MATH-1600-LN01

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: Mondays through Fridays from 10:00 to 10:50 in room U103.
• Final exam: May 17 Friday from 10:00 to 11:40 in room U103 (or by appointment).

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 11:00 to 12:00,
  • on Tuesdays and Thursdays from 2:30 PM to 3:30, and
  • by appointment,
  in room U9C and over Zoom meeting 610-024-2876.

Readings

Continuity and limits

1. General review:
   • 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 7);
     • Optional: Section 1.5 (pages 7&8).
   • Exercises due on January 23 Tuesday (submit these on Canvas or in class):
     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?
   • Exercises from the textbook due on January 24 Wednesday (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, 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).
   • Exercises due on January 24 Wednesday (submit these on Canvas or in class):
     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?
   • Exercises from the textbook due on January 25 Thursday (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.
3. Limits involving infinity:
   • Reading from my notes: Sections 2.3&2.4 (pages 11–13).
   • Reading from the textbook:
     • Section 2.6 through the first paragraph of "Finite Limits as x → ±∞" (page 96);
     • Section 2.6: "Infinite Limits" before Example 13 (pages 102&103).
   • Exercises due on January 25 Thursday (submit these on Canvas or in class):
     1. 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 _____.
     2. 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 _____.
     3. Yes/No: If f(x) always gets larger as x gets larger, does that necessarily mean that lim_x→∞ f(x) = ∞?
   • Exercises from the textbook due on January 26 Friday (submit these through MyLab): 2.6.1, 2.6.2, 2.6.75, 2.6.76, 2.6.77.
4. Continuity informally:
   • Reading from my notes: Chapter 2 through Section 2.1 (pages 9&10).
   • Reading from the textbook: Section 2.5 through "Continuity at a Point" (pages 85–88).
   • Exercises due on January 26 Friday (submit these on Canvas or in class):
     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?
   • Exercises from the textbook due on January 29 Monday (submit these through MyLab): 2.5.1, 2.5.3, 2.5.7, 2.5.9, 2.5.11.
5. Defining continuity:
   • Reading from my notes: Section 2.2 (pages 10&11).
   • Reading from the textbook: Section 2.3: "Examples: Testing the Definition", "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 January 29 Monday (submit these on Canvas or in class): Fill in the blank: Suppose that f is a function and suppose that c is a real number. For simplicity, suppose that f is defined everywhere.
     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, as long as |x − c| < δ, then |f(x) − f(c)| < ε. This means that f is _____ at c.
     2. Instead 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.
   • Exercises from the textbook due on January 30 Tuesday (submit these through MyLab): 2.3.7, 2.3.9, 2.3.11, 2.3.13, 2.3.15, 2.3.17, 2.3.23, 2.3.27.
6. 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: Section 2.6: The rest of "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 January 30 Tuesday (submit these on Canvas or in class):
     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 f is always positive and the limit of 1/f approaching c is 0. (That is, f(x) > 0, and lim_x→c (1/f(x)) = 0.) Then what is the limit of f approaching c? (That is, lim_x→c f(x) = _____.)
     3. Given a function f, define a new function g so that g(t) = f(1/t) for all possible t, and suppose that the limit of g approaching 0⁺ is L. (That is, lim_t→0⁺ f(1/t) = L.) What is the limit of f approaching infinity? (That is, lim_x→∞ f(x) = _____.)
   • Exercises from the textbook due on January 31 Wednesday (submit these through MyLab): 2.5.41, 2.5.45, 2.3.49, 2.2.15, 2.2.19.
7. 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: "Continuous Functions", "Inverse Functions and Continuity", "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).
   • Exercises due on January 31 Wednesday (submit these on Canvas or in class):
     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. Suppose that c and L are real numbers, f is a function defined on (−∞, c), and g is a function on (c, ∞). If h is defined piecewise so that h(x) = f(x) for x < c, while h(c) = L and h(x) = g(x) for x > c, then 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 _____.
     3. What is the limit of (sin x)/x as x → 0?
   • Exercises from the textbook due on February 1 Thursday (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.
8. Calculating with infinity:
   • Reading from the textbook:
     • Section 2.6: The rest of "Finite Limits as x → ±∞", "Limits at Infinity of Rational Functions" (pages 98&99);
     • Section 2.6: Examples 13&14 (pages 103&104);
     • Section 2.6: "Dominant Terms" (pages 106&107).
   • Exercises due on February 1 Thursday (submit these on Canvas or in class):
     1. 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?
     2. What are the limits of e^x as x → ∞ and as x → −∞?
     3. What are the limits of ln x as x → ∞ and as x → 0⁺?
   • Exercises from the textbook due on February 2 Friday (submit these through MyLab): 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.
9. Theorems about continuous functions:
   • Reading from my notes:
     • Optional: Section 2.8 (pages 17&18).
     • Section 2.9 (pages 18&19).
   • 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 February 2 Friday (submit these on Canvas or in class):
     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, ∞).
   • Exercises from the textbook due on February 5 Monday (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

10. Differences and difference quotients:
    • Reading from the textbook: Section 2.1 (pages 51–56).
    • Reading from my notes: Chapter 3 through Section 3.1 (pages 21&22).
    • Exercises due on February 5 Monday (submit these on Canvas or in class): 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.
    • Exercises from the textbook due on February 6 Tuesday (submit these through MyLab): 2.1.1, 2.1.3, 2.1.19, 2.1.21, 2.1.25.
11. Derivatives as limits:
    • Reading from my notes: Section 3.2 (pages 22&23).
    • Reading from the textbook: Chapter 3 through Section 3.1 (pages 116–118).
    • Exercises due on February 6 Tuesday (submit these on Canvas or in class): 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.
    • Exercises from the textbook due on February 7 Wednesday (submit these through MyLab): 3.1.1, 3.1.11, 3.1.13, 3.1.19, 3.1.21, 3.1.23, 3.1.29.
12. Derivative functions:
    • Reading from the textbook:
      • Section 3.2 (pages 120–125);
      • Section 3.3: "Second- and Higher-Order Derivatives" (page 136).
    • Exercises due on February 7 Wednesday (submit these on Canvas or in class): 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.
    • Exercises from the textbook due on February 8 Thursday (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.
13. Differentiating polynomials:
    • Reading from the textbook: Section 3.3 through "Powers, Multiples, Sums, and Differences" (pages 129–132);
    • Exercises due on February 8 Thursday (submit these on Canvas or in class):
      1. If f(x) = mx + b for all x (where m and b are constants), then what is f⁠′⁠(x)?
      2. If f(x) = axⁿ for all x (where a and n are constants), then what is f⁠′⁠(x)?
      3. If f(x) = axⁿ + mx + b for all x (where a, b, m, and n are all constants), then what is f⁠′⁠(x)?
    • Exercises from the textbook due on February 9 Friday (submit these through MyLab): 3.2.1, 3.2.3, 3.2.5, 3.2.13, 3.2.15, 3.3.59.
14. 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:
      • Section 3.3: "Products and Quotients" (pages 133–136), skipping Examples 6.b and 7.b;
      • Skim: Section 3.6 (pages 154–158), focussing on Examples 1, 6.a, 6.b, and 7.
    • Exercises due on February 13 Tuesday (submit these on Canvas or in class): For simplicity, suppose that f and g are differentiable everywhere. What is h⁠′⁠(x) in the following situations? (Write your answers using values of f, g, f⁠′⁠, and/or g⁠′⁠. Do not use d/dx.)
      1. h(x) = f(x) + g(x) for all x.
      2. h(x) = f(x) g(x) for all x.
      3. h(x) = f(x)ⁿ for all x, where n is a non-zero real-valued constant such that f(x)ⁿ is always defined.
      4. h(x) = f(g(x)) for all x.
    • Exercises from the textbook due on February 14 Wednesday (submit these through MyLab): 3.3.71, 3.3.53, 3.3.65, 3.6.87, 3.6.89.
15. Differentials:
    • Reading from my notes:
      • Section 3.6 (pages 26&27);
      • Section 3.8 (pages 28&29).
    • Reading from the textbook: Optional: Section 3.11: "Differentials" (pages 196&197).
    • Exercises due on February 14 Wednesday (submit these on Canvas or in class): 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 du.
    • Exercises not from the textbook due on February 15 Thursday (submit these on Canvas or in class):
      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).
16. Using differentials:
    • Reading from my notes: Section 3.7 (pages 27&28).
    • Exercises due on February 15 Thursday (submit these on Canvas or in class):
      1. 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.
      2. If u and v are differentiable quantities, write a formula for the differential of uv using u, v, du, and/or dv.
      3. If u and v are differentiable quantities, write a formula for the differential of u + v using u, v, du, and/or dv.
    • Exercises from the textbook due on February 16 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.
17. Implicit differentiation:
    • Reading from my notes: Section 3.9 (pages 29&30).
    • Reading from the textbook:
      • Section 3.7 through Example 2 in "Implicitly Defined Functions" (pages 162&163);
      • Optional: The rest of Section 3.7 (pages 163–165).
    • Exercises due on February 16 Friday (submit these on Canvas or in class): Suppose that you have an algebraic equation involving only the variables x and y.
      1. Fill in the blank using a word or words: If you solve this 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 using mathematical symbols: If upon differentiating 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), then the derivative of y with respect to x (when it exists) is dy/dx = _____.
    • Exercises from the textbook due on February 19 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.
18. Implicit and inverse function theorems:
    • Reading from the textbook: 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 February 19 Monday (submit these on Canvas or in class):
      1. Suppose that you have an equation in the variables x and y with a constant on the right-hand side, and when you take the differential of both sides, you get u dx + v dy = 0, where u and v are themselves expressions involving x and/or y. If u ≠ 0, then which variable (x or y) must be a function of which other variable (y or x)?
      2. If f is a differentiable function with f⁠′ ≠ 0 everywhere, then write an expression for (f⁻¹)⁠′⁠(x) using x, f, and f⁻¹.
    • Exercises from the textbook due on February 20 Tuesday (submit these through MyLab): 3.7.50, 3.7.55, 3.7.48, 3.8.5.

Transcendental functions

19. 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);
      • Section 3.8: Most of "The Derivatives of a^u and log_a u", specifically the part about a^u (pages 171&172).
    • Exercises due on February 20 Tuesday (submit these on Canvas or in class):
      1. If e ≈ 2.71828 is the natural base, then write the differential of e^u using e, u, and du.
      2. If b is any constant, then write the differential of b^u using b, ln b, u, and du.
    • Exercises from the textbook due on February 21 Wednesday (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.
20. Logarithmic functions:
    • Reading from the textbook:
      • Skim: Section 1.6: "Properties of Logarithms", "Applications" (pages 42–44);
      • Section 3.8: "Derivative of the Natural Logarithm Function" (pages 170&171);
      • Section 3.8: The rest of "The Derivatives of a^u and log_a u" (pages 172&173);
      • Optional: Section 3.8: "Irrational Exponents and the Power Rule", "The Number e Expressed as a Limit" (pages 173–175).
    • Exercises due on February 21 Wednesday (submit these on Canvas or in class):
      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.
    • Exercises from the textbook due on February 22 Thursday (submit these through MyLab): 1.6.45, 1.6.55, 1.6.69, 3.11.33, 3.8.21, 3.8.27, 3.8.39, 3.8.57, 3.8.75.
21. Logarithmic differentiation:
    • Reading from the textbook: Section 3.8: "Logarithmic Differentiation" (page 173).
    • Exercises due on February 27 Tuesday (submit these on Canvas or in class):
      1. 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?
      2. 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) = ___.
    • Exercises from the textbook due on February 28 Wednesday (submit these through MyLab): 1.6.41, 1.6.43, 1.6.49, 3.8.65, 3.8.47, 3.8.51, 3.6.33, 3.8.89.
22. Trigonometry review:
    • Reading from the textbook:
      • Skim: Section 1.3 (pages 21–27);
      • 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).
    • Exercises due on February 28 Wednesday (submit these on Canvas or in class): Complete the sum-angle formulas:
      1. sin(α + β) = ___.
      2. cos(α + β) = ___.
    • Exercises from the textbook due on February 29 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, 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.
23. Trigonometric operations:
    • Reading from the textbook:
      • 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 February 29 Thursday (submit these on Canvas or in class):
      1. Write the differential of sin u using u, du, and trigonometric operations.
      2. Write the differential of cos u using u, du, and trigonometric operations.
    • Exercises from the textbook due on March 1 Friday (submit these through MyLab): 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.
24. Inverse trigonometric operations:
    • Reading from the textbook: The rest of Section 3.9 (pages 179–182).
    • Exercises due on March 1 Friday (submit these on Canvas or in class):
      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.
    • Exercises from the textbook due on March 4 Monday (submit these through MyLab): 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.
25. 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 March 4 Monday (submit these on Canvas or in class):
      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?
    • Exercises from the textbook due on March 5 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.
26. 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 March 5 Tuesday (submit these on Canvas or in class): 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.
    • Exercises from the textbook due on March 6 Wednesday (submit these through MyLab): 3.5.57, 3.5.58, 3.5.63, 3.5.64.

Applications of differentiation

27. Related rates:
    • Reading from my notes: Optional: 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 March 6 Wednesday (submit this on Canvas or in class): Look at Example 3 on page 186 in Section 3.10 of the textbook. To solve this example, 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.)
    • Exercises from the textbook due on March 7 Thursday (submit these through MyLab): 3.10.1, 3.10.3, 3.10.7, 3.10.11, 3.10.13, 3.10.15, 3.10.30, 3.10.41, 3.10.23, 3.10.27, 3.10.31.
28. Linearization:
    • Reading from my notes: Section 4.3 (pages 35&36).
    • Reading from the textbook:
      • Section 3.11 through "Linearization" (pages 192–195);
      • Optional: Section 3.11 "Error in Differential Approximation" (pages 198&199).
    • Exercises due on March 7 Thursday (submit these on Canvas or in class):
      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.
    • Exercises from the textbook due on March 8 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.
29. Linear estimation:
    • Reading from my notes: Section 4.4 (page 36).
    • Reading from the textbook:
      • Section 3.11 "Estimating with Differentials" (pages 197&198);
      • Section 3.11 "Sensitivity to Change" (page 200).
    • Exercises due on March 20 Wednesday (submit these on Canvas or in class):
      1. 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.
      2. 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?
    • Exercises from the textbook due on March 21 Thursday (submit these through MyLab): 3.4.28, 3.11.51, 3.11.52, 3.11.53, 3.11.57.
30. 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);
      • Section 4.5: The statement and proof of Theorem 7, and the following paragraph (pages 251&252).
    • Exercises due on March 21 Thursday (submit these on Canvas or in class): 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 (and maybe also g) are continuous on the nontrivial compact interval [a, b] (with a < b) and differentiable on its interior interval (a, b), then there is at least one number c in the interval (a, b) such that … something about f⁠′⁠(c) (and maybe also g⁠′⁠(c)). Fill in the blank with an equation indicating what that statement 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 _____.
    • Exercises from the textbook due on March 22 Friday (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.
31. Increasing and decreasing functions:
    • Reading from the textbook: Section 4.3 through "Increasing Functions and Decreasing Functions" (pages 228&229).
    • Exercises due on March 22 Friday (submit these on Canvas or in class): Suppose that I is a nontrivial interval and that f is a function that is differentiable on I. 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.
    • Exercises from the textbook due on March 25 Monday (submit these through MyLab): 4.3.15, 4.3.17, 4.3.71, 4.3.73, 4.3.76.
32. Concavity:
    • Reading from the textbook: Section 4.4 through "Points of Inflection" (pages 233–237).
    • Reading from my notes: Section 4.8 (page 39).
    • Exercises due on March 25 Monday (submit these on Canvas or in class): Suppose that a function f is differentiable on an interval I, and 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.
    • Exercises from the textbook due on March 26 Tuesday (submit these through MyLab): 4.4.97, 4.4.107, 4.4.113, 4.4.117, 4.4.119.
33. L'Hôpital's Rule:
    • Reading from my notes: Section 4.7 (page 39).
    • Reading from the textbook:
      • Section 4.5 through "Indeterminate Forms ∞/∞ , ∞ ⋅ 0 , ∞ − ∞" (pages 246–250);
      • Optional: Section 4.5: The rest of "Proof of L'Hôpital's Rule" (page 251, page 252).
    • Exercises due on March 26 Tuesday (submit these on Canvas or in class): 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 of these five conditions.)
      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.
    • Exercises from the textbook due on March 27 Wednesday (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.
34. Advanced techniques with L'Hôpital's Rule:
    • Reading from the textbook: Section 4.5 "Indeterminate Powers" (pages 250&251).
    • Exercises due on March 27 Wednesday (submit these on Canvas or in class): 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^∞.
    • Exercises from the textbook due on March 28 Thursday (submit these through MyLab): 4.5.37, 4.5.51, 4.5.55, 4.5.59, 4.5.60.

More applications

35. Absolute extrema:
    • Reading from the textbook: Section 4.1: "Finding extrema" (pages 215–217).
    • Exercises due on April 2 Tuesday (submit these on Canvas or in class):
      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?
    • Exercises from the textbook due on April 3 Wednesday (submit these through MyLab): 4.1.11–14, 4.1.23, 4.1.27, 4.1.37, 4.1.39, 4.1.41.
36. Local extrema:
    • Reading from the textbook: The rest of Section 4.3 (pages 229–231).
    • Exercises due on April 3 Wednesday (submit these on Canvas or in class): 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.
    • Exercises from the textbook due on April 4 Thursday (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.
37. The second-derivative test:
    • Reading from the textbook: Section 4.4 "Second Derivative Test for Local Extrema" through the paragraph after the Proof of Theorem 5 (page 237).
    • Exercises due on April 4 Thursday (submit these on Canvas or in class): 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.
    • Exercises from the textbook due on April 5 Friday (submit these through MyLab): 4.4.111, 4.4.112, 4.4.115, 4.4.119, 4.4.121.
38. Graphing:
    • Reading from the textbook:
      • Section 1.4 (pages 29–32);
      • The rest of Section 4.4 (pages 237–242).
    • Exercise due on April 5 Friday (submit this on Canvas or in class): 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) = 60, f(7) = 8, f(12) = 0, and f(15) = 4; and with limits f(−∞) = −10 and f(∞) = 6. What would be an appropriate graphing window to show the graph of this function?
    • Exercises from the textbook due on April 8 Monday (submit these through MyLab): 4.4.1, 4.4.3, 4.4.5, 4.4.7, 4.4.93, 4.4.94, 4.4.95, 4.4.99, 4.4.100.
39. Graphing asymptotes:
    • Reading from my notes: Section 4.9 (pages 40&41).
    • Reading from the textbook: Section 2.6: "Horizontal Asymptotes" and "Oblique Asymptotes" (pages 99–102).
    • Exercises due on April 8 Monday (submit these on Canvas or in class): Suppose that f is differentiable everywhere, and fill in the blanks with expressions involving x and f:
      1. If the graph of y = f(x) has y = 3 as an asymptote as x → ∞, then the limit of _____, as x → ∞, is 3.
      2. If the graph of y = f(x) has y = 2x + 3 as an asymptote as x → ∞, then the limit of _____, as x → ∞, is 2, and the limit of _____, as x → ∞, is 3.
    • Exercises from the textbook due on April 9 Tuesday (submit these through MyLab): 4.4.11, 4.4.19, 4.4.23, 4.4.25, 4.4.39, 4.4.41, 4.4.45, 4.4.59.
40. Applied optimization:
    • Reading from my notes: Section 4.10 (page 41).
    • Reading from the textbook: Section 4.6 through "Examples from Mathematics and Physics" (pages 255–258).
    • Exercises due on April 9 Tuesday (submit these on Canvas or in class):
      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 u, as x approaches 1, is ∞, then is there a maximum value of u, and if so, then what is it?
      3. If u takes only positive values and the limit of u, as x approaches ∞, is 0, then is there a minimum value of u, and if so, then what is it?
    • Exercises from the textbook due on April 10 Wednesday (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.
41. Optimization in economics and finance:
    • Reading from my notes: Section 4.11 (pages 42).
    • Reading from the textbook:
      • Section 3.4: "Derivatives in Economics and Biology" (pages 143–145);
      • Section 4.6: "Examples from Economics" (pages 258&259).
    • Exercises due on April 10 Wednesday (submit these on Canvas or in class):
      1. If cost C is a function of quantity q, then is C/q the marginal cost or the average cost? What about dC/dq?
      2. If you wish to maximize profit, then what do you want the marginal profit to be (typically)?
    • Exercises from the textbook due on April 11 Thursday (submit these through MyLab): 4.6.43, 4.6.45, 4.6.57, 4.6.59.
42. Newton's Method:
    • Reading from my notes: Section 4.5 (page 37).
    • Reading from the textbook: Section 4.7 (pages 266–269).
    • Exercise due on April 11 Thursday (submit this on Canvas or in class): 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 second guess x ≈ x₁ using x₀, f, and f⁠′⁠.
    • Exercises from the textbook due on April 12 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.
43. Antidifferentiation:
    • Reading from the textbook:
      • Section 4.2: "Mathematical Consequences", "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 April 12 Friday (submit these on Canvas or in class): 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. What is ∫ f⁠′⁠(x) dx?
    • Exercises from the textbook due on April 15 Monday (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

44. Riemann sums:
    • Reading from the textbook:
      • Chapter 5 through Section 5.1 (pages 290–298);
      • The rest of Section 5.2 (pages 303–306).
    • Exercises due on April 15 Monday (submit these on Canvas or in class): 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.)
    • Exercises from the textbook due on April 16 Tuesday (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.
45. Riemann integrals:
    • Reading from the textbook: Section 5.3 (pages 307–316).
    • Reading from my notes: Chapter 5 through Section 5.1 (page 43).
    • Exercises due on April 16 Tuesday (submit these on Canvas or in class): Let f and g be functions.
      1. Suppose that ∫⁵_x=3 f(x) dx = 5 and ∫⁵_x=3 g(x) dx = 7. (That is, the integral of f from 3 to 5 is 5, and the integral of g from 3 to 5 is 7.) What is ∫⁵_x=3 (f(x) + g(x)) dx? (That is, what is the integral of f + g from 3 to 5?)
      2. Suppose that ∫⁵_x=3 f(x) dx = 5 and ∫⁸_x=5 f(x) dx = 4. (That is, the integral of f from 3 to 5 is 5, and the integral of f from 5 to 8 is 4.) What is ∫⁸_x=3 f(x) dx? (That is, what is the integral of f from 3 to 8?)
    • Exercises from the textbook due on April 17 Wednesday (submit these through MyLab): 5.3.9, 5.3.11, 5.3.13, 5.3.19, 5.3.23, 5.3.27, 5.3.71.
46. 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 April 17 Wednesday (submit these on Canvas or in class):
      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?
    • Exercises from the textbook due on April 18 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.
47. Integration by substitution:
    • Reading from the textbook: Section 5.5 (pages 332–337);
    • Reading from my notes: Section 5.5 (pages 47&48).
    • Exercises due on April 23 Tuesday (submit these on Canvas or in class):
      1. Fill in the blanks: ∫ e^kx dx = _____; ∫ sin(kx) dx = _____; ∫ cos(kx) dx = _____.
      2. Suppose that F and g are differentiable functions, with f = F⁠′⁠. What is ∫ f(g(x)) g⁠′⁠(x) dx?
    • Exercises from the textbook due on April 24 Wednesday (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.
48. Substitution with definite integrals:
    • Reading from the textbook: Section 5.6 through "Definite Integrals of Symmetric Functions" (pages 339–342).
    • Exercises due on April 24 Wednesday (submit these on Canvas or in class):
      1. 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.
      2. 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.)
    • Exercises from the textbook due on April 25 Thursday (submit these through MyLab): 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.
49. Differential equations:
    • Reading from the textbook: Section 4.8: "Initial Value Problems and Differential Equations", "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 April 25 Thursday (submit these on Canvas or in class): Notice that d(x ln x − x) = ln x dx and that (x ln x − x)|_x=1 = −1. Use these facts below:
      1. Find the general solution of F⁠′⁠(x) = ln x;
      2. Find the particular solution of F⁠′⁠(x) = ln x with F(1) = 3.
    • Exercises from the textbook due on April 26 Friday (submit these through MyLab): 4.8.95, 9.5.97, 4.8.105, 5.5.73, 5.5.75, 4.2.40, 4.2.45, 4.2.47.
50. Planar area:
    • Reading from the textbook:
      • Section 5.4 "Total Area" (pages 327&328);
      • The rest of Section 5.6 (pages 342–345).
    • Exercises due on April 26 Friday (submit these on Canvas or in class):
      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 [a, b], with f ≥ g on [a, b]. 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?
    • Exercises from the textbook due on April 29 Monday (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.
51. Arclength:
    • Reading from the textbook: Section 6.3 (pages 375–379).
    • Exercises due on April 29 Monday (submit these on Canvas or in class):
      1. 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?
      2. Suppose that c and d are real numbers with c ≤ d and g is a function, continuously differentiable on [a, b]. 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?
    • Exercises from the textbook due on April 30 Tuesday (submit these through MyLab): 6.3.1, 6.3.3, 6.3.5, 6.3.7, 6.3.11, 6.3.15.
52. 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 April 30 Tuesday (submit these on Canvas or in class):
      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 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)?
    • Exercises from the textbook due on May 1 Wednesday (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.
53. Surface area of revolution:
    • Section 6.4 (pages 381–384).
    • Reading from my notes: Section 5.7 (page 49).
    • Exercises due on May 1 Wednesday (submit these on Canvas or in class):
      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 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.)
    • Exercises from the textbook due on May 2 Thursday (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:
   • Review date: February 9 Friday.
   • Date taken: February 12 Monday.
   • Corresponding problem sets: 1–9.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
2. Differentiation:
   • Review date: February 23 Friday.
   • Date taken: February 26 Monday.
   • Corresponding problem sets: 10–18.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
3. Transcendental functions:
   • Review date: March 18 Monday.
   • Date taken: March 19 Tuesday.
   • Corresponding problem sets: 19–26.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
4. Applications of differentiation:
   • Review date: March 29 Friday).
   • Date taken: April 1 Monday).
   • Corresponding problem sets: 27–34.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
5. More applications:
   • Review date: April 19 Friday.
   • Date taken: April 22 Monday.
   • Corresponding problem sets: 35–43.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
6. Integration:
   • Review date: May 3 Friday.
   • Date taken: May 6 Monday.
   • Corresponding problem sets: 44–53.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.

Final exam

For the exam, you may use one sheet of notes that you wrote yourself. 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).

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