must lim_n→∞ a_n (the limit of a at infinity) also exist?
         2. Yes/No: If lim_n→∞ a_n (the limit of a at infinity) exists, then must lim_x→∞ f(x) (the limit of f at infinity) also exist?
         3. Yes/No: If both limits exist, must they be the same?
      3. If c is a constant, what is lim_n→∞ (cⁿ/n!) (the limit as n → ∞ of cⁿ divided by n factorial)?
    • Problem set from the textbook due on June 10 Wednesday (submit this through MyLab): 9.1.9, 9.1.13, 9.1.15, 9.1.21, 9.1.23, 9.1.109, 9.1.31, 9.1.39, 9.1.41, 9.1.43, 9.1.45, 9.1.47, 9.1.49, 9.1.51, 9.1.55, 9.1.57, 9.1.59, 9.1.63, 9.1.69, 9.1.83, 9.1.93.

Further lessons

14. Finite series:
    • Reading from my notes (first set): Section 7.2 (pages 56&57).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      1. If a_n = n² for all n, write Σ⁴_n=0 a_n, the sum of a from 0 to 4, explicitly as a sum of 5 constants.
      2. If a is any sequence, write Σ^j_n=i a_n, the sum of a from i to j, as an integral of the function f(x) = a_⌊x⌋.
15. Infinite series:
    • Reading from the textbook:
      • Section 9.2 introduction (pages 508&509);
      • Section 9.2 “The nth-Term Test for a Divergent Series” and the rest (pages 512–515).
    • Reading from my notes (first set): Section 7.3 (pages 57&58).
    • Reading homework due (write this by hand on paper and submit it on Canvas): Fill in each blank with either ‘sequence’ or ‘series’:
      1. lim_n→∞ a_n, the limit of a_n as n goes to infinity, is the limit of an infinite _____.
      2. Σ^∞_n=0 a_n, the sum of a_n as n runs from zero to infinity, is the sum of an infinite _____.
16. Evaluating special series:
    • Reading from my notes (first set): Section 7.4 (pages 58&59).
    • Reading from the textbook: Section 9.2 “Geometric Series” (pages 510–512).
    • Reading homework due (write this by hand on paper and submit it on Canvas): 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 an integer 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 an integer and r is a real number): Σ^∞_n=i rⁿ = _____.
17. The Integral Test:
    • Reading from the textbook: Section 9.3 through “The Integral Test” (pages 518–521).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      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, and for which values does it diverge?
18. Integral estimates for series:
    • Reading from the textbook: the rest of Section 9.3 (pages 521&522).
    • Reading homework due (write this by hand on paper and submit it on Canvas): Suppose that f is a function meeting the conditions of the Integral Test and a is the corresponding sequence, so that a_n = f(n). Also suppose that the infinite integral of f converges. Write down an upper and lower bound of the infinite series of a from i using a finite series of m terms and some infinite integrals. That is, write down a compound inequality _____ ≤ Σ^∞_n=i a_n ≤ _____, where the blanks are expressions with finite series and infinite integrals (but no infinite series).
19. Comparison tests for series:
    • Reading from the textbook: Section 9.4 (pages 524–528).
    • Reading homework due (write this by hand on paper and submit it on Canvas): Suppose that you want to know whether Σ^∞_n=1 (n + (−1)ⁿ)/n², the infinite series of (n + (−1)ⁿ)/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?
20. Alternating series:
    • Section 9.6 introduction (pages 536–538).
    • Reading homework due (write this by hand on paper and submit it on Canvas): Identify which of these series are alternating. (Say Yes or No for each.)
      1. The sum of (−1)ⁿ/(2 + n);
      2. The sum of (2 + (−1)ⁿ)/n;
      3. The sum of (cos n)/n.
21. Absolute convergence:
    • Reading from the textbook:
      • Section 9.5 introduction (pages 529&530);
      • Section 9.6 “Conditional Convergence” & “Rearranging Series” (pages 538–540).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      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?
22. The Ratio and Root Tests:
    • Reading from the textbook: The rest of Section 9.5 (pages 531–534).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      1. Under what circumstances does the Ratio Test not tell you whether a series converges or diverges?
      2. Under what circumstances does the Root Test not tell you whether a series converges or diverges?
      3. If the Ratio Test doesn't tell you, is it possible that the Root Test will?
23. Convergence tests:
    • Reading from the textbook:
    • Section 9.6 “Summary of Tests” (page 540).
    • Reading from my notes (first set): Section 7.5 (pages 59–62).
    • Reading homework due (write this by hand on paper and submit it on Canvas): For each of the following tests for infinite series, state whether (Yes or No) this test can ever tell you that a series converges, whether (Yes or No) it can ever tell you that a series diverges, and whether (Yes or No) it will ever give no answer.
      1. The nth-Term Test (aka the Limit Test);
      2. The Integral Test;
      3. The Direct Comparison Test;
      4. The Limit Comparison Test;
      5. The Alternating-Series Test;
      6. The Absolute-Convergence Test;
      7. The Root Test;
      8. The Ratio Test.
24. Power series:
    • Section 9.7 (pages 543–551).
    • Reading homework due (write this by hand on paper and submit it on Canvas): Which of the following are (or are equivalent to) power series (in the variable x)? (Say Yes or No for each.)
      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
25. Taylor polynomials:
    • Reading from my notes (first set): Chapter 8 through Section 8.1 (pages 63–65).
    • Reading homework due (write this by hand on paper and submit it on Canvas): Suppose that f is a function, a is a real number, f is infinitely differentiable at a, and k is a whole number. Let P_k be the Taylor polynomial of order k generated by f at a; that is, P_k(x) = Σ^k_n=0 f⁽ⁿ⁾(a)(x − a)ⁿ/n!.
      1. The derivatives of f at a and the derivatives of P_k at a must be equal up to what order? That is, f⁽ⁿ⁾(a) = P_k⁽ⁿ⁾(a) for n from 0 to ___ (inclusive).
      2. What are the derivatives of P_k of even higher order? That is, if n > k, then P_k⁽ⁿ⁾(x) = ___.
26. Taylor remainders:
    • Reading from the textbook:
      • Section 9.8 “Taylor Polynomials” (pages 556–558);
      • Section 9.9 through “Estimating the Remainder” (pages 559–562).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      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? That is, the derivative whose value should be near zero is the first derivative (order 1), the second derivative (order 2), the kth derivative (order k), or what?
      2. Suppose that f is a function, a is a number, and f is infinitely differentiable at a. For each whole number k, let R_k be the Taylor remainder of order k generated by f at a; that is, R_k(x) = f(x) − Σ^k_n=0 f⁽ⁿ⁾(a)(x − a)ⁿ/n!. Is it necessarily true for all x that the limit (as k → ∞) of R_k(x) is zero? That is, is it necessarily true that lim_k→∞ R_k(x) = 0?
27. Taylor series:
    • Reading from the textbook: the rest of Section 9.8 (pages 554–556).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      1. Fill in the blank: If f is a function that is infinitely differentiable everywhere, then the Maclaurin series generated by f is the Taylor series generated by f at ___.
      2. Suppose that f is infinitely differentiable at a and let T be the Taylor series generated by f at a; that is, T(x) = Σ^∞_n=0 f⁽ⁿ⁾(a)(x − a)ⁿ/n!. If T(x) converges, must it necessarily converge to f(x)?
28. The Binomial Theorem:
    • Reading from the textbook: Section 9.10 through “The Binomial Series for Powers and Roots” (pages 565–567).
    • Reading from my notes (first set): Section 8.2 (pages 65–67).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      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 so that the series converges), and simplify the expression for the terms.
29. More common Taylor series:
    • Reading from the textbook: The rest of Section 9.9 (pages 562&563).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      1. Write Maclaurin series for e^x, sin x, cos x, and atan x.
      2. Write a Taylor series at x = 1 for ln x.
30. Applications of Taylor series:
    • Reading from the textbook: The rest of Section 9.10 (pages 567–571).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      1. Assuming that a and b are real numbers, fill in the blank using trigonometric operations applied to real numbers (and other operations as appropriate): exp (a + ib) = e^a+ib = ___.
      2. Can any limit using L'Hôpital's Rule be done using Taylor polynomials or series instead? Explain why or why not.
31. Three-dimensional space:
    • Reading from the textbook: Chapter 11 through Section 11.1 introduction (pages 614&615).
    • Reading from my notes (second set): Through Section 1.1 (pages 1–3).
    • Reading homework due (write this by hand on paper and submit it on Canvas): 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?
32. Graphs in three dimensions:
    • The rest of Section 11.1 (pages 616&617).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      1. 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.)
      2. 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)?
33. Vectors:
    • Reading from my notes (second set): Section 1.2 (pages 3&4).
    • Reading from the textbook: Section 11.2 through “Component form” (pages 619–621).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      1. Give a formula for the vector from the point (x₁, y₁) to the point (x₂, y₂).
      2. Give a formula for the point reached by moving along the vector ⟨Δx, Δy⟩ from the point (x, y).
34. Vector algebra:
    • Reading from my notes (second set): Sections 1.3&1.4 (pages 4–7).
    • Reading from the textbook:
      • Section 11.2 “Vector Algebra Operations” (pages 621&622);
      • Section 11.2 “Midpoint of a Line Segment” (pages 624).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      1. If 𝐮, 𝐯, and 𝐰 are vectors, simplify the expression 2(𝐮 + 3𝐯) − 6(𝐯 − 3𝐰) − 18(𝐰 + 𝐮/9).
      2. If P and Q are points, simplify the expression Q − ½(Q − P).
35. Length and angle:
    • Reading from my notes (second set): Section 1.5 (pages 7–9).
    • Reading from the textbook:
      • Section 11.2 “Unit vectors” (pages 623&624);
      • Section 11.2 “Applications” (pages 625&626).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      1. Give a formula for the magnitude (or norm, or length) of the vector ⟨a, b, c⟩.
      2. Suppose that 𝐮 and 𝐯 are vectors; let a be ‖‍𝐮‖‍, let b be ‖‍𝐯‖‍, and let c be ‖‍𝐮 −𝐯‖‍. Express the cosine of the angle between 𝐮 and 𝐯 using a, b, and c. (This is in my notes but not in the textbook.)
36. Projections:
    • Reading from my notes (second set): Section 1.6 (pages 9&10).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      1. State a formula for the vector projection of 𝐮 onto 𝐯 using only their lengths ‖‍𝐮‖ and ‖‍𝐯‖‍, the angle ∠(𝐮, 𝐯) between them, real-number operations, and scalar multiplication involving 𝐮 and/or 𝐯 (so in particular, no dot products).
      2. State a formula for the scalar component of 𝐮 in the direction of 𝐯 using only their lengths ‖‍𝐮‖ and ‖‍𝐯‖‍, the angle ∠(𝐮, 𝐯), and real-number operations (so in particular, no dot products).
37. The dot product:
    • Reading from my notes (second set): Sections 1.7&1.8 (pages 10–12).
    • Reading from the textbook: Section 11.3 (pages 628–634).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      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 vector projection of 𝐮 onto 𝐯 using only dot products and real-number operations (so in particular, no lengths or angles unless expressed using dot products).
38. Area:
    • Reading from my notes (second set): Section 1.9 (pages 12&13).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      1. Consider a triangle, orient two of the three sides of the triangle, and interpet these as vectors 𝐮 and 𝐯. State a formula for the area of this triangle using only the lengths ‖‍𝐮‖ and ‖‍𝐯‖‍, the angle ∠(𝐮, 𝐯), and real-number operations.
      2. 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.
39. The cross product:
    • Reading from my notes (second set): Sections 1.10–1.12 (pages 13–17).
    • Reading from the textbook: Section 11.4 (pages 636–641).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      1. If 𝐮 and 𝐯 are vectors in 2 dimensions, then is 𝐮 × 𝐯 a scalar or a vector?
      2. If 𝐮 and 𝐯 are vectors in 3 dimensions, then is 𝐮 × 𝐯 a scalar or a vector?
40. Parametrized curves:
    • Reading from my notes (second set): Chapter 2 throgh Section 2.1 (page 19).
    • Reading from the textbook: Chapter 10 through Section 10.1 (pages 580–585).
    • Reading homework due (write this by hand on paper and submit it on Canvas): 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?
41. Geometry with vectors:
    • Reading from the textbook: Section 11.5 (pages 642–649).
    • Reading from my notes (second set): Section 2.5 (pages 22&23).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      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⟩.
42. Velocity vectors:
    • Reading from my notes (second set): Sections 2.2&2.3 (pages 19&20).
    • Reading homework due (write this by hand on paper and submit it on Canvas): Let the variable point P represent position of some object, let the variable scalar t represent time, let the variable vector 𝐯 represent the object's velocity, and let the variable vector 𝐚 represent its acceleration (in the technical sense, that is vector acceleration).
      1. Express 𝐯 using P and t, and concepts of Calculus.
      2. Express 𝐚 using P, t, and/or 𝐯 and concepts of Calculus.
43. Calculus with parametrized curves:
    • Reading from the textbook: Section 10.2 through “Tangents and Areas” (pages 588–590).
    • Reading from my notes (second set): Section 2.6 (pages 23–25).
    • Reading homework due (write this by hand on paper and submit it on Canvas): 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!)
44. Arclength of parametrized curves:
    • Reading from the textbook: the rest of Section 10.2 (pages 590–596).
    • Reading from my notes (second set): Section 2.7 (pages 25&26).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      1. If a curve is parametrized by x = f(t) and y = g(t) for a ≤ t ≤ b (and assuming, as usual, that the parametrization is continuously differentiable and one-to-one), then what integral in the variable t gives the length of this curve?
      2. How does changing the parametrization of a curve affect its arclength?
45. Polar coordinates:
    • Reading from the textbook: Section 10.3 through “Definition of Polar Coordinates” (pages 598&599).
    • Reading from my notes (second set): Section 2.8 (pages 26&27).
    • Reading homework due (write this by hand on paper and submit it on Canvas): True or false:
      1. For every point P in the coordinate plane, for every pair (r, θ) of real numbers that gives P in polar coordinates, r ≥ 0 and 0 ≤ θ < 2π.
      2. For every point P in the coordinate plane, for at least one pair (r, θ) of real numbers that gives P in polar coordinates, r ≥ 0 and 0 ≤ θ < 2π.
46. Equations in polar coordinates:
    • Reading from the textbook: The rest of Section 10.3 (pages 599–601).
    • Reading homework due (write this by hand on paper and submit it on Canvas): Let P be a point in the plane, suppose that (x, y) represents P in rectangular coordinates, and suppose that (r, θ) represents P in polar coordinates.
      1. Express x and y using r and θ.
      2. Express r² in terms of x and y.
      3. Expres sin θ and cos θ in terms of x, y, and r.
47. Graphs in polar coordinates:
    • Reading from the textbook: Section 10.4 (pages 602–605).
    • Reading from my notes (second set): Section 2.9 (page 27).
    • Reading homework due (write this by hand on paper and submit it on Canvas): 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?
48. Area in polar coordinates:
    • Reading from the textbook: Section 10.5 through “Area in the Plane” (pages 606–608).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      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.)
49. Length in polar coordinates:
    • Reading from the textbook: Section 10.5 “Length of a Polar Curve” (pages 608&609).
    • Reading homework due (write this by hand on paper and submit it on Canvas):
      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).)

Quizzes

1. Integration review:
   • Date available: May 29 Friday.
   • Date due: June 1 Monday.
   • Corresponding problem sets: 1–4.
   • Help allowed: Your notes, self-contained calculator.
   • NOT allowed: Textbook, my notes, other people, websites, online apps, etc.
   • Work to show: Submit a picture of your work on Canvas: at least one intermediate step for each result.
2. Advanced integration techniques:
   • Date available: June 5 Friday.
   • Date due: June 8 Monday.
   • Corresponding problem sets: 5–9.
   • Help allowed: Your notes, self-contained calculator.
   • NOT allowed: Textbook, my notes, other people, websites, online apps, etc.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result, including any substitutions you make.
3. More to come.

Final exam

For the exam, you may use one sheet of notes that you wrote yourself; please take a scan or a picture of this (both sides) and submit it on Canvas. However, you may not use your textbook, my notes, or anything else not written by you. You certainly should not talk to other people! Calculators are allowed (although you shouldn't really need one), but not communication devices (like cell phones).

The exam consists of questions similar in style and content to those in the practice exam (DjVu TBA).

The final exam is proctored. If you're near any of the three main SCC campuses (Lincoln, Beatrice, Milford), then you can schedule the exam at one of the Testing Centers; it will automatically be ready for you at Lincoln, but let me know if you plan to take it at Beatrice or Milford, so that I can have it ready for you there. If you have access to a computer with a webcam and mike, then you can take it using ProctorU for a small fee; let me know if you want to do this so that I can send you an invitation to schedule it. If you're near Lincoln, then we may be able to schedule a time for you to take the exam with me in person. If none of these will work for you, then contact me as soon as possible!

The permanent URI of this web page is https://tobybartels.name/MATH-1700/2026SS/.