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

• Summary of differential calculus;
• Rules of differentiation;
• Integration.

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 through “Infinite Limits” (pages 105–107);
     • 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.)
   • Problem set 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.
   • Problem set 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).
   • Problem set 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?
   • Problem set 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.)
    • Problem set 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).
    • Problem set 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.)
    • Problem set 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.

Exponential applications

13. Exponential functions:
    • Reading from the textbook:
      • Skim: Section 1.5 (pages 62–69);
      • Skim: Section 3.1 (pages 181–185);
      • Section 3.2 through “The Derivative of e^x” (pages 187&188).
    • Reading from my notes: My online notes on Exponents and logarithms.
    • Reading homework due on June 10 Wednesday (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.
    • Problem set from the textbook due on June 11 Thursday (submit this through MyLab): 3.2.13, 3.2.28, 3.2.49, 3.2.57, 3.4.25, 3.4.39, 3.4.47, 3.4.51, 3.3.17, 3.3.18, 3.3.31, 3,3,32.
14. Logarithmic functions:
    • Reading from my notes: The second half of Rules of differentiation (page 2).
    • Reading from the textbook:
      • Skim: The rest of Section 1.6 (page 72, pages 73–81);
      • The rest of Section 3.2 (pages 188–194).
    • Reading homework due on June 11 Thursday (write this by hand on paper and submit it on Canvas):
      1. 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) = ___.
      2. Write the differential of ln u using u and du.
      3. If b is any constant, then write the differential of log_b u using b, u, and du.
    • Problem set from the textbook due on June 12 Friday (submit this through MyLab): 3.2.15, 3.2.19, 3.2.20, 3.2.43, 3.2.44, 3.2.47, 3.2.55, 3.4.31, 3.4.33, 3.4.49, 3.4.83, 3.3.19, 3.3.33, 3.3.34, 3.3.56, 3.3.91.
15. Applications of exponents and logarithms:
    • Reading from the textbook: Section 3.7 (pages 228–233).
    • Reading homework due on June 12 Friday (write this by hand on paper and submit it on Canvas):
      1. Suppose that x = f(p) for a smooth function p.
         1. Write the rate of change of x with respect to p, using p, f(p), and/or f‍′‍(p);
         2. Write the relative rate of change of x with respect to p, using p, f(p), and/or f‍′‍(p);
         3. Write the elasticity of x with respect to p, using p, f(p), and/or f‍′‍(p).
      2. Suppose that a and k are constants.
         1. If x = a − kp, then what is the rate of change of x with respect to p?
         2. If x = ae^−kp, then what is the relative rate of change of x with respect to p?
         3. If x = ap^−k, then what is the elasticity of x with respect to p?
    • Problem set from the textbook due on June 16 Tuesday (submit this through MyLab): 3.2.67, 3.2.71, 3.4.97, 3.7.9, 3.7.11, 3.7.13, 3.7.25, 3.7.27, 3.7.33, 3.7.35, 3.7.37, 3.7.38, 3.7.49.
16. Calculating limits:
    • Reading from my notes: My online notes on Limits.
    • Reading from the textbook:
      • Section 2.1 “Limits: An Algebraic Approach” (pages 97–101);
      • Section 4.3 (pages 274–282).
    • Reading homework due on June 16 Tuesday (write this by hand on paper and submit it on Canvas): Suppose that D is a direction of a variable x (such as x → 3 or x → ∞), and f and g are smooth functions that are defined in that direction of x. L'Hôpital's Rule says that under certain conditions, lim_D (f(x)‍/‍g(x)) = lim_D (f‍′‍(x)‍/‍g‍′‍(x)) if the right-hand side exists. Which of these are conditions under which L'Hôpital's Rule applies? (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.
    • Problem set from the textbook due on June 17 Wednesday (submit this through MyLab): 2.1.41, 2.1.43, 2.1.51, 2.1.54, 2.1.61, 2.1.63, 2.2.17, 2.2.19, 2.2.21, 2.2.23, 4.3.11, 4.3.13, 4.3.15, 4.3.19, 4.3.23, 4.3.25, 4.3.27, 4.3.30, 4.3.31, 4.3.35, 4.3.36, 4.3.37, 4.3.39, 4.3.41, 4.3.43, 4.3.45, 4.3.49, 4.3.51, 4.3.61, 4.3.63, 4.3.66.

Graph-related applications

17. Tangent lines and local extrema:
    • Reading from the textbook: Chapter 4 through Section 4.1 (pages 240–252).
    • Reading homework due on June 17 Wednesday (write this by hand on paper and submit it on Canvas): Suppose that f is a continuous function, and f is smooth except possibly at a number c (so f′(c) might or might not exist). For each of the following circumstances, 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.
    • Problem set from the textbook due on June 18 Thursday (submit this through MyLab): 2.5.59, 3.4.37, 3.4.40, 3.3.63, 3.3.65, 4.1.9, 4.1.11, 4.1.15, 4.1.17, 4.1.19, 4.1.21, 4.1.23, 4.1.33, 4.1.37, 4.1.43, 4.1.45, 4.1.69–4.1.74.
18. Concavity and inflections:
    • Reading from the textbook: Section 4.2 (pages 257–268).
    • Reading homework due on June 18 Thursday (write this by hand on paper and submit it on Canvas): Suppose that a < b, so that (a, b) is open interval, and f is a smooth function defined on (a, b). Fill in each blank with ‘upward’ or ‘downward’:
      1. If the derivative f‍′ is increasing on (a, b), then f is concave _____ on (a, b);
      2. If the derivative f‍′ is decreasing on (a, b), then f is concave _____ on (a, b);
      3. If f‍″ is positive on (a, b), then f is concave _____ on (a, b);
      4. If f‍″ is negative on (a, b), then f is concave _____ on (a, b).
    • Problem set from the textbook due on June 19 Friday (submit this through MyLab): 4.2.9, 4.2.13–16, 4.2.25, 4.2.27, 4.2.31, 4.2.33, 4.2.35, 4.2.37, 4.2.39.
19. Graphing:
    • Reading from the textbook:
      • Section 2.2 “Locating Vertical Asymptotes” (pages 107–109);
      • The rest of Section 2.2 (pages 112–114);
      • Section 4.4 (pages 283–292).
    • Reading homework due on June 19 Friday (write this by hand on paper and submit it on Canvas): 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 lim_x→∞ f‍′‍(x) = 2, and lim_x→∞ (f(x) − 2x) = 3, then y = f(x) has _____ as an asymptote.
    • Problem set from the textbook due on June 23 Tuesday (submit this through MyLab): 2.2.51, 2.2.53, 2.2.55, 2.2.57, 2.2.61, 2.2.63, 4.4.9, 4.4.19, 4.4.21, 4.4.27, 4.4.34, 4.4.39.
20. Optimization:
    • Reading from my notes: My online notes on optimization.
    • Reading from the textbook:
      • Section 4.5 (pages 296–302);
      • Section 4.6 (pages 304–313).
    • Reading homework due on June 23 Tuesday (write this by hand on paper and submit it on Canvas): Let x and y be quantities that take real-number values.
      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 you wish to maximize profit, then what do you want the marginal profit to be (typically)?
    • Problem set from the textbook due on June 24 Wednesday (submit this through MyLab): 4.5.9, 4.5.11, 4.5.13, 4.5.15, 4.5.19, 4.5.21, 4.5.26, 4.5.27, 4.5.31, 4.5.33, 4.5.35, 4.5.43, 4.5.47, 4.5.51, 4.5.55, 4.5.67, 4.5.69, 4.5.71, 4.6.19, 4.6.21, 4.6.25, 4.6.31, 4.6.32, 4.6.41, 4.6.43.

Integration

21. Riemann integration:
    • Reading from my notes: Integrals through “Definite integrals” (page 1).
    • Reading from the textbook: Section 5.4 (pages 358–365).
    • Reading homework due on June 24 Wednesday (write this by hand on paper and submit it on Canvas):
      1. Consider the interval [0, 100], let this interval be partitioned into 5 equally spaced subintervals, and let f be a function defined on [0, 100].
         1. Write down the left Riemann sum of f over this partition;
         2. Write down the right Riemann sum of f over this partition.
         (Since you don't know what function f is, your answers will involve unevaluated values of f).
      2. Let f and g be functions.
         1. 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?)
         2. 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?)
    • Problem set from the textbook due on June 25 Thursday (submit this through MyLab): 5.4.7, 5.4.13, 5.4.17, 5.4.19, 5.4.21, 5.4.31, 5.4.32, 5.4.33, 5.4.35, 5.4.37, 5.4.39, 5.4.40, 5.4.41.
22. Antidifferentiation:
    • Reading from my notes: Integrals:
      • “Antidifferentials” (page 2);
      • “Integration techniques” (page 3).
    • Reading from the textbook:
      • Chapter 5 through Section 5.1 (pages 322–332);
      • Chapter 5 through Section 5.2 (pages 335–344).
    • Reading homework due on June 25 Thursday (write this by hand on paper and submit it on Canvas): For simplicity, suppose that f is a smooth function defined everywhere.
      1. Fill in the blank with a mathematical expression: ∫ f‍′‍(x) dx = _____. (If you introduce a new variable, state what it means.)
      2. Fill in the blank: If k is a non-zero constant, then ∫ e^kx dx = _____.
      3. Fill in each blank with a single adjective: ∫_a^b f(x) dx is the _____ integral of f from a to b, while ∫ f(x) dx is a(n) _____ integral of f.
    • Problem set from the textbook due on June 26 Friday (submit this through MyLab): 5.1.9, 5.1.11, 5.1.13, 5.1.15, 5.1.17, 5.1.19, 5.1.21, 5.1.23, 5.1.43, 5.1.45, 5.1.47, 5.1.51, 5.1.67, 5.1.69, 5.1.71, 5.2.23, 5.2.25, 5.2.29, 5.2.31, 5.2.35, 5.2.41, 5.2.59, 5.2.63, 5.2.65, 5.2.67.
23. The Fundamental Theorem of Calculus:
    • Reading from my notes: Integrals:
      • “The fundamental theorem of calculus” (page 2);
      • “Summary” (page 3).
    • Reading from the textbook: Section 5.5 (pages 369–377).
    • Reading homework due on June 26 Friday (write this by hand on paper and submit it on Canvas):
      1. If g is a smooth function defined everywhere, then what is ∫_a^b g‍′‍(t) dt?
      2. If f is continuous everywhere, define F so that ∫ f(x) dx = F(x) + C; what is ∫_a^b f(t) dt?
      3. Suppose you wish to integrate 2x√(x² + 1) with respect to x from x = 0 to x = 1, using the substitution u = x² + 1 (so that du = 2x dx). Explain the mistake in this calculation: ∫₀¹ 2x√(x² + 1) dx = ∫₀¹ u^½ du = (⅔u^3⁄2)|‍₀¹ = ⅔(1)^3⁄2 − ⅔(0)^3⁄2 = ⅔. (For the record, the correct value of the integral is actually 4⁄3 √2 − ⅔ ≈ 1.219.)
    • Problem set from the textbook due on June 30 Tuesday (submit this through MyLab): 5.5.13, 5.5.17, 5.5.21, 5.5.25, 5.5.29, 5.5.33, 5.5.35, 5.5.37, 5.5.39, 5.5.41, 5.5.45, 5.5.57, 5.5.59, 5.5.61.
24. Differential equations:
    • Reading from the textbook: Section 5.3 (pages 347–354).
    • Reading homework due on June 30 Tuesday (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.
    • Problem set from the textbook due on July 1 Wednesday (submit this through MyLab): When they write y(0), they really mean y|_x=0, that is the value of y when the value of x is 0: 5.3.9, 5.3.11, 5.3.13, 5.3.15, 5.3.17, 5.3.19, 5.3.25, 5.3.39, 5.3.43, 5.3.89, 5.3.91.
25. More applications of integration:
    • Reading from the textbook:
      • Section 6.1 (pages 388–395);
      • Section 6.2 (pages 398–407).
    • Reading homework due on July 1 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 f and g are continuous functions with f(x) ≥ g(x) whenever a ≤ x ≤ 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. If f is the probability density function of a continuous random variable, then what is the probability that the value of the variable is between a and b? (Assume that a < b rather than the reverse.)
      3. If you have an income stream of f(t) dollars per year, where t is time in years since the stream begins, and the income earns a continuous interest rate of 100r% per year, then what is the value (in dollars) of this income stream after T years?
    • Problem set from the textbook due on July 2 Thursday (submit this through MyLab): 6.1.41, 6.1.43, 6.1.45, 6.1.47, 6.1.49, 6.1.51, 6.1.53, 6.1.55, 6.1.84, 6.1.85, 6.2.21, 6.2.25, 6.2.45, 6.2.47, 6.2.67, 6.2.69, 6.2.73.

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 that's not simply finding a differential or derivative for each result.
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. Exponential applications:
   • 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. In #5, you may use any method that works to calculate the result exactly, even if the instruction tells you to use a particular method.
5. Graph-related 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; show at least what equations you solve or what tests you make 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!

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