# MATH-1600-LN01

Welcome to the permanent home page for Section LN01 of MATH-1600 (Calculus 1) at Southeast Community College in the Spring term of 2023. I am Toby Bartels, your instructor.

• Official syllabus (DjVu).
• Course policies (DjVu).
• Class hours: Mondays through Fridays from 10:00 to 10:50 in LNK V110.
• Final exam: May 5 Friday from 10:00 to 11:40 in LNK V110 (or by appointment).

## Contact information

Feel free to send a message at any time, even nights and weekends (although I'll be slower to respond then).

The official textbook for the course is the 4th Edition of University Calculus: Early Transcendentals by Hass et al published by Addison Wesley (Pearson). You automatically get an online version of this textbook through Canvas, although you can use a print version instead if you like. This comes with access to Pearson MyLab, integrated into Canvas, on which many of the assignments appear. I also have a supplemental text (DjVu) containing my notes on the material.

### Continuity and limits

1. General review:
• Skim: Through Section 1.2 (through page 18);
• Skim: Section 1.6 through "Finding Inverses" (pages 38–41).
• Through Section 1.4 (through page 7);
• Optional: Section 1.5 (pages 7&8).
• Exercises due on January 10 Tuesday (submit these here 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 11 Wednesday (submit these through MyLab in the Next item): 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:
• 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 11 Wednesday (submit these here 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 limxcf(x) exists?
3. Yes/No: If limxc+f(x) and limxcf(x) both exist and are equal, then must limxcf(x) also exist?
• Exercises from the textbook due on January 12 Thursday (submit these through MyLab in the Next item): 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).
• 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 12 Thursday (submit these here 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 limx→∞f(x) = ∞?
• Exercises from the textbook due on January 13 Friday (submit these through MyLab in the Next item): 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 13 Friday (submit these here 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 17 Tuesday (submit these through MyLab in the Next item): 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 17 Tuesday (submit these here 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 18 Wednesday (submit these through MyLab in the Next item): 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).
• 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 18 Wednesday (submit these here 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 limxc (1/f(x)) = 0.) Then what is the limit of f approaching c? (That is, limxcf(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, limt→0+f(1/t) = L.) What is the limit of f approaching infinity? (That is, limx→∞f(x) = _____.)
• Exercises from the textbook due on January 19 Thursday (submit these through MyLab in the Next item): 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).
• 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 19 Thursday (submit these here on Canvas or in class):
1. Suppose that c is a real number, g is a continuous function on (−∞, c], and h is a continuous function on (c, ∞). If f is defined piecewise so that f(x) = g(x) for x ≤ c while f(x) = h(x) for x > c, then fill in the blank with an equation involving values or limits of g and/or h: f is continuous at c if and only if _____.
2. 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)?
3. What is the limit of (sin x)/x as x → 0?
• Exercises from the textbook due on January 20 Friday (submit these through MyLab in the Next item): 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:
• 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 January 20 Friday (submit these here 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 ex 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 January 23 Monday (submit these through MyLab in the Next item): 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.
• Optional: Section 2.8 (pages 17&18).
• Section 2.9 (pages 18&19).
• 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 January 23 Monday (submit these here 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 January 24 Tuesday (submit these through MyLab in the Next item): 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.
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 January 24 Tuesday (submit these here 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 January 25 Wednesday (submit these through MyLab in the Next item): 2.1.1, 2.1.3, 2.1.19, 2.1.21, 2.1.25.
Quiz 1, covering the material in Problem Sets 1–10, is on January 30 Monday.

### Differentiation

1. 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 January 25 Wednesday (submit these here 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. 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 January 26 Thursday (submit these through MyLab in the Next item): 3.1.1, 3.1.11, 3.1.13, 3.1.19, 3.1.21, 3.1.23, 3.1.29.
2. Derivative functions:
• Section 3.2 (pages 120–125);
• Section 3.3: "Second- and Higher-Order Derivatives" (page 136).
• Exercises due on January 26 Thursday (submit these here 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 January 27 Friday (submit these through MyLab in the Next item): 3.2.27, 3.2.29, 3.2.30, 3.2.31, 3.2.34, 3.2.35, 3.2.37, 3.2.39, 3.2.41, 3.2.45, 3.2.47, 3.2.49.
3. Differentiating polynomials:
• Reading from the textbook: Section 3.3 through "Powers, Multiples, Sums, and Differences" (pages 129–132);
• Exercises due on January 31 Tuesday (submit these here 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 1 Wednesday (submit these through MyLab in the Next item): 3.2.1, 3.2.3, 3.2.5, 3.2.13, 3.2.15, 3.3.59.
4. Rules for differentiation:
• Reading from my notes: Section 3.3 (pages 23&24).
• Reading from the textbook: Section 3.3: "Products and Quotients" (pages 133–136), skipping Examples 6.b and 7.b.
• Exercises due on February 1 Wednesday (submit these here on Canvas or in class): Write answers using prime notation, not d/dx.
1. If f and g are differentiable everywhere and h(x) = f(x) g(x) for all x, then what is h⁠′⁠(x)?
2. If f and g are differentiable everywhere, g(x) ≠ 0 for all x, and h(x) = f(x)/g(x) for all x, then what is h⁠′⁠(x)?
• Exercises from the textbook due on February 2 Thursday (submit these through MyLab in the Next item): 3.3.71, 3.3.53, 3.3.65.
5. The Chain Rule:
• Section 3.4 (pages 24&25);
• Optional: Section 3.5 (pages 25&26).
• Reading from the textbook: Skim: Section 3.6 (pages 154–158), focussing on Examples 1, 6.a, 6.b, and 7.
• Exercises due on February 2 Thursday (submit these here on Canvas or in class):
1. If f and g are any functions, then their composite f ∘ g is guaranteed to be differentiable at c if f is differentiable at _____ and g is differentiable at _____.
2. If f and g are differentiable everywhere and h(x) = f(g(x)) for all x, then what is h⁠′⁠(x)?
• Exercises from the textbook due on February 3 Friday (submit these through MyLab in the Next item): 3.6.87, 3.6.89.
6. Differentials:
• 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 3 Friday (submit these here 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 from the textbook due on February 6 Monday (submit these through MyLab in the Next item): 3.11.19, 3.11.20, 3.11.21, 3.11.23.
7. Using differentials:
• Reading from my notes: Section 3.7 (pages 27&28).
• Exercises due on February 6 Monday (submit these here 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 7 Tuesday (submit these through MyLab in the Next item): 3.3.1, 3.3.3, 3.3.5, 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.
8. Implicit differentiation:
• Reading from my notes: Section 3.9 (pages 29&30).
• 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 7 Tuesday (submit these here 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 8 Wednesday (submit these through MyLab in the Next item): 3.11.23, 3.7.1, 3.7.3, 3.7.7, 3.7.21, 3.7.29, 3.7.31.
9. 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 8 Wednesday (submit these here 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 9 Thursday (submit these through MyLab in the Next item): 3.7.55, 3.8.1, 3.8.5, 3.8.9.
10. Exponential functions:
• 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 au and logau", specifically the part about au (pages 171&172).
• Exercises due on February 9 Thursday (submit these here on Canvas or in class):
1. If e ≈ 2.71828 is the natural base, then write the differential of eu using e, u, and du.
2. If b is any constant, then write the differential of bu using b, ln b, u, and du.
• Exercises from the textbook due on February 10 Friday (submit these through MyLab in the Next item): 1.5.11, 1.5.15, 1.5.19, 3.11.31, 3.3.29, 3.3.31, 3.3.35, 3.3.51, 3.6.35, 3.6.37.
11. Logarithmic functions:
• 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 au and logau" (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 10 Friday (submit these here 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 logbu using b, u, and du.
• Exercises from the textbook due on February 13 Monday (submit these through MyLab in the Next item): 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.
12. Logarithmic differentiation:
• Reading from the textbook: Section 3.8: "Logarithmic Differentiation" (page 173).
• Exercises due on February 13 Monday (submit these here 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 (ux) = ___.
• Exercises from the textbook due on February 14 Tuesday (submit these through MyLab in the Next item): 1.6.41, 1.6.43, 1.6.49, 3.8.65, 3.8.47, 3.8.51, 3.6.33, 3.8.89.
Quiz 2, covering the material in Problem Sets 11–22, is on February 20 Monday.

### Applications of differentiation

1. Trigonometry review
• 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 14 Tuesday (submit these here 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 15 Wednesday (submit these through MyLab in the Next item): 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.
2. Trigonometric operations:
• 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 15 Wednesday (submit these here 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 February 16 Thursday (submit these through MyLab in the Next item): 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.
3. Inverse trigonometric operations:
• Reading from the textbook: The rest of Section 3.9 (pages 179–182).
• Exercises due on February 16 Thursday (submit these here 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 February 17 Friday (submit these through MyLab in the Next item): 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.
4. 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 February 22 Wednesday (submit these here 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 February 23 Thursday (submit these through MyLab in the Next item): 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.
5. 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 February 23 Thursday (submit these here 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 February 24 Friday (submit these through MyLab in the Next item): 3.5.57, 3.5.58, 3.5.63, 3.5.64.
6. 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 February 24 Friday (submit this here 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 February 27 Monday (submit these through MyLab in the Next item): 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.
7. Linearization:
• Reading from my notes: Section 4.3 (pages 35&36).
• Section 3.11 through "Linearization" (pages 192–195);
• Optional: Section 3.11 "Error in Differential Approximation" (pages 198&199).
• Exercises due on February 27 Monday (submit these here 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 February 28 Tuesday (submit these through MyLab in the Next item): 3.11.1, 3.11.2, 3.11.3, 3.11.5, 3.11.7, 3.11.9, 3.11.11, 3.11.15.
8. Linear estimation:
• Section 3.11 "Estimating with Differentials" (pages 197&198);
• Section 3.11 "Sensitivity to Change" (page 200).
• Exercises due on February 28 Tuesday (submit these here 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 1 Wednesday (submit these through MyLab in the Next item): 3.4.28, 3.11.51, 3.11.52, 3.11.53, 3.11.57.
9. Mean-value theorems:
• Reading from my notes: Section 4.5 (pages 37&38).
• 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 1 Wednesday (submit these here 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 2 Thursday (submit these through MyLab in the Next item): 4.2.1, 4.2.5, 4.2.9, 4.2.11, 4.2.13, 4.2.21, 4.2.25.
10. Increasing and decreasing functions:
• Reading from the textbook: Section 4.3 through "Increasing Functions and Decreasing Functions" (pages 228&229).
• Exercises due on March 2 Thursday (submit these here 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 3 Friday (submit these through MyLab in the Next item): 4.3.15, 4.3.17, 4.3.71, 4.3.73, 4.3.76.
11. Constant functions:
• Section 4.2: "Mathematical Consequences", "Finding Velocity and Position from Acceleration" (pages 223–224);
• Optional: The rest of Section 4.2 (pages 224–226).
• Exercises due on March 3 Friday (submit these here on Canvas or in class): Suppose that f and g are differentiable on some interval I. Fill in each blank with a single word:
1. If f⁠′⁠(x) = 0 for every x in I, then f is _____ on I;
2. If f⁠′⁠(x) = g⁠′⁠(x) for every x in I and f(c) = g(c) for some c in I, then f and g are _____ on I.
3. If f⁠′ is constant on I, then f is _____ on I.
• Exercises from the textbook due on March 6 Monday (submit these through MyLab in the Next item): 4.2.29, 4.2.31, 4.2.39, 4.2.43, 4.2.48.
12. Concavity:
• Reading from the textbook: Section 4.4 through "Points of Inflection" (pages 233–237).
• Reading from my notes: Section 4.7 (page 39).
• Exercises due on March 6 Monday (submit these here 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 7 Tuesday (submit these through MyLab in the Next item): 4.4.97, 4.4.107, 4.4.113, 4.4.117, 4.4.119.
Quiz 3, covering the material in Problem Sets 23–34, is on March 10 Friday.

1. L'Hôpital's Rule:
• Reading from my notes: Section 4.6 (page 39).
• 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 8 Wednesday (submit these here 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 limD(f(x)⁠/⁠g(x)) = limD(f⁠′⁠(x)⁠/⁠g⁠′⁠(x)) if the latter exists? (Say Yes or No for each of these five conditions.)
1. limDf(x) and limDg(x) are both zero;
2. limDf(x) is a nonzero real number while limDg(x) is zero;
3. limDf(x) is zero while limDg(x) is a nonzero real number;
4. limDf(x) and limDg(x) are both non-zero real numbers;
5. limDf(x) and limDg(x) are both infinite.
• Exercises from the textbook due on March 9 Thursday (submit these through MyLab in the Next item): 4.5.1, 4.5.3, 4.5.5, 4.5.11, 4.5.13, 4.5.15, 4.5.21.
2. Advanced techniques with L'Hôpital's Rule:
• Reading from the textbook: Section 4.5 "Indeterminate Powers" (pages 250&251).
• Exercises due on March 20 Monday (submit these here 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 21 Tuesday (submit these through MyLab in the Next item): 4.5.37, 4.5.51, 4.5.55, 4.5.59, 4.5.60.
3. Absolute extrema:
• Reading from the textbook: Section 4.1: "Finding extrema" (pages 215–217).
• Exercises due on March 21 Tuesday (submit these here 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 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 March 22 Wednesday (submit these through MyLab in the Next item): 4.1.11–14, 4.1.23, 4.1.27, 4.1.37, 4.1.39, 4.1.41.
4. Local extrema:
• Reading from the textbook: The rest of Section 4.3 (pages 229–231).
• Exercises due on March 22 Wednesday (submit these here 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 possible 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 March 23 Thursday (submit these through MyLab in the Next item): 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.
5. 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 March 23 Thursday (submit these here 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 March 24 Friday (submit these through MyLab in the Next item): 4.4.111, 4.4.112, 4.4.115, 4.4.119, 4.4.121.
6. Graphing:
• Section 1.4 (pages 29–32);
• The rest of Section 4.4 (pages 237–242).
• Exercise due on March 24 Friday (submit this here 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 March 27 Monday (submit these through MyLab in the Next item): 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.
7. Graphing asymptotes:
• Reading from my notes: Section 4.8 (pages 40&41).
• Reading from the textbook: Section 2.6: "Horizontal Asymptotes" and "Oblique Asymptotes" (pages 99–102).
• Exercises due on March 27 Monday (submit these here 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 March 28 Tuesday (submit these through MyLab in the Next item): 4.4.11, 4.4.19, 4.4.23, 4.4.25, 4.4.39, 4.4.41, 4.4.45, 4.4.59.
8. Applied optimization:
• Reading from my notes: Section 4.9 (page 41).
• Reading from the textbook: Section 4.6 through "Examples from Mathematics and Physics" (pages 255–258).
• Exercises due on March 28 Tuesday (submit these here 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 1, is 0, then is there a minimum value of u, and if so, then what is it?
• Exercises from the textbook due on March 29 Wednesday (submit these through MyLab in the Next item): 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.
9. Optimization in economics and finance:
• Reading from my notes: Section 4.10 (pages 42).
• Section 3.4: "Derivatives in Economics and Biology" (pages 143–145);
• Section 4.6 "Examples from Economics" (pages 258&259).
• Exercises due on March 29 Wednesday (submit these here 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 March 30 Thursday (submit these through MyLab in the Next item): 4.6.43, 4.6.45, 4.6.57, 4.6.59.
10. Newton's Method:
• Reading from my notes: Section 4.4 (page 37).
• Reading from the textbook: Section 4.7 (pages 266–269).
• Exercise due on March 30 Thursday (submit this here 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 March 31 Friday (submit these through MyLab in the Next item): 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.
Quiz 4, covering the material in Problem Sets 35–44, is on April 3 Monday.

### Integration

1. Summation notation:
• The bottom of page 478 and page 479 (§9.1: introduction);
• From page 293 through the formulas at the top of page 296 (§5.2: Finite Sums and Sigma Notation).
• Exercise due on April 4 Tuesday (submit this here on Canvas or in class): What is the sum, as k takes integer values from 1 to n, of k? Write both the direct symbolic way of writing this sum using the summation operator ∑ and the formula for the result using n.
• Exercises from the textbook due on April 5 Wednesday (submit these through MyLab in the Next item): TBA.
2. Riemann sums:
• Chapter 5 through Section 5.1 (pages 290–298);
• Section 5.2 through Example 2 (pages 300&301);
• Section 5.2 "Riemann Sums" (pages 304–306).
• Exercises due on April 5 Wednesday (submit these here on Canvas or in class): Consider the interval [0, 100], and let this interval be partitioned into 10 subintervals, with endpoints 0, 11, 13, 24, 28, 33, 35, 49, 56, 60, and 100. Also, let this partition be tagged with the numbers 7, 12, 16, 25, 30, 34, 37, 55, 57, and 80.
1. State the norm (aka mesh) of this partition.
2. If f is a function defined on [0, 100], then write down the Riemann sum for f over this tagged partition.
• Exercises from the textbook due on April 6 Thursday (submit these through MyLab in the Next item): 5.2.1, 5.2.7, 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.
3. 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 6 Thursday (submit these here on Canvas or in class): Let f and g be functions.
1. Suppose that ∫5x=3f(x) dx = 5 and ∫5x=3g(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 ∫5x=3 (f(x) + g(x)) dx? (That is, what is the integral of f + g from 3 to 5?)
2. Suppose that ∫5x=3f(x) dx = 5 and ∫8x=5f(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 ∫8x=3f(x) dx? (That is, what is the integral of f from 3 to 8?)
• Exercises from the textbook due on April 7 Friday (submit these through MyLab in the Next item): 5.3.9, 5.3.11, 5.3.13, 5.3.27, 5.3.71.
4. Antidifferentiation:
• 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 7 Friday (submit these here on Canvas or in class):
1. If f(x) = sin(x2 + e3x) for all x, then what is ∫ f⁠′⁠(x) dx? (If you work out a formula for f⁠′⁠, then you're working too hard.)
2. Fill in the blanks: ∫abf(x) dx is the _____ integral of f from a to b, while ∫ f(x) dx is the _____ integral of f (as a function of x).
• Exercises from the textbook due on April 10 Monday (submit these through MyLab in the Next item): 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.
5. 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 10 Monday (submit these here on Canvas or in class):
1. If f is continuous everywhere, then what is the derivative of ∫0xf(t) dt with respect to x?
2. If f is continuously differentiable everywhere, then what is ∫abf⁠′⁠(t) dt?
3. If f is continuous everywhere, define F so that ∫ f(x) dx = F(x) + C; what is ∫abf(t) dt?
• Exercises from the textbook due on April 11 Tuesday (submit these through MyLab in the Next item): 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.
6. Integration by substitution:
• Reading from the textbook: Section 5.5 (pages 332–337);
• Reading from my notes: Section 5.5 (pages 46&ndashTBA).
• Exercises due on April 11 Tuesday (submit these here on Canvas or in class):
1. Fill in the blanks: ∫ ekx 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 12 Wednesday (submit these through MyLab in the Next item): 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.
7. Substitution with definite integrals:
• Reading from the textbook: Section 5.6 through "Definite Integrals of Symmetric Functions" (pages 339–342).
• Reading from my notes: TBA.
• Exercises due on April 12 Wednesday (submit these here on Canvas or in class):
1. Suppose that F and g are differentiable functions, with f = F⁠′⁠. Assuming that f and g⁠′ are continuous, what is ∫abf(g(x)) g⁠′⁠(x) dx?
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: ∫0π/2 sin x cos x dx = ∫0π/2u du = (½u2)|0π/2 = ½(π/2)2 − ½(0)2 = π2/8. (The correct value of the integral is actually ½.)
• Exercises from the textbook due on April 13 Thursday (submit these through MyLab in the Next item): 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.
8. Differential equations:
• Reading from the textbook: Section 4.8: "Initial Value Problems and Differential Equations", "Antiderivatives and Motion" (pages 274&275).
• Section 5.4 (page 46);
• Chapter 6 through Section 6.3 (pages 49–51), especially Section 6.3 (page 49).
• Exercises due on April 13 Thursday (submit these here 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) = 0.
• Exercises from the textbook due on April 14 Friday (submit these through MyLab in the Next item): 4.8.95, 9.5.97, 4.8.105, 5.5.73, 5.5.75, 4.2.40, 4.2.45, 4.2.47.
9. Planar area:
• Section 5.4 "Total Area" (pages 327&328);
• The rest of Section 5.6 (pages 342–345).
• Exercises due on April 14 Friday (submit these here 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 17 Monday (submit these through MyLab in the Next item): 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.
10. Arclength:
• Reading from the textbook: Section 6.3 (pages 375–379).
• Exercises due on April 17 Monday (submit these here 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 18 Tuesday (submit these through MyLab in the Next item): 6.3.1, 6.3.3, 6.3.5, 6.3.7, 6.3.11, 6.3.15.
11. Volume of revolution:
• Chapter 6 through Section 6.1 (pages 356–363);
• Section 6.2 (pages 367–372).
• Exercises due on April 18 Tuesday (submit these here 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, 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 April 19 Wednesday (submit these through MyLab in the Next item): 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.
12. Surface area of revolution:
• Section 6.4 (pages 381–384).
• Reading from my notes: Section 5.7 (pages 47&48).
• Exercises due on April 19 Wednesday (submit these here 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 April 20 Thursday (submit these through MyLab in the Next item): 6.4.9, 6.4.13, 6.4.15, 6.4.17, 6.4.19, 6.4.21.
Quiz 5, covering the material in Problem Sets 45–56, is on April 24 Monday.

## Quizzes

1. Continuity and limits:
• Review date: January 27 Friday (in class).
• Date taken: January 30 Monday (in class).
• Corresponding problem sets: 1–10.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
2. Differentiation:
• Review date: February 17 Friday (in class).
• Date taken: February 20 Monday (in class).
• Corresponding problem sets: 11–22.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
3. Applications of differentiation:
• Review date: March 9 Thursday (in class).
• Date taken: March 10 Friday (in class).
• Corresponding problem sets: 23–34.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
• Review date: March 31 Friday (in class).
• Date taken: April 3 Monday (in class).
• Corresponding problem sets: 35–44.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
5. Integration:
• Review date: April 21 Friday (in class).
• Date taken: April 24 Monday (in class).
• Corresponding problem sets: 45–56.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.

## Final exam

There is a comprehensive final exam on May 5 Friday, in our normal classroom at the normal time but lasting until 11:40. (You can also arrange to take it at a different time May 1–5.) To speed up grading at the end of the term, the exam is multiple choice and filling in blanks, with no partial credit.

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 (TBA).

This web page and the files linked from it (except for the official syllabus) were written by Toby Bartels, last edited on 2023 March 13. Toby reserves no legal rights to them.

The permanent URI of this web page is `http://tobybartels.name/MATH-1600/2023SP/`.