# MATH-1700-ES31

Welcome to the permanent home page for Section ES31 of MATH-1700 (Calculus 2) at Southeast Community College in the Spring term of 2020. I am Toby Bartels, the instructor.

Here is material about the administration of the course:

• Canvas page (where you must log in for full access, available while the course is in session).
• Help with DjVu (if you have trouble reading the files below).
• Course policies (DjVu).
• Class hours: Mondays through Fridays from 11:00 to 11:50 in ESQ 100D.
Information to contact me:
• Name: Toby Bartels, PhD;
• Canvas messages.
• Email: TBartels@Southeast.edu.
• Voice mail: 1-402-323-3452.
• Text messages: 1-402-805-3021.
• Office hours:
• Mondays and Wednesdays from 9:30 to 10:30,
• Tuesdays and Thursdays from 2:30 PM to 4:00, and
• by appointment,
in ESQ 112 and online. (I am often available outside of those times; feel free to send a message any time and to check for me in the office whenever it's open.)

The official textbook for the course is the 4th Edition of University Calculus: Early Transcendentals by Hass et al published by Addison Wesley (Pearson). There are also two packets of course notes (DjVu):
• The first set, on single-variable Calculus (including material from Calculus 1);
• The second set, on multivariable Calculus (including material for Calculus 3).
1. General review:
• Date due: January 14 Tuesday.
• Reading from my notes (first set): Review at least Chapter 4 through Section 4.3 (pages 39–42).
• Reading from the textbook: Review at least Sections 5.4 and 5.5.
• Exercises 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 ∫bx=af(x) dx, that is the definite integral of f(x) with respect to x as x runs from a to b?
2. Work:
• Date due: January 15 Wednesday.
• Reading from the textbook: Section 6.5 (pages 386–389).
• Exercises 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 (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.
3. Moments:
• Date due: January 16 Thursday.
• Reading from the textbook: Section 6.6 (pages 392–400).
• Exercises due:
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 (, ȳ) in terms of the total mass M and the moments Mx and My.
4. Differential equations:
• Date due: January 17 Friday.
• Reading from my notes (first set):
• Section 4.4 (page 42);
• Chapter 5 through Section 5.2 (pages 45–47);
• Optional: The rest of Chapter 5 (pages 47&48).
• Reading from the textbook: Section 7.2 "Separable Differential Equations" (pages 416–418).
• Exercises due: 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 _____.
5. Exponential growth:
• Date due: January 21 Tuesday.
• Reading from the textbook:
• Section 7.2 through "Exponential Change" (pages 415&416);
• The rest of Section 7.2 (pages 418–422).
• Exercises due:
1. Fill in the blanks with a vocabulary word: 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.
6. Integration by parts:
• Date due: January 22 Wednesday.
• Reading from my notes (first set): Section 4.5 (pages 42&43).
• Reading from the textbook: Chapter 8 through Section 8.1 (pages 436–442).
• Exercises due: Use the formula ∫ u dv = uv − ∫ v du for integration by parts.
1. To find ∫ x ex 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?
7. Partial fractions:
• Date due: January 23 Thursday.
• Reading from the textbook: Section 8.4 (pages 456–461).
• Exercises due: 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.)
8. Trigonometric integration:
• Date due: January 24 Friday.
• Reading from the textbook: Section 8.2 through "Eliminating Square Roots" (pages 445–447).
• Exercises due:
1. To find ∫ sin2x dx, what trigonometric identity would you use? That is, sin2x = _____?
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. ∫ sin2x cos2x dx,
2. ∫ sin2x cos3x dx,
3. ∫ sin3x cos2x dx,
4. ∫ sin3x cos3x dx.
9. Tricky trigonometric integration:
• Date due: January 27 Monday.
• Reading from the textbook: The rest of Section 8.2 (pages 447–449).
• Exercises due: Rewrite the following expressions using only powers of sin x and/or cos x (and possibly multiplication):
1. sec2x,
2. sec x tan x,
3. csc2x,
4. csc x cot x.
10. Trigonometric substitution:
• Date due: January 28 Tuesday.
• Reading from the textbook: Section 8.3 (pages 451–454).
• Exercise due: 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.
11. Integration using computers and tables:
• Date due: January 29 Wednesday.
• 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, \$149 per year to make intensive computations online);
• Wolfram Mathematica (\$161 to download, \$81 per year online);
• Wolfram Alpha (free for some uses online, \$57 per year for advanced features, \$3 for a smartphone app with intermediate features);
• David Scherfgen's integral calculator (free to use online, not a full CAS, but shows steps).
• Exercise due: Which entry in the table of integrals in the back of the textbook (pages T1–T6) tells you how to integrate ∫ x2(x2 + 3)−1/2 dx? (Hints: That table doesn't use negative exponents; it uses fractions instead. And it doesn't use fractional exponents; it uses roots instead.)
12. Numerical integration:
• Date due: January 30 Thursday.
• Reading from the textbook: Section 8.6 through "Simpson's Rule" (pages 469–472).
• Exercises due:
1. Which numerical method of integration approximates a function with a piecewise-constant function? (Hint: This is not from Section 8.6; you already know it from Chapter 5.)
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?
13. Error estimation:
• Date due: February 4 Tuesday.
• The rest of Section 8.6 (pages 473–476).
• Exercises due: Suppose that you are attempting to approximate the integral of a function f on an interval [a, b] and f is at least three times differentiable on [a, b]. Also suppose that M2 ≥ |f″(x)| and M3 ≥ f″′(x) whenever x ∈ [a, b]. Answer the following questions using only a, b, M2, and M3 (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 largest possible 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 largest possible number of subintervals that you might need?
14. Improper integrals:
• Date due: February 5 Wednesday.
• Reading from the textbook: Section 8.7 through "Improper Integrals with a CAS" (pages 478–484).
• Exercise 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.
15. Comparison tests:
• Date due: February 6 Thursday.
• The rest of Section 8.7 (pages 484–486).
• Exercises due: 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?
16. Sequences:
• Date due: February 7 Friday.
• Reading from the textbook:
• Chapter 9 through Section 9.1 "Representing Sequences" (pages 495–497);
• Section 9.1 "Recursive Definitions" (pags 502&503).
• Exercises due: 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. an = n! (the factorial of n);
2. an = (−1)n;
3. a0 = 0, a1 = 1, an+2 = an + an+1.
17. Limits of infinite sequences:
• Date due: February 11 Tuesday.
• Reading from my notes (first set): Chapter 6 through Section 6.1 (pages 49&50).
• Reading from the textbook: The rest of Section 9.1 (pages 497–502, 503&504).
• Exercises due: Suppose that f is a function defined everywhere, and a is a sequence given by an = f(n).
1. If limx→∞f(x) (the limit of f at infinity) exists, then must limn→∞an (the limit of a at infinity) be the same?
2. If limn→∞an (the limit of a at infinity) exists, then must limx→∞f(x) (the limit of f at infinity) be the same?
3. If c is a constant, what is limn→∞cn/n! (the limit as n → ∞ of cn divided by n factorial)?
18. Finite series:
• Date due: February 12 Wednesday.
• Reading from my notes (first set): Section 6.2 (pages 50&51).
• Exercises due:
1. If an = n2 for all n, write Σ4n=0an, the sum of a from 0 to 4, explicitly as a sum of 5 constants.
2. If a is any sequence, write Σjn=ian, the sum of a from i to j, as an integral of the function f(x) = ax.
19. Infinite series:
• Date due: February 13 Thursday.
• 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 6.3 (pages 51&52).
• Exercises due: Fill in each blank with either ‘sequence’ or ‘series’:
1. limn→∞an, the limit of an as n goes to infinity, is the limit of an infinite _____.
2. Σn=0an, the sum of an as n runs from zero to infinity, is the sum of an infinite _____.
20. Evaluating special series:
• Date due: February 14 Friday.
• Reading from my notes (first set): Section 6.4 (pages 52&53).
• Reading from the textbook: Section 9.2 "Geometric Series" (pages 510–512).
• Exercises due: Finish these formulas and attach any conditions necessary for them to be true:
1. The sum of bn+1 − bn as n runs from i to infinity (where i is a natural number and b is an infinite sequence of real numbers): Σn=i (bn+1 − bn) = _____.
2. The sum of rn as n runs from i to infinity (where i is a natural number and r is a real number): Σn=irn = _____.
21. The Integral Test:
• Date due: February 17 Monday.
• Reading from the textbook: Section 9.3 through "The Integral Test" (pages 518–521).
• Exercises due:
1. Does the Integral Test apply to the function f(x) = sin2x)? Why or why not?
2. For which values of p does Σn=1 (1/np), the sum of 1/np as n runs from 1 to infinity, converge, and for which values does it diverge?
22. Integral estimates for series:
• Date due: February 18 Tuesday.
• Reading from the textbook: the rest of Section 9.3 (pages 521&522).
• Exercises due: Suppose that f is a function meeting the conditions of the Integral Test and a is the corresponding sequence, so that an = 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 _____ ≤ Σn=ian ≤ _____, where the blanks are expressions with finite series and infinite integrals (but no infinite series).
23. Comparison tests for series:
• Date due: February 19 Wednesday.
• Reading from the textbook: Section 9.4 (pages 524–528).
• Exercises due: Suppose that you want to know whether Σn=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?
24. Alternating series:
• Date due: February 20 Thursday.
• Section 9.6 introduction (pages 536–538).
• Exercises due: 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.
25. Absolute convergence:
• Date due: February 25 Tuesday.
• Reading from the textbook:
• Section 9.5 introduction (pages 529&530);
• Section 9.6 "Conditional Convergence" & "Rearranging Series" (pages 538–540).
• Exercises 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?
26. The Ratio and Root Tests:
• Date due: February 26 Wednesday.
• Reading from the textbook: The rest of Section 9.5 (pages 531–534).
• Exercises due:
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?
27. Convergence tests:
• Date due: February 27 Thursday.
• Reading from the textbook:
• Section 9.6 "Summary of Tests" (page 540).
• Reading from my notes (first set): Section 6.5 (pages 53–56).
• Exercises due: 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.
28. Power series:
• Date due: February 28 Friday.
• Section 9.7 (pages 543–551).
• Exercises due: Which of the following are (or are equivalent to) power series (in the variable x)? (Say Yes or No for each.)
1. Σn=0n2(x − 3)n
2. Σn=5 (2x − 3)n
3. Σn=0 (√x − 3)n
4. 5 + 7x − 3x3
29. Taylor polynomials:
• Date due: March 3 Tuesady.
• Reading from my notes (first set): Chapter 7 through Section 7.1 (pages 57–59).
• Exercises due: Suppose that f is a function, a is a real number, f is infinitely differentiable at a, and k is a whole number. Let Pk be the Taylor polynomial of order k generated by f at a; that is, Pk(x) = Σkn=0f(n)(a)(x − a)n/n!.
1. The derivatives of f at a and the derivatives of Pk at a must be equal up to what order? That is, f(n)(a) = Pk(n)(a) for n from 0 to ___ (inclusive).
2. What are the derivatives of Pk of even higher order? That is, if n > k, then Pk(n)(x) = ___.
30. Taylor remainders:
• Date due: March 4 Wednesday.
• Reading from the textbook:
• Section 9.8 "Taylor Polynomials" (pages 556–558);
• Section 9.9 through "Estimating the Remainder" (pages 559–562).
• Exercises 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? 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 Rk be the Taylor remainder of order k generated by f at a; that is, Rk(x) = f(x) − Σkn=0f(n)(a)(x − a)n/n!. Is it necessarily true for all x that the limit (as k → ∞) of Rk(x) is zero? That is, is it necessarily true that limk→∞Rk(x) = 0?
31. Taylor series:
• Date due: March 5 Thursday.
• Reading from the textbook: the rest of Section 9.8 (pages 554–556).
• Exercises due:
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) = Σn=0f(n)(a)(x − a)n/n!. If T(x) converges, must it necessarily converge to f(x)?
32. The Binomial Theorem:
• Date due: March 6 Friday.
• 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 7.2 (pages 59–61).
• Exercises 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 so that the series converges), and simplify the expression for the terms.
33. More common Taylor series:
• Date due: March 9 Monday.
• Reading from the textbook: The rest of Section 9.9 (pages 562&563).
• Exercises due:
1. Write Maclaurin series for ex, sin x, cos x, and atan x.
2. Write a Taylor series at x = 1 for ln x.
34. Applications of Taylor series:
• Date due: March 11 Wednesday.
• Reading from the textbook: The rest of Section 9.10 (pages 567–571).
• Exercises due:
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) = ea+ib = ___.
2. Can any limit using L'Hôpital's Rule be done using Taylor polynomials or series instead? Explain why or why not.
35. Three-dimensional space:
• Date due: March 12 Thursday.
• 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).
• Exercise due: 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?
36. Graphs in three dimensions:
• Date due: March 17 Tuesday.
• The rest of Section 11.1 (pages 616&617).
• Exercises due:
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)?
37. Vectors:
• Date due: April 1 Wednesday.
• 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).
• Exercises due:
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).
38. Vector algebra:
• Date due: April 2 Thursday.
• 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).
• Exercises due:
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).
39. Length and angle:
• Date due: April 3 Friday.
• 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).
• Exercises due:
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.)
40. Projections:
• Date due: April 7 Tuesday.
• Reading from my notes (second set): Section 1.6 (pages 9&10).
• Exercises due:
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 the lengths |u| and |v|, the angle ∠(u, v), and real-number operations (so in particular, no dot products).
41. The dot product:
• Date due: April 8 Wednesday.
• Reading from my notes (second set): Sections 1.7&1.8 (pages 10–12).
• Reading from the textbook: Section 11.3 (pages 628–634).
• Exercises 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 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).
42. Area:
• Date due: April 9 Thursday.
• Reading from my notes (second set): Section 1.9 (pages 12&13).
• Exercises due:
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.
43. The cross product:
• Date due: April 10 Friday.
• Reading from my notes (second set): Sections 1.10–1.12 (pages 13–17).
• Reading from the textbook: Section 11.4 (pages 636–641).
• Exercises due:
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?
44. Parametrized curves:
• Date due: April 13 Monday.
• Reading from my notes (second set): Chapter 2 throgh the first half of Section 2.1 (page 19).
• Reading from the textbook: Chapter 10 through Section 10.1 (pages 580–585).
• Exercises 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?
45. Geometry with vectors:
• Date due: April 14 Tuesday.
• Reading from the textbook: Section 11.5 (pages 642–649).
• Reading from my notes (second set): Section 2.2 (page 21).
• Exercises 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⟩.
46. Velocity vectors:
• Date due: April 15 Wednesday.
• Reading from my notes (second set): The rest of Section 2.1 (pages 19&20).
• Exercises due: 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.
47. Calculus with parametrized curves:
• Date due: April 21 Tuesday.
• Reading from the textbook: Section 10.2 through "Tangents and Areas" (pages 588–590).
• Reading from my notes (second set): Section 2.3 (pages 21–23).
• Exercises 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!)
48. Arclength of parametrized curves:
• Date due: April 22 Wednesday.
• Reading from the textbook: the rest of Section 10.2 (pages 590–596).
• Reading from my notes (second set): Section 2.4 (pages 23&24).
• Exercises due:
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?
49. Polar coordinates:
• Date due: April 23 Thursday.
• Reading from the textbook: Section 10.3 through "Definition of Polar Coordinates" (pages 598&599).
• Reading from my notes (second set): Section 2.5 (pages 24&25).
• Exercises due: 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.
50. Equations in polar coordinates:
• Date due: April 24 Friday.
• Reading from the textbook: The rest of Section 10.3 (pages 599–601).
• Exercises due: 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.
51. Graphs in polar coordinates:
• Date due: April 27 Monday.
• Reading from the textbook: Section 10.4 (pages 602–605).
• Reading from my notes (second set): Section 2.6 (pages 25&26).
• Exercises 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?
52. Area in polar coordinates:
• Date due: April 28 Tuesday.
• Reading from the textbook: Section 10.5 through "Area in the Plane" (pages 606–608).
• Exercises 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.)
53. Length in polar coordinates:
• Date due: April 29 Wednesday.
• Reading from the textbook: Section 10.5 "Length of a Polar Curve" (pages 608&609).
• Exercises 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).)
That's it!
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