MATH-1600-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: Online only.
• Final exam: By appointment July 28 – August 1.

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: on Mondays, Wednesdays, and Fridays from 2:00 PM to 3:00, or by appointment; in room U9C, or over Zoom meeting 610-024-2876.

Readings

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

• Objectives:
  • Understand what to expect from this course;
  • Know how to submit assignments.
• Reading: The course policies (DjVu, same as above).
• Questions due on May 27 Tuesday or ASAP thereafter (submit these on Canvas):
  1. If you want to submit an assignment that you've written out by hand, how will you send me a picture of it?
  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.)
• Exercises from the textbook due on May 28 Wednesday or ASAP thereafter (submit these 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.

Most of the dates below are wrong!

Continuity and limits

1. General review:
   • Objectives:
     • Review function notation;
     • Use evaluation notation.
   • Reading from the textbook:
     • Skim: Through Section 1.2 (through page 18);
     • Skim: Section 1.6 through "Finding Inverses" (pages 38–41).
   • Reading from my notes:
     • Through Section 1.4 (through page 6);
     • Optional: Section 1.5 (page 7).
   • Exercises due on May 28 Wednesday (submit these on Canvas): Show at least one intermediate step for each answer.
     1. If f(x) = x² for all x and u = 2x + 3, then what is f(u)?
     2. If x + y = 1 and x − y = 3, then what are x and y?
     3. If y = 3x + 2, then what is y|‍_x=4?
   • Discuss this on Canvas.
   • Exercises from the textbook due on May 29 Thursday (submit these through MyLab): 1.1.7, 1.1.8, 1.1.13, 1.1.23, 1.1.25, 1.1.75, 1.2.5.
2. Limits informally:
   • Reading from the textbook:
     • Section 2.2 through "An Informal Description of the Limit of a Function" (pages 58–61);
     • Section 2.4 through "Limits at Endpoints of an Interval" (pages 78–80);
     • Section 2.6 through the first paragraph of "Finite Limits as x → ±∞" (page 97);
     • Section 2.6 "Infinite Limits" before Example 13 (pages 102&103).
   • Reading from my notes: Sections 2.3&2.4 (pages 11–13).
   • Exercises due on May 29 Thursday (submit these on Canvas):
     1. Fill in the blank: If f(x) can be made arbitrarily close to L by making x sufficiently close to (but still distinct from) c, then L is the _____ of f(x) as x approaches c.
     2. Yes/No: If f(x) exists whenever x ≠ c but f(c) does not exist, then is it possible that lim_x→c f(x) exists?
     3. Yes/No: If lim_x→c⁺ f(x) and lim_x→c⁻ f(x) both exist and are equal, then must lim_x→c f(x) also exist?
     4. Fill in the blank: If f(x) can be made arbitrarily large by making x sufficiently close to (but still distinct from) c, then the limit of f(x) as x approaches c is _____.
     5. Fill in the blank: If f(x) can be made arbitrarily close to L by making x sufficiently large, then L is the limit of f(x) as x approaches _____.
     6. Yes/No: If f(x) always gets larger as x gets larger, does that necessarily mean that lim_x→∞ f(x) = ∞?
   • Discuss this on Canvas.
   • Exercises from the textbook due on May 30 Friday (submit these through MyLab): 2.2.1, 2.2.2, 2.2.7, 2.2.8, 2.2.9, 2.2.10, 2.4.1, 2.4.3, 2.4.5, 2.6.1, 2.6.2, 2.6.75, 2.6.76, 2.6.77.
3. Continuity:
   • Reading from my notes: Chapter 2 through Section 2.2 (pages 9–11).
   • Reading from the textbook:
     • Section 2.5 through "Continuity at a Point" (pages 85–88);
     • Section 2.3 "Examples: Testing the Definition" and "Finding Deltas Algebraically for Given Epsilons" (pages 70–74); note that nearly all of these examples are for continuous functions, so pretend that they're using the definition of continuity from my notes, which actually makes things slightly simpler (since you can ignore the ‹0 <› part).
   • Exercises due on May 30 Friday (submit these on Canvas):
     1. If f(x) can be made arbitrarily close to f(c) by making x sufficiently close to (but still distinct from) c, then f is _____ at c.
     2. Suppose that f(x) exists whenever x ≠ c but f(c) does not exist. Is it possible that f is continuous at c?
     3. Suppose that f is a function, and suppose that c is a real number. For simplicity, suppose that f is defined everywhere. Fill in the blanks:
        1. Also suppose that, no matter what positive real number ε I give you, you can respond with a positive real number δ so that, no matter what real number x I give you, if |‍x − c|‍ < δ, then |‍f(x) − f(c)|‍ < ε. This means that f is _____ at c.
        2. Instead of the stuff in part A, suppose that I can find a positive real number ε so that, no matter what positive real number δ you respond with, I can find a real number x, such that |‍x − c|‍ < δ but |‍f(x) − f(c)|‍ ≥ ε. This means that f has a _____ at c.
   • Discuss this on Canvas.
   • Exercises from the textbook due on June 2 Monday (submit these through MyLab): 2.5.1, 2.5.3, 2.5.7, 2.5.9, 2.5.11, 2.3.7, 2.3.9, 2.3.11, 2.3.13, 2.3.15, 2.3.17, 2.3.23, 2.3.27.
4. Defining limits:
   • Reading from my notes: Section 2.5 (pages 13&14).
   • Reading from the textbook:
     • Section 2.5 "Continuous Extension to a Point" (pages 93&94);
     • Optional: Section 2.3 through "Definition of Limit" (pages 69&70);
     • Optional: Section 2.4 "Precise Definitions of One-Sided Limits" (pages 80 and 81);
     • Optional: The rest of Section 2.6 "Finite Limits as x → ±∞" through Example 1 (pages 97&98);
     • Optional: Section 2.6 "Precise Definitions of Infinite Limits" (pages 104&105).
   • Exercises due on June 2 Monday (submit these on Canvas):
     1. Suppose that f is a function defined everywhere except at c, and define a new function g so that g(x) = f(x) whenenver x ≠ c but g(c) = L. If g is continuous at c, then L is the _____ of f approaching c.
     2. Suppose that lim_x→c (f(x) for x > c) = L. What is the limit of f approaching c from the right? That is, lim_x→c⁺ f(x) = _____.
     3. Suppose that f(x) > 0 for all x, and lim_x→c (1‍/‍f(x)) = 0. What is the limit of f approaching c? That is, lim_x→c f(x) = _____.
     4. Suppose that lim_t→0⁺ f(1‍/‍t) = L. What is the limit of f approaching infinity? That is, lim_x→∞ f(x) = _____.
   • Discuss this on Canvas.
   • Exercises from the textbook due on June 3 Tuesday (submit these through MyLab): 2.5.41, 2.5.45, 2.3.49, 2.2.15, 2.2.19.
5. Evaluating limits and checking continuity:
   • Reading from my notes: Sections 2.6&2.7 (pages 15–17).
   • Reading from the textbook:
     • The rest of Section 2.2 (pages 61–65);
     • Section 2.5 from "Continuous Functions" to "Continuity of Composites of Functions" (pages 88–91);
     • Section 2.3 "Using the Definition to Prove Theorems" (page 74);
     • Section 2.4 "Limits Involving (sin θ)‍/‍θ" (pages 81–83);
     • The rest of Section 2.6 "Finite Limits as x → ±∞" through "Limits at Infinity of Rational Functions" (pages 98&99);
     • Section 2.6 "Infinite Limits" Examples 13&14 (pages 103&104);
     • Section 2.6 "Dominant Terms" (pages 106&107).
   • Exercises due on June 3 Tuesday (submit these on Canvas):
     1. If you're taking the limit of a rational expression as x → c, and you get 0‍/‍0 when you evaluate the expression at x = c, then what factor can you cancel from the numerator and denominator to simplify your expression (and then evaluate the limit)?
     2. If you're taking the limit, as x → ∞, of a rational expression whose numerator has degree m and whose denominator has degree n, then what should you factor out of both numerator and denominator to guarantee that you can evaluate the limit by doing calculations with infinity?
     3. Suppose that c and L are real numbers; and h is a piecewise-defined function with h(x) = f(x) for x < c, h(c) = L, and h(x) = g(x) for x > c; where f is a function defined on (−∞, c) and g is a function defined on (c, ∞). Fill in the blank with an equation or equations involving c, L, and/or values and/or limits of f and/or g: h is continuous at c if and only if _____.
     4. What is the limit of (sin x)‍/‍x as x → 0?
     5. What are the limits of e^x as x → ∞ and as x → −∞?
     6. What are the limits of ln x as x → ∞ and as x → 0⁺?
     7. What are the limits of sin x and cos x as x → ∞ and as x → −∞? (This is a trick question!)
   • Discuss this on Canvas.
   • Exercises from the textbook due on June 4 Wednesday (submit these through MyLab): 2.5.13, 2.5.15, 2.5.19, 2.5.21, 2.5.25, 2.5.27, 2.5.29, 2.2.25, 2.2.29, 2.2.35, 2.2.37, 2.2.43, 2.4.25, 2.2.53, 2.2.57, 2.2.65, 2.4.11, 2.4.17, 2.6.9, 2.6.11, 2.6.15, 2.6.19, 2.6.25, 2.6.27, 2.6.29, 2.6.35, 2.6.41, 2.6.45, 2.6.49, 2.6.53, 2.6.57.
6. Theorems about continuous functions:
   • Reading from my notes:
     • Optional: Section 2.8 (pages 17&18).
     • Section 2.9 (pages 18–20).
   • Reading from the textbook:
     • Section 2.5 "Intermediate Value Theorem for Continuous Functions" (pages 91–93);
     • Section 4.1 through "Local (Relative) Extreme Values" (pages 212–215).
   • Exercises due on June 4 Wednesday (submit these on Canvas):
     1. For each of the following circumstances, state whether a continuous function f defined on [0, 1] must have a root (aka a zero, a solution to f(x) = 0) or might not have a root under those circumstances:
        1. f(0) < 0 and f(1) < 0,
        2. f(0) < 0 and f(1) > 0,
        3. f(0) > 0 and f(1) < 0,
        4. f(0) > 0 and f(1) > 0.
     2. For each of the following intervals, state whether a continuous function defined on that interval must have a maximum on the interval or might not have a maximum on the interval:
        1. [0, 1],
        2. [0, ∞),
        3. (0, 1],
        4. (0, ∞).
   • Discuss this on Canvas.
   • Exercises from the textbook due on June 5 Thursday (submit these through MyLab): 2.5.55, 2.5.57, 2.5.59, 4.1.1, 4.1.3, 4.1.5, 4.1.7, 4.1.9, 4.1.15, 4.1.17, 4.1.19.

Differentiation

7. Difference quotients and derivatives:
   • Reading from the textbook: Section 2.1 (pages 51–56).
   • Reading from my notes: Chapter 3 through Section 3.2 (pages 21&23).
   • More reading from the textbook: Chapter 3 through Section 3.1 (pages 116–118).
   • Exercises due on June 5 Thursday (submit these on Canvas):
     1. Suppose that f is a function, and for simplicity, assume that f is defined everywhere. Let y = f(x).
        1. Write down a formula for Δy, using f, x, and Δx.
        2. Write down a formula for the average rate of change of f on [a, b], using f, a, and b.
        3. Write down a formula for the average rate of change of y with respect to x, using f, x, and Δx.
     2. Suppose that f is a function and c is a number in the domain of f.
        1. Write down a formula for f‍′‍(c) (assuming that it exists) as a limit of an expression involving values of f.
        2. Fill in the blank with a single word: If f‍′‍(c) exists, then it is the _____ of f at c.
        3. The line through the point (c, f(c)) whose slope is f‍′‍(c) (if that exists) is _____ to the graph of f at that point.
   • Discuss this on Canvas.
   • Exercises from the textbook due on June 6 Friday (submit these through MyLab): 2.1.1, 2.1.3, 2.1.19, 2.1.21, 2.1.25, 3.1.1, 3.1.11, 3.1.13, 3.1.19, 3.1.21, 3.1.23, 3.1.29.
8. Derivative functions:
   • Reading from the textbook:
     • Section 3.2 (pages 120–125);
     • Section 3.3 through "Powers, Multiples, Sums, and Differences" (pages 129–132);
     • Section 3.3 "Second- and Higher-Order Derivatives" (page 136).
   • Exercises due on June 6 Friday (submit these on Canvas): Let f be a function.
     1. The function f‍′ is the _____ of f.
     2. If the domain of f‍′ is the same as the domain of f, then f is _____.
     3. The derivative of f‍′ is the _____ derivative of f.
     4. If f(x) = mx + b for all x (where m and b are constants), then what is f‍′‍(x)?
     5. If f(x) = axⁿ for all x (where a and n are constants), then what is f‍′‍(x)?
     6. If f(x) = axⁿ + mx + b for all x (where a, b, m, and n are all constants), then what is f‍′‍(x)?
   • Discuss this on Canvas.
   • Exercises from the textbook due on June 10 Tuesday (submit these through MyLab): 3.2.27, 3.2.29, 3.2.30, 3.2.31, 3.2.34, 3.2.35, 3.2.37, 3.2.39, 3.2.41, 3.2.45, 3.2.47, 3.2.49, 3.2.1, 3.2.3, 3.2.5, 3.2.13, 3.2.15, 3.3.59.
9. Rules for differentiation:
   • Reading from my notes:
     • Section 3.3 (pages 23&24);
     • Section 3.4 (pages 24&25);
     • Optional: Section 3.5 (pages 25&26).
   • Reading from the textbook:
     • Skim Section 3.3 "Products and Quotients" (pages 133–136), focussing on the picture and Example 7.a;
     • Skim: Section 3.6 (pages 154–158), focussing on Figures 3.26 and 3.27 and Examples 1, 6.a, 6.b, and 7.
   • Exercises due on June 10 Tuesday (submit these on Canvas): Let f and g be functions, and let c and n be numbers. Fill in the blanks using f, g, f‍′‍, g‍′‍, c, and/or n. Do not use d‍/‍dx or similar notation yet.)
     1. If h(x) = f(x) + g(x) for all possible x, then h‍′‍(c) = _____ if this exists.
     2. If h(x) = f(x) − g(x) for all possible x, then h‍′‍(c) = _____ if this exists.
     3. If h(x) = f(x) g(x) for all possible x, then h‍′‍(c) = _____ if this exists.
     4. If h(x) = f(x)‍/‍g(x) for all possible x, then h‍′‍(c) = _____ if this exists.
     5. If h(x) = f(x)ⁿ for all possible x, then h‍′‍(c) = _____ if this exists.
     6. If h(x) = f(g(x)) for all possible x, then h‍′‍(c) = _____ if this exists.
   • Discuss this on Canvas.
   • Exercises from the textbook due on June 11 Wednesday (submit these through MyLab): 3.3.71, 3.3.53, 3.3.65, 3.6.87, 3.6.89.
10. Differentials:
    • Reading from my notes:
      • Sections 3.6&3.7 (pages 26–28);
      • Optional: Section 3.8 (pages 28&29).
    • Exercises due on June 11 Wednesday (submit these on Canvas): Let u be a differentiable quantity.
      1. Fill in the blank: The ______ of u is du.
      2. If f is a fixed differentiable function, write a formula for the differential of f(u) using f‍′‍, u, and/or du.
      3. If n is a constant and u is a differentiable quantity, write a formula for the differential of uⁿ using n, u, and/or du.
      4. If u and v are differentiable quantities, write a formula for the differential of uv using u, v, du, and/or dv.
      5. If u and v are differentiable quantities, write a formula for the differential of u + v using u, v, du, and/or dv.
    • Discuss this on Canvas.
    • Exercises not from the textbook due on June 12 Thursday (submit these on Canvas):
      1. Find the differential of 2x² − 7x + 3. (Answer.)
      2. Find the differential of 2x² + 5x + 1.
      3. Find d(y² + 6y − 10). (Answer.)
      4. Find d(t² + 4t + 7).
      5. Find d(2r³ + 1). (Answer.)
      6. Find d(−2a³ + 5).
      7. Find d(4 + 4‍/‍x). (Answer.)
      8. Find d(6‍/‍x − 2).
      9. Find d(5 + 3√x). (Answer.)
      10. Find d(3 − 7√x).
      11. Find the differential of 10√(n + 5). (Answer.)
      12. Find the differential of 16√(k + 9).
      13. Find the differential of 3x‍/‍(x + 2). (Answer.)
      14. Find the differential of 5x‍/‍(3 + x).
11. Using differentials:
    • Reading from my notes: Section 3.9 (pages 29&30).
    • Exercises due on June 12 Thursday (submit these on Canvas): Suppose y = f(x) for a differentiable function f.
      1. Write f‍′‍(x) using x, y, dx, and/or dy.
      2. Write down all of the notations you know of for the value of the second derivative of f at x, using f, x, and/or y and differentiation operators. Be as precise as possible. (I realize that this is rather open-ended, so be prepared to revise it after my feedback.)
    • Discuss this on Canvas.
    • Exercises from the textbook due on June 13 Friday (submit these through MyLab): 3.3.1, 3.3.3, 3.3.17, 3.3.18, 3.3.19, 3.3.23, 3.3.25, 3.3.41, 3.6.23, 3.6.31, 3.6.85.
12. Implicit differentiation:
    • Reading from the textbook:
      • Section 3.7 through Example 2 in "Implicitly Defined Functions" (pages 162&163);
      • Section 3.7 "Lenses, Tangent Lines, and Normal Lines" (pages 164&165);
      • Section 3.8 through "Derivatives of Inverses of Differentiable Functions" (pages 167–169).
    • Reading from my notes: Section 3.10 (pages 30–32).
    • Exercises due on June 13 Friday (submit these on Canvas):
      1. Suppose that you have an algebraic equation involving only the variables x and y, and upon taking the differentials of both sides of this equation, you get u dx + v dy = 0, where u and v are algebraic expressions involving only x and y (but not dx or dy).
         1. Fill in the blank with a vocabulary word: If you solve the original equation for y and get a unique solution, then this defines y explicitly as a function of x; but even if you cannot or do not solve it, the equation may still define y _____ as a function of x.
         2. Fill in the blank with an algebraic expression using x, y, u, and/or v: If y is a function of x, then the derivative of y with respect to x is dy‍/‍dx = _____ (if this exists).
         3. Answer Yes or No: If the expression in the answer for part B exists for every value of x and y (no division by zero etc), then must y be a function of x?
      2. If f is a differentiable function with f‍′‍ ≠ 0 everywhere, then write an expression for (f⁻¹)‍′‍(x) using x, f, f⁻¹, and/or f‍′‍.
    • Discuss this on Canvas.
    • Exercises from the textbook due on June 16 Monday (submit these through MyLab): 3.11.23, 3.7.1, 3.7.3, 3.7.7, 3.7.21, 3.7.29, 3.7.31, 3.7.48, 3.7.50, 3.7.55, 3.8.5.

Transcendental functions

13. Exponential functions:
    • Reading from the textbook:
      • Skim: Section 1.5 (pages 33–37);
      • Skim: Section 1.6 "Logarithmic Functions" (pages 41&42);
      • Section 3.3 "Derivatives of Exponential Functions" (pages 132&133);
      • Most of Section 3.8 "The Derivatives of a^u and log_a u", specifically the part about a^u (pages 171&172).
    • Exercises due on June 16 Monday (submit these on Canvas):
      1. If e ≈ 2.71828 is the natural base, write the differential of e^u using e, u, and du.
      2. If b is any constant, write the differential of b^u using b, ln b, u, and du.
    • Discuss this on Canvas.
    • Exercises from the textbook due on June 17 Tuesday (submit these through MyLab): 1.5.11, 1.5.15, 1.5.19, 3.11.31, 3.3.5, 3.3.29, 3.3.31, 3.3.35, 3.3.51, 3.6.35, 3.6.37.
14. Logarithmic functions:
    • Reading from the textbook:
      • Skim: Section 1.6 "Properties of Logarithms" and "Applications" (pages 42–44);
      • Section 3.8 "Derivative of the Natural Logarithm Function" (pages 170&171);
      • The rest of Section 3.8 "The Derivatives of a^u and log_a u" and "Logarithmic Differentiation" (pages 172&173);
      • Optional: Section 3.8 "Irrational Exponents and the Power Rule" and "The Number e Expressed as a Limit" (pages 173–175).
    • Exercises due on June 17 Tuesday (submit these on Canvas):
      1. Write the differential of ln u using u and du.
      2. If b is any constant, then write the differential of log_b u using b, u, and du.
      3. Suppose that you have an explicit formula y = f(x) and need to find a formula for dy‍/‍dx = f‍′‍(x). If you decide to, instead of doing this directly, use logarithmic differentiation, then what would be your first step before differentiating anything?
      4. Fill in the blanks to break down these expressions using properties of logarithms. (Assume that u and v are both positive.)
         1. ln (uv) = ___.
         2. ln (u‍/‍v) = ___.
         3. ln (u^x) = ___.
    • Discuss this on Canvas.
    • Exercises from the textbook due on June 18 Wednesday (submit these through MyLab): 1.6.41, 1.6.43, 1.6.45, 1.6.49, 1.6.55, 1.6.69, 3.11.33, 3.8.21, 3.8.27, 3.8.39, 3.8.47, 3.8.51, 3.8.57, 3.8.65, 3.8.75, 3.8.89, 3.6.33.
15. Trigonometric operations:
    • Reading from the textbook:
      • Skim: Section 1.3 (pages 21–27);
      • Section 3.5 through "Derivative of the Cosine Function" (pages 148–150);
      • Section 3.5 "Derivatives of the Other Basic Trigonometric Functions" (pages 151&152).
    • Exercises due on June 18 Wednesday (submit these on Canvas):
      1. Complete the sum-angle formulas:
         1. sin(α + β) = ___;
         2. cos(α + β) = ___.
      2. Write the differential of sin u using u, du, and trigonometric operations.
      3. Write the differential of cos u using u, du, and trigonometric operations.
    • Discuss this on Canvas.
    • Exercises from the textbook due on June 19 Thursday (submit these through MyLab): 1.3.5, 1.3.7, 1.3.9, 1.3.11, 1.3.31, 1.3.33, 1.3.47, 1.3.49, 3.11.25, 3.11.26, 3.11.27, 3.11.29, 3.5.1, 3.5.3, 3.5.5, 3.5.11, 3.5.13, 3.5.15, 3.5.19, 3.5.23, 3.5.31, 3.5.35, 3.6.25, 3.6.39, 3.6.43, 3.6.47, 3.6.65, 3.8.63, 3.8.93, 3.7.11.
16. Inverse trigonometric operations:
    • Reading from the textbook:
      • Skim: The rest of Section 1.6 (pages 44–48);
      • Skim: Section 3.9 through "Inverses of tan x, cot x, sec x, and csc x" (pages 177–179);
      • Read: The rest of Section 3.9 (pages 179–182).
    • Exercises due on June 19 Thursday (submit these on Canvas):
      1. Simplify arcsin x + arccos x (where arcsin may also be written as sin⁻¹ and other ways, and similarly for arccos).
      2. Write the differential of arctan u (where arctan may also be written as tan⁻¹ and other ways) using u, du, and algebraic (not trigonometric) operations.
    • Discuss this on Canvas.
    • Exercises from the textbook due on June 20 Friday (submit these through MyLab): 1.6.71, 1.6.72, 1.6.73, 1.6.74, 3.9.1, 3.9.3, 3.9.5, 3.9.7, 3.9.8, 3.9.9, 3.9.11, 3.9.14, 3.9.15, 3.9.18, 3.9.19, 3.11.35, 3.11.36, 3.11.37, 3.9.21, 3.9.23, 3.9.25, 3.9.31, 3.9.35, 3.9.37, 3.9.39.
17. Using derivatives with respect to time:
    • Reading from my notes: Chapter 4 through Section 4.1 (pages 33&34).
    • Reading from the textbook: Section 3.4 through "Motion Along a Line" (pages 139–143).
    • Exercises due on June 20 Friday (submit these on Canvas):
      1. If an object's position P varies with time t, then the derivative dP‍/‍dt (if it exists) is the object's instantaneous _____.
      2. The absolute value of the velocity is the _____.
      3. In a technical sense, is an object's acceleration the time derivative of its speed or of its velocity?
    • Discuss this on Canvas.
    • Exercises from the textbook due on June 24 Tuesday (submit these through MyLab): 3.4.1, 3.4.3, 3.4.5, 3.4.7, 3.4.9, 3.4.13, 3.4.17, 3.4.18, 3.4.19, 3.4.23.
18. Harmonic motion:
    • Reading from the textbook: Section 3.5 "Simple Harmonic Motion" (pages 150&151).
    • Reading from my notes: Section 4.2 (pages 34&35).
    • Exercises due on June 24 Tuesday (submit these on Canvas): Suppose that a physical object is undergoing simple harmonic motion with an angular frequency of ω. Set the origin at the equilibrium point, and set the initial time when the object is at its maximum position. If this maximum position is A, then write down, as a function of time t (and using the constants ω and A):
      1. the object's position x,
      2. its velocity v, and
      3. its acceleration a.
      Check that a = −ω²x holds.
    • Discuss this on Canvas.
    • Exercises from the textbook due on June 25 Wednesday (submit these through MyLab): 3.5.57, 3.5.58, 3.5.63, 3.5.64, 3.6.99, 3.6.100.

Applications of differentiation

19. Related rates:
    • Reading from my notes: Review: The end of Section 3.9 in the middle of page 30.
    • Reading from the textbook: Section 3.10 (pages 184–188).
    • Exercises due on June 25 Wednesday (submit this on Canvas): Look at Example 3 on page 186 in Section 3.10 of the textbook. In the course of solving this, the textbook writes down five equations that are derived from the set-up (rather than from other equations):
      1. s² = x² + y²;
      2. x = 0.8;
      3. y = 0.6;
      4. dy‍/‍dt = −60; and
      5. ds‍/‍dt = 20.
      For each of these equations, in the context of this example, state (Yes or No) whether it makes sense to differentiate the equation with respect to time, that is to take the time derivative of both sides of the equation. (You can answer this from only understanding the set-up to the example; even if the textbook never differentiates an equation to solve the problem, it might still make sense to do so, or it might not.)
    • Discuss this on Canvas.
    • Exercises from the textbook due on June 26 Thursday (submit these through MyLab): 3.10.1, 3.10.3, 3.10.7, 3.10.11, 3.10.13, 3.10.15, 3.10.23, 3.10.27, 3.10.30, 3.10.31, 3.10.41.
20. Linear approximation:
    • Reading from my notes: Sections 4.3&4.4 (pages 35&36).
    • Reading from the textbook:
      • Section 3.11 through "Linearization" (pages 192–195);
      • Section 3.11 "Estimating with Differentials" (pages 197&198);
      • Optional: Section 3.11 "Error in Differential Approximation" (pages 198&199);
      • Section 3.11 "Sensitivity to Change" (page 200).
    • Exercises due on June 26 Thursday (submit these on Canvas):
      1. If a is a real number and f is a function that is differentiable at a, then give a formula for the linear approximation to f near a.
      2. If L is the linear approximation to f near a, then give L(a) and L‍′‍(a) in terms of values of f and its derivative.
      3. If f is a differentiable function, then about how much does the value (output) of f change at that point if you increase the argument (input) from x by about Δx? (Your answer should involve only f(x) and/or f‍′‍(x), as well as the change Δx or dx.
      4. If dy‍/‍dx = −3 when x = a, while dy‍/‍dx = 2 when x = b, then is the quantity y more or less sensitive to small changes in x when x ≈ a compared to when x ≈ b?
    • Discuss this on Canvas.
    • Exercises from the textbook due on June 27 Friday (submit these through MyLab): 3.11.1, 3.11.2, 3.11.3, 3.11.5, 3.11.7, 3.11.9, 3.11.11, 3.11.15, 3.4.28, 3.11.51, 3.11.52, 3.11.53, 3.11.57.
21. Mean-value theorems:
    • Reading from my notes: Section 4.6 (pages 37&38).
    • Reading from the textbook:
      • Section 4.2 through "A Physical Interpretation" (pages 220–223);
      • The statement and proof of Theorem 7 from Section 4.5 and the following paragraph (pages 251&252).
    • Exercises due on June 27 Friday (submit these on Canvas): There are three increasingly general versions of the Mean Value Theorem: Rolle's, Lagrange's (the usual form), and Cauchy's. Each of them says that if f is continuous on the nontrivial compact interval [a, b] and differentiable on its interior interval (a, b) (and maybe some other conditions), then there is at least one number c in the interval (a, b) such that … something about f‍′‍(c). Fill in the blank with an equation indicating what that something is:
      1. Rolle: If f is as described above and f(a) = f(b), then some c exists in (a, b) such that _____.
      2. Lagrange: If f is as described above, then some c exists in (a, b) such that _____.
      3. Cauchy: If f and g are as described above and g‍′‍(x) ≠ 0 whenever a < x < b, then some c exists in (a, b) such that _____.
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 1 Tuesday (submit these through MyLab): 4.2.1, 4.2.5, 4.2.9, 4.2.11, 4.2.13, 4.2.21, 4.2.25.
22. Monotony and concavity:
    • Reading from the textbook:
      • Section 4.3 through "Increasing Functions and Decreasing Functions" (pages 228&229);
      • Section 4.4 through "Points of Inflection" (pages 233–237).
    • Reading from my notes: Section 4.8 (page 39).
    • Exercises due on July 1 Tuesday (submit these on Canvas): Suppose that I is a nontrivial interval and that f is a function that is differentiable on I.
      1. Fill in each blank with an order relation (<, >, ≤, or ≥):
         1. If f‍′‍(x) ___ 0 for every x in I, then f is (strictly) increasing on I;
         2. If f‍′‍(x) ___ 0 for every x in I, then f is (strictly) decreasing on I;
         3. If f is increasing on I, then f‍′‍(x) ___ 0 for every x in I;
         4. If f is decreasing on I, then f‍′‍(x) ___ 0 for every x in I;
      2. Fill in each blank with ‘upward’ or ‘downward’:
         1. If the derivative f‍′ is increasing on I, then f is concave _____ on I;
         2. If the derivative f‍′ is decreasing on I, then f is concave _____ on I;
         3. If f is twice differentiable on I and f‍″ is positive on I, then f is concave _____ on I;
         4. If f is twice differentiable on I and f‍″ is negative on I, then f is concave _____ on I.
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 2 Wednesday (submit these through MyLab): 4.3.15, 4.3.17, 4.3.71, 4.3.73, 4.3.76, 4.4.97, 4.4.107, 4.4.113, 4.4.117, 4.4.119.
23. L'Hôpital's Rule:
    • Reading from my notes: Section 4.7 (page 39).
    • Reading from the textbook:
      • Section 4.5 through "Indeterminate Powers" (pages 246–251);
      • Optional: The rest of Section 4.5 "Proof of L'Hôpital's Rule" (page 251, page 252).
    • Exercises due on July 2 Wednesday (submit these on Canvas):
      1. If D is any direction in the variable x, and if f‍′‍(x)‍/‍g‍′‍(x) exists in that direction, then under which of the following conditions does L'Hôpital's Rule guarantee that lim_D (f(x)‍/‍g(x)) = lim_D (f‍′‍(x)‍/‍g‍′‍(x)) if the latter exists? (Say Yes or No for each.)
         1. lim_D f(x) and lim_D g(x) are both zero;
         2. lim_D f(x) is a nonzero real number while lim_D g(x) is zero;
         3. lim_D f(x) is zero while lim_D g(x) is a nonzero real number;
         4. lim_D f(x) and lim_D g(x) are both non-zero real numbers;
         5. lim_D f(x) and lim_D g(x) are both infinite.
      2. Given the following indeterminate forms, if you want to use L'Hôpital's Rule, for which of these would you first find the limit of the natural logarithm? (Say Yes or No for each.)
         1. 0 ⋅ ∞;
         2. ∞ − ∞;
         3. 0⁰;
         4. 1^∞.
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 3 Thursday (submit these through MyLab): 4.5.1, 4.5.3, 4.5.5, 4.5.11, 4.5.13, 4.5.15, 4.5.21, 4.5.37, 4.5.51, 4.5.55, 4.5.59, 4.5.60.

More applications

24. Absolute extrema:
    • Reading from the textbook: Section 4.1 "Finding extrema" (pages 215–217).
    • Exercises due on July 3 Thursday (submit these on Canvas):
      1. If a function f whose domain is [−1, 1] has an absolute maximum at 0, then what are the two possibilities for f‍′‍(0)?
      2. If a function f whose domain is [−1, 1] has a nonzero derivative everywhere on its domain, then what are the two possible places where it might have an absolute minimum?
      3. Suppose a function f is defined on (at least) the interval [0, 10], and the only places in [0, 10] where f‍′ is zero or undefined are at 0, 3, and 5. Suppose that f(0) = 2, f(3) = 7, f(5) = 3, and f(10) = 0.
         1. Where is the maximum of f on [0, 10]?
         2. What is the maximum of f on [0, 10]?
         3. Where is the minimum of f on [0, 10]?
         4. What is the minimum of f on [0, 10]?
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 7 Monday (submit these through MyLab): 4.1.11–14, 4.1.23, 4.1.27, 4.1.37, 4.1.39, 4.1.41.
25. Local extrema:
    • Reading from the textbook:
      • The rest of Section 4.3 (pages 229–231);
      • Section 4.4 "Second Derivative Test for Local Extrema" through the paragraph after the Proof of Theorem 5 (page 237).
    • Exercises due on July 7 Monday (submit these on Canvas):
      1. Suppose that I is an interval in the real line, c is a number in the interior of I (so not an endpoint of I), and f is a function defined on (at least) I. Also suppose that f is continuous on I and differentiable on I except possibly at c. (So f must be continuous at c, but may or may not be differentiable there.) For each of the following circumstances (for values of x in I), state whether f has a local maximum at c, a local minimum at c, both, or neither.
         1. If f‍′‍(x) < 0 when x < c, while also f‍′‍(x) < 0 when x > c;
         2. If f‍′‍(x) < 0 when x < c, while instead f‍′‍(x) > 0 when x > c;
         3. If f‍′‍(x) > 0 when x < c, while instead f‍′‍(x) < 0 when x > c;
         4. If f‍′‍(x) > 0 when x < c, while also f‍′‍(x) > 0 when x > c.
      2. Suppose that I is an interval in the real line, c is a number in the interior of I (so not an endpoint of I), and f is a function that is twice differentiable on (at least) I. For each of the following circumstances, state whether f must have a local maximum at c, f must have a local minimum at c, or the given information is not enough to tell.
         1. If f‍′‍(c) = 0 and f‍″‍(c) < 0.
         2. If f‍′‍(c) = 0 and f‍″‍(c) = 0.
         3. If f‍′‍(c) = 0 and f‍″‍(c) > 0.
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 8 Tuesday (submit these through MyLab): 4.3.1, 4.3.3, 4.3.5, 4.3.7, 4.3.13, 4.3.19, 4.3.23, 4.3.29, 4.3.33, 4.3.43, 4.4.111, 4.4.112, 4.4.115, 4.4.119, 4.4.121.
26. Graphing:
    • Reading from the textbook:
      • Section 1.4 (pages 29–32);
      • Section 2.6 "Horizontal Asymptotes" and "Oblique Asymptotes" (pages 99–102);
      • The rest of Section 4.4 (pages 237–242).
    • Reading from my notes: Section 4.9 (pages 40&41).
    • Exercises due on July 8 Tuesday (submit these on Canvas):
      1. Suppose that a function f is continuous everywhere; has critical points at x = −20, 0, 7, and 12; potential inflection points at −20, −3, 7, and 15; with values f(−20) = −5, f(−3) = 4, f(0) = 30, f(7) = 8, f(12) = 0, and f(15) = 4; and with limits f(−∞) = ∞ and f(∞) = 6. What would be an appropriate graphing window to show the graph of this function?
      2. Fill in the blanks with linear equations in x and/or y:
         1. If f is discontinuous at 4 and lim_x→4⁺ f(x) = ∞, then y = f(x) has _____ as an asymptote;
         2. If lim_x→∞ f(x) = 3, then y = f(x) has _____ as an asymptote;
         3. If f is differentiable at sufficiently large inputs, lim_x→∞ f‍′‍(x) = 2, and lim_x→∞ (f(x) − 2x) = 3, then y = f(x) has _____ as an asymptote.
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 9 Wednesday (submit these through MyLab): 4.4.1, 4.4.3, 4.4.5, 4.4.7, 4.4.11, 4.4.19, 4.4.23, 4.4.25, 4.4.39, 4.4.41, 4.4.45, 4.4.59, 4.4.93, 4.4.94, 4.4.95, 4.4.99, 4.4.100.
27. Applied optimization:
    • Reading from my notes: Sections 4.10&4.11 (pages 41&42).
    • Reading from the textbook:
      • Section 3.4 "Derivatives in Economics and Biology" (pages 143–145);
      • Section 4.6 (pages 255–259).
    • Exercises due on July 9 Wednesday (submit these on Canvas):
      1. If y = f(x), where f is a differentiable function, and x can take any value, then what should f‍′‍(x) be to maximize y?
      2. If the limit of y, as x approaches 1, is ∞, then is there a maximum value of y, and if so, then what is it?
      3. If y takes only positive values and the limit of y, as x approaches ∞, is 0, then is there a minimum value of y, and if so, then what is it?
      4. If cost C is a function of quantity q, then is C‍/‍q the marginal cost or the average cost? What about dC‍/‍dq?
      5. If you wish to maximize profit, then what do you want the marginal profit to be (typically)?
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 10 Thursday (submit these through MyLab): 4.6.1, 4.6.3, 4.6.7, 4.6.9, 4.6.11, 4.6.13, 4.6.15, 4.6.29, 4.6.31, 4.6.43, 4.6.45, 3.4.21, 4.6.57, 4.6.59, 4.6.62.
28. Newton's Method:
    • Reading from my notes: Section 4.5 (page 37).
    • Reading from the textbook: Section 4.7 (pages 266–269).
    • Exercises due on July 10 Thursday (submit these on Canvas):
      1. If you are attempting to use Newton's Method to solve f(x) = 0, and your first guess is x ≈ x₀, then write down a formula for your next guess x ≈ x₁ using x₀, f, and f‍′‍.
      2. Yes or No: If f is differentiable everywhere, f‍′‍(x) is never zero, and f(x) = 0 has a solution, then is Newton's Method guaranteed to find a solution?
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 11 Friday (submit these through MyLab): 4.7.1, 4.7.3, 4.7.5, 4.7.11, 4.7.13, 4.7.14, 4.7.31, 4.7.32, 4.7.33.
29. Antidifferentiation:
    • Reading from the textbook:
      • Section 4.2 "Mathematical Consequences" and "Finding Velocity and Position from Acceleration" (pages 223–224);
      • Optional: The rest of Section 4.2 (pages 224–226);
      • Section 4.8 through "Finding Antiderivatives" (pages 271–274);
      • Section 4.8 "Indefinite Integrals" (pages 276&277).
    • Reading from my notes: Section 5.2 (pages 44&45).
    • Exercises due on July 11 Friday (submit these on Canvas): For simplicity, suppose that f and g are differentiable everywhere.
      1. Fill in the blank with a single word: If f‍′‍(x) = 0 for every x, then f is _____.
      2. Fill in the blank with a single word: If f‍′‍(x) = g‍′‍(x) for every x and f(c) = g(c) for some c, then f and g are _____.
      3. Fill in the blank with a single word: If f‍′ is constant, then f is _____.
      4. Fill in the blank with a mathematical expression: ∫ f‍′‍(x) dx = _____. (If you introduce a new variable, state what it means.)
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 15 Tuesday (submit these through MyLab): 4.2.29, 4.2.31, 4.2.39, 4.2.43, 4.2.48, 4.8.1, 4.8.3, 4.8.5, 4.8.9, 4.8.11, 4.8.13, 4.8.15, 4.8.17, 4.8.19, 4.8.21, 4.8.23, 4.8.27, 4.8.29, 4.8.35, 4.8.39, 4.8.41, 4.8.45, 4.8.49, 4.8.51, 4.8.55, 4.8.61, 4.8.65, 4.8.83.

Integration

30. Riemann integration:
    • Reading from the textbook:
      • Chapter 5 through Section 5.1 (pages 290–298);
      • Section 5.2 through Example 3 (pages 300–302).
      • Section 5.2 "Riemann Sums" (pages 304–306);
      • Section 5.3 (pages 307–316).
    • Reading from my notes: Chapter 5 through Section 5.1 (page 43).
    • Exercises due on July 15 Tuesday (submit these on Canvas):
      1. Consider the interval [0, 100], and let this interval be partitioned into 5 subintervals, with endpoints 0, 13, 28, 35, 56, and 100. Also, let this partition be tagged with the numbers 7, 24, 35, 53, and 80.
         1. State the norm/mesh of this partition;
         2. If f is a function defined on [0, 100], write down the Riemann sum for f over this tagged partition (since you don't know what f is, your answer will involve unevaluated values of f).
      2. Suppose that the integral of f from 3 to 5 is 5, and the integral of g from 3 to 5 is 7. (That is, ∫₃⁵ f(x) dx = 5 and ∫₃⁵ g(x) dx = 7.) What is the integral of f + g from 3 to 5? (That is, what is ∫₃⁵ (f(x) + g(x)) dx?)
      3. Suppose that the integral of f from 3 to 5 is 5, and the integral of f from 5 to 8 is 4. (That is, ∫₃⁵ f(x) dx = 5 and ∫₅⁸ f(x) dx = 4.) What is the integral of f from 3 to 8? (That is, what is ∫₃⁸ f(x) dx?)
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 16 Wednesday (submit these through MyLab): 5.1.1, 5.1.2, 5.1.4, 5.1.5, 5.1.7, 5.1.8, 5.1.9, 5.1.11, 5.1.13, 5.1.14, 5.1.15, 5.1.16, 5.1.17, 5.1.19, 5.2.37, 5.2.39, 5.2.41, 5.2.42, 5.3.9, 5.3.11, 5.3.13, 5.3.19, 5.3.23, 5.3.27, 5.3.71.
31. The Fundamental Theorem of Calculus:
    • Reading from the textbook: Section 5.4 through "The Relationship Between Integration and Differentiation" (pages 320–327).
    • Reading from my notes: Section 5.3 (pages 45&46).
    • Exercises due on July 16 Wednesday (submit these on Canvas):
      1. If f is continuous everywhere, then what is the derivative of ∫₀^x f(t) dt with respect to x?
      2. If g is continuously differentiable everywhere, then what is ∫_a^b g‍′‍(t) dt?
      3. If f is continuous everywhere, define F so that ∫ f(x) dx = F(x) + C; what is ∫_a^b f(t) dt?
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 17 Thursday (submit these through MyLab): 5.4.1, 5.4.7, 5.4.9, 5.4.11, 5.4.13, 5.4.15, 5.4.23, 5.4.29, 5.4.39, 5.4.43, 5.4.47, 5.4.51, 5.4.79.
32. Integration by substitution:
    • Reading from the textbook:
      • Section 5.5 (pages 332–337);
      • Section 5.6 through "Definite Integrals of Symmetric Functions" (pages 339–342).
    • Reading from my notes: Section 5.5 (pages 47&48).
    • Exercises due on July 17 Thursday (submit these on Canvas):
      1. Fill in the blanks:
         1. ∫ e^kx dx = _____;
         2. ∫ sin(kx) dx = _____;
         3. ∫ cos(kx) dx = _____.
      2. Suppose that F and g are differentiable functions, with f = F‍′‍. What is ∫ f(g(x)) g‍′‍(x) dx?
      3. Suppose that f and g are functions; for simplicity, assume that they're both continuously differentiable everywhere. Write ∫_a^b f(g(x)) g‍′‍(x) dx as an integral in which g‍′ does not appear.
      4. Suppose you wish to integrate sin x cos x dx from x = 0 to x = π‍/‍2, using the substitution u = sin x (so that du = cos x dx). Explain the mistake in this calculation: ∫₀^π‍/‍2 sin x cos x dx = ∫₀^π‍/‍2 u du = (½u²)|‍₀^π‍/‍2 = ½(π‍/‍2)² − ½(0)² = π²‍/‍8. (For the record, the correct value of the integral is actually ½.)
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 18 Friday (submit these through MyLab): 5.5.1, 5.5.3, 5.5.5, 5.5.7, 5.5.15, 5.5.17, 5.5.21, 5.5.25, 5.5.27, 5.5.31, 5.5.35, 5.5.39, 5.5.47, 5.5.55, 5.5.61, 5.6.1, 5.6.3, 5.6.5, 5.6.7, 5.6.9, 5.6.13, 5.6.19, 5.6.37, 5.6.41, 5.6.45.
33. Differential equations:
    • Reading from the textbook: Section 4.8 "Initial Value Problems and Differential Equations" and "Antiderivatives and Motion" (pages 274&275).
    • Reading from my notes:
      • Section 5.4 (page 46);
      • Chapter 6 through Section 6.3 (pages 51–53), especially Section 6.3.
    • Exercises due on July 18 Friday (submit these on Canvas): Notice that d(x ln x − x) = ln x dx and that (x ln x − x)|‍_x=1 = −1. (You should be able to check these, but you don't have to.) Use these facts here:
      1. Find the general solution of F‍′‍(x) = ln x;
      2. Find the particular solution of F‍′‍(x) = ln x with F(1) = 3.
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 22 Tuesday (submit these through MyLab): 4.8.95, 4.8.97, 4.8.105, 5.5.73, 5.5.75, 4.2.40, 4.2.45, 4.2.47.
34. Planar area and arclength:
    • Reading from the textbook:
      • Section 5.4 "Total Area" (pages 327&328);
      • The rest of Section 5.6 (pages 342–345);
      • Section 6.3 (pages 375–379).
    • Exercises due on July 22 Tuesday (submit these on Canvas):
      1. Suppose that a and b are real numbers with a ≤ b and f and g are functions, both continuous on [a, b], with f ≥ g on [a, b]. What is the area of the region of the (x, y)-plane bounded by x = a, x = b, y = f(x), and y = g(x)?
      2. Suppose that c and d are real numbers with c ≤ d and f and g are functions, both continuous on [c, d], with f ≥ g on [c, d]. What is the area of the region of the (x, y)-plane bounded by x = f(y), x = g(y), y = c, and y = d?
      3. Suppose that a and b are real numbers with a ≤ b and f is a function, continuously differentiable on [a, b]. What is the length of the curve in the (x, y)-plane given by y = f(x) and bounded by x = a and x = b?
      4. Suppose that c and d are real numbers with c ≤ d and g is a function, continuously differentiable on [c, d]. What is the length of the curve in the (x, y)-plane given by x = g(y) and bounded by y = c and y = d?
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 23 Wednesday (submit these through MyLab): 5.6.49, 5.6.53, 5.6.57, 5.6.59, 5.6.62, 5.6.69, 5.6.71, 5.6.77, 5.6.83, 5.6.89, 5.6.101, 6.3.1, 6.3.3, 6.3.5, 6.3.7, 6.3.11, 6.3.15.
35. Volume of revolution:
    • Reading from the textbook:
      • Chapter 6 through Section 6.1 (pages 356–363);
      • Section 6.2 (pages 367–372).
    • Exercises due on July 23 Wednesday (submit these on Canvas):
      1. Suppose that a and b are real numbers with a ≤ b, and r and R are functions, both continuous on [a, b], with R ≥ r ≥ 0 on [a, b]. What is the volume of the solid obtained by revolving, around the x-axis, the region of the (x, y)-plane bounded by x = a, x = b, y = r(x), and y = R(x)?
      2. Suppose that c and d are real numbers with c ≤ d, and r and R are functions, both continuous on [c, d], with R ≥ r ≥ 0 on [c, d]. What is the volume of the solid obtained by revolving, around the y-axis, the region of the (x, y)-plane bounded by x = r(y), x = R(y), y = c, and y = d?
      3. Suppose that a and b are real numbers with 0 ≤ a ≤ b, and h and H are functions, both continuous on [a, b], with H ≥ h on [a, b]. What is the volume of the solid obtained by revolving, around the y-axis, the region of the (x, y)-plane bounded by x = a, x = b, y = h(x), and y = H(x)?
      4. Suppose that c and d are real numbers with 0 ≤ c ≤ d, and h and H are functions, both continuous on [c, d], with H ≥ h on [c, d]. What is the volume of the solid obtained by revolving, around the x-axis, the region of the (x, y)-plane bounded by x = h(y), x = H(y), y = c, and y = d?
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 24 Thursday (submit these through MyLab): 6.1.1, 6.1.5, 6.1.9, 6.1.13, 6.1.15, 6.1.19, 6.1.23, 6.1.27, 6.1.37, 6.1.47, 6.1.53, 6.2.1, 6.2.3, 6.2.5, 6.2.9, 6.2.15, 6.2.21, 6.2.25, 6.2.27, 6.2.31, 6.2.39.
36. Surface area of revolution:
    • Section 6.4 (pages 381–384).
    • Reading from my notes: Section 5.7 (page 49).
    • Exercises due on July 24 Thursday (submit these on Canvas):
      1. Suppose that a and b are real numbers with a ≤ b and f is a function, continuously differentiable on [a, b], with f ≥ 0 on [a, b]. What is the area of the surface obtained by revolving, around the x-axis, the curve in the (x, y)-plane given by y = f(x) and bounded by x = a and x = b?
      2. Suppose that c and d are real numbers with c ≤ d and g is a function, continuously differentiable on [c, d], with g ≥ 0 on [c, d]. What is the area of the surface obtained by revolving, around the y-axis, the curve in the (x, y)-plane given by x = g(y) and bounded by y = c and y = d?
      3. Suppose that a and b are real numbers with 0 ≤ a ≤ b and f is a function, continuously differentiable on [a, b]. What is the area of the surface obtained by revolving, around the y-axis, the curve in the (x, y)-plane given by y = f(x) and bounded by x = a and x = b? (This is not in the textbook, but it's in my notes.)
      4. Suppose that c and d are real numbers with 0 ≤ c ≤ d and g is a function, continuously differentiable on [c, d]. What is the area of the surface obtained by revolving, around the x-axis, the curve in the (x, y)-plane given by x = g(y) and bounded by y = c and y = d? (This is not in the textbook, but it's in my notes.)
    • Discuss this on Canvas.
    • Exercises from the textbook due on July 25 Friday (submit these through MyLab): 6.4.9, 6.4.13, 6.4.15, 6.4.17, 6.4.19, 6.4.21.

Quizzes

1. Continuity and limits:
   • Date available: June 6 Friday.
   • Date due: June 9 Monday.
   • Corresponding problem sets: 1–6.
   • 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 except for those in #2 and #9.
2. Differentiation:
   • Date available: June 20 Friday.
   • Date due: June 23 Monday.
   • Corresponding problem sets: 7–12.
   • Help allowed: Your notes, 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 except for that in #2. Be sure to show what limit you take in #1 (where you use the definition of derivative).
3. Transcendental functions:
   • Date available: June 27 Friday.
   • Date due: June 30 Monday.
   • Corresponding problem sets: 13–18.
   • Help allowed: Your notes, 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.
4. Applications of differentiation:
   • Date available: July 11 Friday.
   • Date due: July 14 Monday.
   • Corresponding problem sets: 19–23.
   • Help allowed: Your notes, 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 except for those in #4.
5. More applications:
   • Date available: July 18 Friday.
   • Date due: July 21 Monday.
   • Corresponding problem sets: 24–29.
   • Help allowed: Your notes, 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 except in #1 (where your answer for each part should follow from your answers to previous parts).
6. Integration:
   • Date available: July 25 Friday.
   • Date due: July 28 Monday.
   • Corresponding problem sets: 30–36.
   • Help allowed: Your notes, 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.

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 book 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).

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-1600/2025SS/.