MATH-1400-WBP81

Course administration

• Canvas page (where you must log in).
• Help with DjVu (if you have trouble reading the DjVu files on this page).
• Official syllabus (DjVu).
• Course policies (DjVu).
• Class hours: Asynchronous online.
• Final exam: By appointment July 6–10.

Contact information

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

Readings

Try to read this introduction on the first day of the session:

• Objectives:
  • Understand what to expect from this course;
  • Know how to submit assignments.
• Reading:
  • My course policies (DjVu, same as above);
  • My online introduction.
• Reading homework due on May 19 Tuesday or ASAP thereafter (write this by hand on paper and submit it on Canvas or another way):
  1. If you want to submit something that you've written by hand on paper, how will you send me a picture of it? (submit on Canvas, attach to an email, etc).
  2. How will you get the final exam proctored? (Lincoln Testing Center, ProctorU, etc).
  You can change your mind about these later! (But let me know if you change your mind about #2.)
• Problem set from the textbook due on May 20 Wednesday or ASAP thereafter (submit this through MyLab): O.1.1, O.1.2, O.1.3, O.1.4, O.1.5, O.1.6, O.1.7, O.1.8.

Continuity and limits

1. Review of functions:
   • Objectives:
     • Review function notation;
     • Review different kinds of functions.
   • Reading from the textbook:
     • Skim: Through Section 1.4 (through page 58), especially through Section 1.1 (through page 13);
     • Skim: Section 1.6 “Inverse Functions” (pages 72&73).
   • Reading homework due on May 20 Wednesday (write this by hand on paper and submit it 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 when x is 4?
   • Problem set from the textbook due on May 21 Thursday (submit this through MyLab): 1.1.15, 1.1.17, 1.1.39, 1.1.46, 1.1.49, 1.1.51, 1.1.58, 1.1.63, 1.1.73, 1.1.83, 1.2.49, 1.3.15, 1.3.33, 1.3.49, 1.3.71, 1.4.8, 1.6.39.
2. Limits informally:
   • Reading from the textbook:
     • Section 2.1 “Limits: A Graphical Approach” (pages 93–97);
     • Section 2.2 “Infinite Limits” (pages 106&107);
     • Section 2.2 “Limits at Infinity” (pages 109–112).
   • Reading homework due on May 21 Thursday (write this by hand on paper and submit it 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) = ∞?
   • Problem set from the textbook due on May 22 Friday (submit this through MyLab): 2.1.13, 2.1.21, 2.1.25, 2.1.47, 2.1.95, 2.2.9, 2.2.11, 2.2.15.
3. Continuity informally:
   • Reading from the textbook: Section 2.3 through “Continuity Properties” (pages 118–122).
   • Reading homework due on May 22 Friday (write this by hand on paper and submit it 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?
   • Problem set from the textbook due on May 26 Tuesday (submit this through MyLab): 2.3.11, 2.3.19, 2.3.27, 2.3.35, 2.3.39, 2.3.61, 2.3.69, 2.3.71, 2.3.75, 2.3.95.
4. Differences and differentials:
   • Reading from my notes: Summary of differential calculus through “Differences and differentials of linear expressions” (pages 1&2).
   • Reading homework due on May 26 Tuesday (write this by hand on paper and submit it on Canvas):
     1. Write the difference Δ_a^b f(x) using values of the function f.
     2. Write Δ(u + v) using Δu and Δv.
     3. Write d(u + v) using du and dv.
     4. Write d(au + bv) using du and dv, assuming that a and b are constant.
   • Problem set from my notes due on May 27 Wednesday (write this by hand on paper and submit it on Canvas): 1, 3, 5, 7, 8, 9, 10 (from page 7, with some answers on page 8).

Differentiation

5. Differentials:
   • Reading from my notes:
     • The rest of Summary of differential calculus through “Strategy for calculating differentials” (pages 2–4);
     • The first half of Rules of differentiation (page 1).
   • Reading homework due on May 27 Wednesday (write this by hand on paper and submit it on Canvas):
     1. Let u be a smooth quantity. Fill in the blank: The ______ of u is du.
     2. If n is a constant and u is a smooth quantity, write a formula for the differential of uⁿ using n, u, and/or du.
     3. If k is a constant and u is a smooth quantity, write a formula for the differential of ku using k, u, and/or du.
     4. If u and v are smooth quantities, write a formula for the differential of u − v using u, v, du, and/or dv.
   • Problem set from my notes due on May 28 Thursday (write this by hand on paper and submit it on Canvas): 11, 13, 15, 16, 17, 19, 25, 27, 29, 30, 33, 37, 39, 43, 45, 47, 55, 61, 65 (from page 7, with some answers on page 8).
6. Derivatives:
   • Reading from my notes: The rest of Summary of differential calculus (pages 4–6).
   • Reading from the textbook:
     • Section 2.4 (pages 130–141), but don't worry too much about the four-step process for calculating a derivative as a limit, because you don't need that if you use the rules for differentiation from my notes;
     • Optional: Section 2.5 (pages 145–152), which approaches the rules for differentiation from a different perspective than the one in my notes.
   • Reading homework due on May 28 Thursday (write this by hand on paper and submit it on Canvas): Suppose y = f(x) for a smooth 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.)
   • Exercises from the textbook due on May 29 Friday (submit this through MyLab): 2.5.9, 2.5.11, 2.5.13, 2.5.17, 2.5.19, 2.5.21, 2.5.25, 2.5.33, 2.5.35, 2.5.41, 2.5.45, 2.5.47, 2.5.51, 4.2.17, 4.2.19.
7. Trickier derivatives:
   • Optional reading from the textbook, which again approaches the rules for differentiation from a different perspective than the one in my notes:
     • Section 3.4 (pages 204–212);
     • Section 3.3 (pages 196–201).
   • Reading homework due on May 29 Friday (write this by hand on paper and submit it on Canvas):
     1. If f is a smooth function and u is a smooth quantity, write a formula for the differential of f(u) using f‍′‍, u, and/or du.
     2. If u and v are smooth quantities, write a formula for the differential of uv using u, v, du, and/or dv;
     3. If u and v are smooth quantities, write a formula for the differential of u‍/‍v using u, v, du, and/or dv.
   • Exercises from the textbook due on June 2 Tuesday (submit this through MyLab): 3.4.17, 3.4.21, 3.4.27, 3.4.29, 3.4.47, 3.3.13, 3.3.15, 3.3.25, 3.3.29, 3.3.89.
8. Implicit differentiation:
   • Reading from the textbook: Section 3.5 before Example 3 (pages 215–219).
   • Optional reading from my notes: Review the example at the end of “Derivatives” (the middle of page 5).
   • Reading homework due on June 2 Tuesday (write this by hand on paper and submit it on Canvas): Suppose that you have an algebraic equation involving only the variables x and y.
     1. Fill in the blank with a vocabulary word: If you solve the 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. Upon taking the differentials of both sides of this equation, suppose that you get A dx + B dy = 0, where A and B are algebraic expressions involving only x and y (but not dx or dy). Fill in the blank with an algebraic expression using x, y, A, and/or B: If y is a function of x, then the derivative of y with respect to x is dy‍/‍dx = _____ (if this exists).
   • Exercises from the textbook due on June 3 Wednesday (submit this through MyLab) (assume that they mean dy/‍dx when they write y′): 3.5.17, 3.5.19, 3.5.21, 3.5.23, 3.5.25, 3.5.27.

Basic applications

9. Marginal analysis:
   • Reading from the textbook: Section 2.7 (pages 162–169).
   • Reading homework due on June 3 Wednesday (write this by hand on paper and submit it on Canvas): Let x be the quantity of goods sold by a business in a period of time, and let C be the cost to the business of producing and distributing the goods.
     1. If the goods are all sold at the same price p, then what is the business's revenue, and what is the business's profit?
     2. Is C‍/‍x the marginal cost or the average cost? What about dC‍/‍dx?
   • Exercises from the textbook due on June 4 Thursday (submit this through MyLab): 2.7.9, 2.7.11, 2.7.13, 2.7.15, 2.7.17, 2.7.21, 2.7.23, 2.7.25, 2.7.33, 2.7.39, 2.7.41.
10. Related rates:
    • Reading from the textbook: Section 3.6 (pages 222–225).
    • Reading homework due on June 4 Thursday (write this by hand on paper and submit it on Canvas): Look at Example 3.7.3 on page 224 of the textbook. In the course of solving this, the textbook writes down four equations that are derived from the set-up (rather than from other equations):
      1. x² + y² = 25;
      2. x = −3;
      3. y = 4;
      4. dx‍/‍dt = 0.4.
      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 you never need to differentiate an equation to solve the problem, it might still make sense to do so, or it might not.)
    • Exercises from the textbook due on June 5 Friday (submit this through MyLab): 3.6.9, 3.6.11, 3.6.13, 3.6.17, 3.6.19, 3.6.25, 3.6.27, 3.6.29.
11. Linear approximation:
    • Reading from the textbook:
      • Section 2.6 “Increments” (pages 155–157);
      • Section 2.6 “Approximations Using Differentials” (pages 158–160).
    • Reading homework due on June 5 Friday (write this by hand on paper and submit it on Canvas): Suppose that y = f(x), and the value of x undergoes a change Δx.
      1. Express the increment Δy exactly, using x and/or Δx, as well as values of f.
      2. Give an approximation for Δy using only f(x) and/or f′(x), as well as the change in x (Δx or dx).
    • Exercises from the textbook due on June 9 Tuesday (submit this through MyLab): 2.6.9, 2.6.13, 2.6.27, 2.6.29, 2.6.43, 2.6.45, 2.6.47.
12. Solving inequalities:
    • Reading from the textbook: Section 2.3 “Solving Inequalities Using Continuity Properties” (pages 123–126).
    • Reading homework due on June 9 Tuesday (write this by hand on paper and submit it on Canvas): Suppose that you have an inequality in the variable x that you wish to solve. You investigate the inequality and discover the following facts about it:
      • both sides are always defined;
      • the left-hand side is always continuous;
      • the right-hand side is discontinuous when x is 2 but is otherwise continuous;
      • the two sides are equal when x is −3⁄2 and only then;
      • the original inequality is true when x is −3⁄2 or 3 but false when x is −2, 0, or 2.
      What are the solutions to the inequality? (You can give your answer as a statement solved for x, a solution set for x in interval and/or list notation, or a labelled one-dimensional graph of this solution set.)
    • Exercises from the textbook due on June 10 Wednesday (submit this through MyLab): 2.3.55, 2.3.47, 2.3.49, 2.3.51, 2.3.53.

Applications of differentiation

13. Calculating limits:
    • Reading from the textbook:
      • Section 2.1 “Limits: An Algebraic Approache” (pages 97–101);
      • Section 4.3 (pages 274–282).
    • Reading homework due on June 26 Thursday (write this by hand on paper and submit it on Canvas): TBA.
    • Exercises from the textbook due on June 27 Friday (submit this through MyLab): TBA.
14. 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).
    • Reading homework due on June 16 Monday (write this by hand on paper and submit it on Canvas): Let e ≈ 2.71828 be the natural base.
      1. Which of the following functions is equal to its own derivative? (Answer Yes or No for each.)
         1. f(x) = e^x;
         2. f(x) = 2^x;
         3. f(x) = 2e^x;
         4. f(x) = e^2x.
      2. Write the differential of e^u using e, u, and du.
      3. If b is any constant, write the differential of b^u using b, ln b, u, and du.
    • Exercises from the textbook due on June 17 Tuesday (submit this 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.
15. 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).
    • Reading homework due on June 17 Tuesday (write this by hand on paper and submit it 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) = ___.
    • Exercises from the textbook due on June 18 Wednesday (submit this 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.
16. 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).
    • Reading homework due on June 27 Friday (write this by hand on paper and submit it 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 _____.
    • Exercises from the textbook due on July 1 Tuesday (submit this through MyLab): 4.2.1, 4.2.5, 4.2.9, 4.2.11, 4.2.13, 4.2.21, 4.2.25.
17. 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).
    • Reading homework due on July 1 Tuesday (write this by hand on paper and submit it 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.
    • Exercises from the textbook due on July 2 Wednesday (submit this 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.
18. 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).
    • Reading homework due on July 2 Wednesday (write this by hand on paper and submit it 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^∞.
    • Exercises from the textbook due on July 3 Thursday (submit this 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).
    • Reading homework due on July 3 Thursday (write this by hand on paper and submit it 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]?
    • Exercises from the textbook due on July 7 Monday (submit this 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).
    • Reading homework due on July 7 Monday (write this by hand on paper and submit it 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.
    • Exercises from the textbook due on July 8 Tuesday (submit this 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).
    • Reading homework due on July 8 Tuesday (write this by hand on paper and submit it 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.
    • Exercises from the textbook due on July 9 Wednesday (submit this 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).
    • Reading homework due on July 9 Wednesday (write this by hand on paper and submit it on Canvas):
      1. If y = f(x), where f is a smooth 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)?
    • Exercises from the textbook due on July 10 Thursday (submit this 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).
    • Reading homework due on July 10 Thursday (write this by hand on paper and submit it 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?
    • Exercises from the textbook due on July 11 Friday (submit this 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).
    • Reading homework due on July 11 Friday (write this by hand on paper and submit it 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.)
    • Exercises from the textbook due on July 15 Tuesday (submit this 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).
    • Reading homework due on July 15 Tuesday (write this by hand on paper and submit it 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?)
    • Exercises from the textbook due on July 16 Wednesday (submit this 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).
    • Reading homework due on July 16 Wednesday (write this by hand on paper and submit it 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?
    • Exercises from the textbook due on July 17 Thursday (submit this 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).
    • Reading homework due on July 17 Thursday (write this by hand on paper and submit it 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 smooth 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.)
    • Exercises from the textbook due on July 18 Friday (submit this 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.
    • Reading homework due on July 18 Friday (write this by hand on paper and submit it 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.
    • Exercises from the textbook due on July 22 Tuesday (submit this 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).
    • Reading homework due on July 22 Tuesday (write this by hand on paper and submit it 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?
    • Exercises from the textbook due on July 23 Wednesday (submit this 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).
    • Reading homework due on July 23 Wednesday (write this by hand on paper and submit it 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?
    • Exercises from the textbook due on July 24 Thursday (submit this 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).
    • Reading homework due on July 24 Thursday (write this by hand on paper and submit it 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.)
    • Exercises from the textbook due on July 25 Friday (submit this through MyLab): 6.4.9, 6.4.13, 6.4.15, 6.4.17, 6.4.19, 6.4.21.

Quizzes

1. Limits and continuity:
   • Date available: May 29 Friday.
   • Date due: June 1 Monday.
   • Corresponding problem sets: 1–4.
   • Help allowed: Your notes, self-contained calculator.
   • NOT allowed: Textbook, my notes, other people, websites, online apps, etc.
   • Work to show: Submit a picture of your work on Canvas: in problems involving graphs, copy the graph onto your paper and circle the region or regions of the graph that are relevant to your answers (or write an explanation of them); in problems involving equations, show at least what numerical calculations you make.
2. Differentiation:
   • Date available: June 5 Friday.
   • Date due: June 8 Monday.
   • Corresponding problem sets: 5–8.
   • Help allowed: Your notes, self-contained calculator.
   • NOT allowed: Textbook, my notes, other people, websites, online apps, etc.
   • Notation warning: In #4, y′ should be dy/‍dx.
   • Work to show: Submit a picture of your work on Canvas, at least one intermediate step for each result that's not simply finding a differential or derivative.
3. Basic applications:
   • Date available: June 12 Friday.
   • Date due: June 15 Monday.
   • Corresponding problem sets: 9–12.
   • 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.
4. Applications of differentiation:
   • Date available: June 19 Friday.
   • Date due: June 22 Monday.
   • Corresponding problem sets: 13–16.
   • 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.
5. More applications:
   • Date available: June 26 Friday.
   • Date due: June 29 Monday.
   • Corresponding problem sets: 17–20.
   • 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.
6. Integration:
   • Date available: July 2 Thursday.
   • Date due: July 6 Monday.
   • Corresponding problem sets: 21–25.
   • 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.

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 TBA).

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

The permanent URI of this web page is https://tobybartels.name/MATH-1400/2026SS8/.