Readings and homework

Here are the assigned readings and exercises (Reading 1, Reading 2, Reading 3, Reading 4, Reading 5, Reading 6, Reading 7, Reading 8, Reading 9, Reading 10, Reading 11, Reading 12, Reading 13, Reading 14, Reading 15, Reading 16, Reading 17, Reading 18, Reading 19, Reading 20, Reading 21, Reading 22, Reading 23, Reading 24, Reading 25, Reading 26, Reading 27, Reading 28, Reading 29, Reading 30, Reading 31, Reading 32, Reading 33, Reading 34, Reading 35, Reading 36, Reading 37, Reading 38); but anything whose assigned date is in the future is subject to change!

1. General review:
   • Date assigned: October 3 Tuesday.
   • Date due: October 4 Wednesday.
   • Reading from the textbook:
     • Pages 1–11 (§1.1);
     • Pages 14–18 (§1.2);
     • Pages 38–41 (§1.6: One-to-one Functions, Inverse Functions, Finding Inverses).
   • Reading from my notes: pages 2–5 (§1).
   • Problems due:
     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?
     4. Given that 2.2² = 4.84, 2.25² = 5.0625, and 2.3² = 5.29, what is √5 rounded to one digit after the decimal point?
2. Continuity and limits informally:
   • Date assigned: October 4 Wednesday.
   • Date due: October 5 Thursday.
   • Reading from the textbook:
     • Pages 58–60 (§2.2: Limits of Function Values);
     • From page 78 through Example 2 on page 80 (§2.4: introduction; Approaching a Limit from One Side);
     • From page 85 through Example 5 on page 88 (§2.5: introduction; Continuity at a Point; the beginning of Continuous Functions);
     • Page 96 (§2.6: introduction; the beginning of Finite Limits as x → ±∞);
     • From the bottom of page 101 through the end of Example 13 at the top of page 103 (§2.6: Infinite Limits).
   • Problems due:
     • From §2.2 (page 66): 7;
     • Additional problems:
       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 _____, as x approaches c, of f(x).
       2. 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.
       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. Suppose that f(x) exists whenever x ≠ c but f(c) does not exist. Is it possible that f is continuous at c?
3. Epsilontics:
   • Date assigned: October 5 Thursday.
   • Date due: October 6 Friday.
   • Reading from my notes: pages 6–11 (§§2.1–2.4).
   • Reading from the textbook:
     • From page 69 through the end of Example 5 at the top of page 74 (all of §2.3 except Using the Definition to Prove Theorems);
     • Optional: The rest of page 80 and through Example 4 on page 81 (§2.4: Precise Definitions of One-Sided Limits);
     • Optional: Page 97 through Example 1 (§2.6: most of the rest of Finite Limits as x → ±∞);
     • Optional: The rest of page 103 and Example 15 on page 104 (§2.6: Precise Definitions of Infinite Limits).
   • Problems due (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.
4. Evaluating limits and checking continuity:
   • Date assigned: October 6 Friday.
   • Date due: October 9 Monday.
   • Reading from the textbook:
     • Pages 61–65 (the rest of §2.2);
     • The rest of page 74 and page 75 (§2.3: Using the Definition to Prove Theorems);
     • The rest of page 81 and on through page 83 (§2.4: Limits Involving (sin θ)/θ);
     • The rest of page 88 and on through Example 9 on page 91 (§2.5: the rest of Continuous Functions; Inverse Functions and Continuity; Composites);
     • Pages 93&94 (§2.5: Continuous Extension to a Point);
     • The bottom of page 97 and page 98 through Example 3 (§2.6: the rest of Finite Limits as x → ±∞; Limits at Infinity of Rational Functions);
     • From the bottom of page 105 to page 107 (§2.6: Dominant Terms; Summary).
   • Reading from my notes: pages 11&12 (§2.5).
   • Problems due: Fill in the blanks using mathematical symbols:
     1. If a function f is continous at a real number c, then lim_x→c f(x) = _____;
     2. If lim_x→c (1/f(x)) = 0 and f(x) is always positive, then lim_x→c f(x) = _____;
     3. If lim_t→0⁺ f(1/t) = L, then lim_x→∞ f(x) = _____.
5. Theorems about continuous functions:
   • Date assigned: October 9 Monday.
   • Date due: October 10 Tuesday.
   • Reading from the textbook:
     • From the middle of page 209 to the top of page 212 (§4.1: introduction; Local (Relative) Extreme Values);
     • The rest of page 91 and page 92 (§2.5: Intermediate Value Theorem for Continuous Functions).
   • Problems due:
     1. For each of the following intervals, state whether a continuous function defined on that interval must have a maximum or might not have a maximum:
        1. [0, 1],
        2. [0, ∞),
        3. (0, 1],
        4. (0, ∞);
     2. For each of the following circumstances, state whether a continuous function f defined on [0,1] must have a root (aka a zero) 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.
6. Derivatives as limits:
   • Date assigned: October 11 Wednesday.
   • Date due: October 12 Thursday.
   • Reading from the textbook:
     • Pages 51–56 (§2.1);
     • Pages 115–117 (§3.1).
   • Problems due: 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.
7. Derivative functions:
   • Date assigned: October 12 Thursday.
   • Date due: October 13 Friday.
   • Reading from my notes: page 13 (§3 introduction, §3.1).
   • Reading from the textbook:
     • Pages 119–124 (§3.2);
     • Page 135 (§3.3: Second- and Higher-Order Derivatives).
   • Problems due: 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.
8. Basic rules:
   • Date assigned: October 13 Friday.
   • Date due: October 16 Monday.
   • Reading from my notes: page 14 and through the first full paragraph on page 15 (§3.2).
   • Online video on examples of derivatives.
   • Reading from the textbook (optional):
     • From page 128 through the end of Example 4 on the top of page 131 (§3.3: introduction; Powers, Multiples, Sums, and Differences);
     • From the middle of page 132 through to page 134 (§3.3: Products and Quotients).
   • Problems due:
     1. If f(x) = xⁿ for all x (where n is a constant), then what is f′(x)?
     2. If f and g are differentiable everywhere and h(x) = f(x) g(x) for all x, then what is h′(x)?
     3. 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)?
9. The Chain Rule:
   • Date assigned: October 16 Monday.
   • Date due: October 17 Tuesday.
   • Reading from my notes: the rest of page 15 (§3.3).
   • Reading from the textbook:
     • Pages 153–157 (§3.6), at least Examples 1, 6.a, 6.b, and 7;
     • Optional: pages 166–168 (§3.8: Derivatives of Inverses of Differentiable Functions).
   • Problems due:
     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 functions, then write a formula for (f ∘ g)′ using (all or some of) f, g, f′, and g′.
10. Differentials:
    • Date assigned: October 18 Wednesday.
    • Date due: October 19 Thursday.
    • Reading from my notes: Pages 16–18 (§3.4&3.5).
    • Reading from the textbook (optional): From the definition on page 193 through Example 5 at the top of page 195 (§3.11: Differentials).
    • Problems due: 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.
      3. If n is a constant, write a formula for the differential of uⁿ using n, u, and du.
11. Implicit differentiation:
    • Date assigned: October 19 Thursday.
    • Date due: October 20 Friday.
    • Reading from my notes: Page 19 and the very top of page 20 (most of §3.6).
    • Reading from the textbook:
      • The middle of page 161 (§3.7: introduction);
      • Optional: From the bottom of page 161 through Example 4 on page 163 (§3.7: Implicitly Defined Functions; Derivatives of Higher Order).
    • Problems due: Suppose that you have an algebraic equation involving only the variables x and y.
      1. Fill in the blank using 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 their differentials), then the derivative of y with respect to x (when it exists) is dy/dx = _____.
12. Exponential functions:
    • Date assigned: October 20 Friday.
    • Date due: October 23 Monday.
    • Reading from the textbook:
      • Optional review: Pages 33–37 (§1.5);
      • Pages 131&132 (§3.3: Derivatives of Exponential Functions);
      • The bottom half of page 170 and the top half of page 171 (most of The Derivatives of a^u and log_a u in §3.8, specifically the part about a^u).
    • Problems due:
      1. If e ≈ 2.71828 is the natural base, then write the differential of e^u using e, u, and du;
      2. If b is any constant, then write the differential of b^u using b, ln b, u, and du.
13. Logarithmic functions:
    • Date assigned: October 23 Monday.
    • Date due: October 24 Tuesday.
    • Reading from the textbook:
      • Optional review: From the bottom of page 41 through page 44 (§1.6: Logarithmic Functions, Properties of Logarithms, Applications);
      • From the bottom of page 168 through Example 6 on page 172 (§3.8: Derivative of the Natural Logarithm Function, The Derivatives of a^u and log_a u, Logarithmic Differentiation);
      • Optional: Pages 172–174 (§3.8: Irrational Exponents and the Power Rule, The Number e Expressed as a Limit).
    • Problems due:
      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.
14. Trigonometric operations:
    • Date assigned: October 25 Wednesday.
    • Date due: October 26 Thursday.
    • Reading from the textbook:
      • Optional: Pages 21–27 (§1.3);
      • Pages 147&148 (§3.5: Derivative of the Sine Function, Derivative of the Cosine Function);
      • Pages 150&151 (§3.5: Derivatives of the Other Basic Trigonometric Functions.
    • Problems due:
      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.
15. Inverse trigonometric operations:
    • Date assigned: October 26 Thursday.
    • Date due: October 27 Friday.
    • Reading from the textbook:
      • Optional: From the bottom of page 44 through page 48 (§1.6: Inverse Trigonometric Functions; The Arcsine and Arccosine Functions; Identities Involving Arcsine and Arccosine);
      • Optional: Pages 176&177 (§3.9: Inverses of tan x, cot x, sec x, and csc x);
      • From the bottom of page 177 through page 180 (the rest of §3.9).
    • Problems due:
      1. Simplify asin x + acos x (where asin x may also be written as arcsin x, Sin⁻¹ x, or sin⁻¹ x and acos x may also be written as arccos x, Cos⁻¹ x, or cos⁻¹ x);
      2. Write the differential of atan u (which may also be written as arctan u, Tan⁻¹ u, or tan⁻¹ u) using u, du, and algebraic operations.
16. Using derivatives with respect to time:
    • Date assigned: October 27 Friday.
    • Date due: October 30 Monday.
    • Reading from my notes: Pages 20&21 (§3.7).
    • Reading from the textbook:
      • From page 138 through the end of Example 4 on page 142 (§3.4: Instantaneous Rates of Change; Motion Along a Line);
      • Examples 3&4 on pages 149&150 (§3.5: Simple Harmonic Motion).
    • Problems due:
      1. If an object's position s varies with time t, then the derivative ds/dt (if it exists) is the object's instantaneous _____.
      2. In a technical sense, is an object's acceleration the time derivative of its speed or of its velocity?
      3. If an object is undergoing periodic motion and its acceleration is proportional to its displacement from its average position, then it is undergoing simple _____ motion.
17. Related rates:
    • Date assigned: October 30 Monday.
    • Date due: October 31 Tuesday.
    • Reading from my notes: The top of pages 20 (the remainder of §3.6).
    • Reading from the textbook: Pages 182–187 (§3.10).
    • Problem due: Look at Example 3 on page 184 of the textbook. As the textbook solves this example, they begin by writing down five equations: x = 0.8 mi, y = 0.6 mi, dy/dt = −60 mi/hr, ds/dt = 20 mi/hr, and s² = x² + y². In the context of that problem, for which of these equations does it make sense to differentiate the equation with respect to time, that is to take the time derivative of both sides of the equation? (List all equations for which this would make sense in context, which may be any number of these equations from zero to five of them.)
18. Sensitivity and linear approximation:
    • Date assigned: November 1 Wednesday.
    • Date due: November 2 Thursday.
    • Reading from my notes: Pages 21&22 (§3.8).
    • Reading from the textbook:
      • Pages 190–193 (§3.11: introduction, Linearization);
      • From pages 195 through the top of page 197 (§3.11: Estimating with Differentials, Error in Differential Approximation);
      • The bottom of page 197 and page 198 (§3.11: Sensitivity to Change);
      • The bottom half of page 143 (Example 3.4.6 and the paragraph immediately before it).
    • Problems due:
      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.
19. Mean-value theorems:
    • Date assigned: November 2 Thursday.
    • Date due: November 3 Friday.
    • Reading from my notes: the bottom of page 23 and the top of page 24 (most of §3.10).
    • Reading from the textbook:
      • Pages 217–219 (§4.2: introduction, Rolle's Theorem, The Mean Value Theorem, A Physical Interpretation);
      • The bottom half of page 246 and the top half of page 247 (the statement and proof in §4.5 of Theorem 7: Cauchy's Mean Value Theorem).
    • Problems due: 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 as described above such that _____;
      2. Lagrange: If f is as described above, then some c exists as described above such that _____;
      3. Cauchy: If f and g are as described above and g′(x) ≠ 0 whenever a < x < b, then some c exists as described above such that _____.
20. More theorems about derivatives:
    • Date assigned: November 3 Friday.
    • Date due: November 6 Monday.
    • Reading from the textbook:
      • The top half of page 225 (§4.3: Increasing Functions and Decreasing Functions: Corollary 3);
      • Page 220 and the top half of page 221 (§4.2: Mathematical Consequences, Finding Velocity and Position from Acceleration).
    • Reading from my notes: the middle of page 23 (the beginning of §3.10).
    • Problems due: Suppose that a function f is differentiable on some interval I. Fill in each blank with a single word:
      1. If f′ is positive on I (meaning that f′(x) > 0 for all x in I), then f is _____ on I;
      2. If f′ is negative on I, then f is _____ on I;
      3. If f′ is zero on I, then f is _____ on I;
      4. If f′ is constant on I, then f is _____ on I.
21. L'Hôpital's Rule:
    • Date assigned: November 6 Monday.
    • Date due: November 7 Tuesday.
    • Reading from my notes (optional): §3.11 on page 24.
    • Reading from the textbook:
      • From page 241 through Example 8 on page 246 (§4.5: introduction; Indeterminate Form 0/0; Indeterminate Forms ∞/∞, ∞ ⋅ 0, ∞ − ∞; Indeterminate Powers);
      • Optional: the proof on the bottom of page 247.
    • Problems due: If D is any direction in the variable x, 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? (List all answers for which this theorem applies, which may be any number of these answers from zero to five of them.)
      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 infinite while lim_D g(x) is zero;
      4. lim_D f(x) is infinite while lim_D g(x) is a nonzero real number;
      5. lim_D f(x) and lim_D g(x) are both infinite.
22. Absolute extrema:
    • Date assigned: November 8 Wednesday.
    • Date due: November 9 Thursday.
    • Reading from the textbook: Pages 212–214 (§4.1: Finding extrema).
    • Problems due:
      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?
23. Local extrema:
    • Date assigned: November 9 Thursday.
    • Date due: November 10 Friday.
    • Reading from the textbook:
      • From the bottom half of page 225 through page 228 (the rest of §4.3);
      • The middle half of page 233 (§4.4: Second Derivative Test for Local Extrema until the paragraph before Example 7).
    • Problems due: Suppose that f is differentiable on an open interval and that c is a number in that interval. (Fill in each blank with a word more specific than ‘extremum’.)
      1. If f′(x) < 0 when x < c while f′(x) > 0 when x > c, then f has a local _____ at c.
      2. If f′(x) > 0 when x < c while f′(x) < 0 when x > c, then f has a local _____ at c.
      3. If f′(c) = 0 while f′′(c) > 0, then f has a local _____ at c.
      4. If f′(c) = 0 while f′′(c) < 0, then f has a local _____ at c.
24. Concavity:
    • Date assigned: November 10 Friday.
    • Date due: November 13 Monday.
    • Reading from the textbook: From page 230 through the end of Example 6 on the top of page 233 (§4.4: Concavity; Points of Inflection).
    • Reading from my notes: Section 3.12 (most of page 24 and the very top of page 25).
    • Problems due: 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.
25. Graphing:
    • Date assigned: November 13 Monday.
    • Date due: November 14 Tuesday.
    • Reading from the textbook:
      • From the rest of page 233 through page 238 (the rest of §4.4);
      • Optional: Pages 29–32 (§1.4).
    • Reading from my notes: Section 3.13 (the middle of page 25).
    • Problems due: 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.
26. Applied optimization:
    • Date assigned: November 15 Wednesday.
    • Date due: November 16 Thursday.
    • Reading from the textbook:
      • The rest of pages 142&143 (§3.4: Derivatives in Economics);
      • Pages 250–254 (§4.6).
    • Reading from my notes: Sections 3.14&3.15 (pages 25–27).
    • Problems due:
      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 cost C is a function of quantity q, then is C/q the marginal cost or the average cost? What about dC/dq?
      3. If you wish to maximize profit, then what do you want the marginal profit to be (typically)?
27. Newton's Method:
    • Date assigned: November 16 Thursday.
    • Date due: November 17 Friday.
    • Reading from my notes: Section 3.9 (page 23).
    • Reading from the textbook: Pages 261–263 (§4.7).
    • Problem due: If you are attempting to use Newton's Method to solve f(x) = 0, and your first guess is x ≈ x₁, then write down a formula for your second guess x ≈ x₂ using x₁, f, and f′.
28. Summation notation:
    • Date assigned: November 17 Friday.
    • Date due: November 20 Monday.
    • Reading from the textbook: From page 293 through the formulas at the top of page 296 (§5.2: Finite Sums and Sigma Notation).
    • Reading from my notes: Pages 36–39 (§6 introduction, §6.1), especially pages 38&39.
    • Problem due: 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.
29. Riemann sums:
    • Date assigned: November 20 Monday.
    • Date due: November 21 Tuesday.
    • Reading from the textbook:
      • Pages 283–291 (§5.1);
      • Pages 297–299 (§5.2: Riemann Sums).
    • Problems due: Suppose that f is a function defined on [a, b]. Partition this interval into n equally spaced intervals, and write down (using summation notation) expressions for Riemann sums for f over this partition:
      1. using left endpoints;
      2. using midpoints;
      3. using right endpoints.
30. Riemann integrals:
    • Date assigned: November 21 Tuesday.
    • Date due: November 27 Monday.
    • Reading from the textbook:
      • From page 296 through the end of Example 5 on page 297 (§5.2: Limits of Finite Sums);
      • Pages Pages 300–308 (§5.3).
    • Reading from my notes: Page 28 (§4.1).
    • Problems due:
      1. Suppose that ∫⁵_x=3 f(x) dx = 5 and ∫⁵_x=3 g(x) dx = 7. What is ∫⁵_x=3 (f(x) + g(x)) dx?
      2. Suppose that ∫⁵_x=3 f(x) dx = 5 and ∫⁸_x=5 f(x) dx = 4. What is ∫⁸_x=3 f(x) dx?
31. Antidifferentiation:
    • Date assigned: November 28 Tuesday.
    • Date due: November 29 Wednesday.
    • Reading from the textbook:
      • From page 265 through Example 4 on page 268 (§4.8: introduction, Finding Antiderivatives);
      • Pages 270&271 (§4.8: Indefinite Integrals).
    • Reading from my notes:
      • The top half of page 29 (§4.2);
      • The bottom half of page 30 (§4.4).
    • Problems due:
      1. If f(x) = sin(x² + e^3x) 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: ∫_a^b f(x) dx is the _____ integral of f from a to b, while ∫ f(x) dx is the _____ integral of f.
32. The Fundamental Theorem of Calculus:
    • Date assigned: November 29 Wednesday.
    • Date due: November 30 Thursday.
    • Reading from the textbook: From page 312 through the middle of page 318 (all of §5.4 except for Total Area).
    • Reading from my notes:
      • The rest of pages 29&30 (§4.3),
      • The top of page 31 (§4.5).
    • Problems due:
      1. If f is continuous everywhere, then what is the derivative of ∫₀^x f(t) dt with respect to x?
      2. If f is continuously differentiable everywhere, then what is ∫_a^b f′(t) dt?
      3. If f is continuous everywhere, then let F(x) be ∫f(x) dx; what is ∫_a^b f(t) dt?
33. Integration by substitution:
    • Date assigned: November 30 Thursday.
    • Date due: December 1 Friday.
    • Reading from the textbook:
      • Pages 323–329 (§5.5);
      • From page 331 through Example 3 on page 333 (§5.6: introduction, The Substitution Formula, Definite Integrals of Symmetric Functions).
    • Problems due: Suppose that F and g are differentiable functions, with f = F′.
      1. What is ∫ f(g(x)) g′(x) dx?
      2. Assuming that f and g′ are continuous, what is ∫_a^b f(g(x)) g′(x) dx?
34. Differential equations:
    • Date assigned: December 1 Friday.
    • Date due: December 4 Monday.
    • Reading from the textbook: The rest of page 268 and through the end of Example 5 on the top of page 270 (§4.8: Initial Value Problems and Differential Equations, Antiderivatives and Motion).
    • Reading from my notes:
      • The middle of page 31 (§4.6),
      • pages 33–35 (§5), especially §5.2 on page 35.
    • Problems due: Knowing that ∫ ln x dx is x ln x − x and (x ln x − x)|_x=1 is −1, solve the following problems:
      1. Solve f'(x) = ln x in general;
      2. Solve f'(x) = ln x with f(1) = 0.
35. Planar area:
    • Date assigned: December 5 Tuesday.
    • Date due: December 6 Wednesday.
    • Reading from the textbook:
      • The rest of page 318 and through page 320 (§5.4: Total Area);
      • The rest of page 333 and through page 337 (§5.6: Areas Between Curves, Integration with Respect to y).
    • Problems due:
      1. Assuming that f(x) ≥ g(x) whenever a ≤ x ≤ b, where f and g are continuous functions, 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. Assuming that f(y) ≥ g(y) whenever c ≤ y ≤ d, where f and g are continuous functions, 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?
36. Arclength:
    • Date assigned: December 6 Wednesday.
    • Date due: December 7 Thursday.
    • Reading from the textbook: Pages 366–370 (§6.3).
    • Problems due:
      1. Assuming that f is a continuously differentiable function, 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. Assuming that g is a continuously differentiable function, 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?
37. Volume of revolution:
    • Date assigned: December 7 Thursday.
    • Date due: December 8 Friday.
    • Reading from the textbook:
      • Pages 347–355 (§6.1);
      • Pages 358–363 (§6.2).
    • Problems due:
      1. Assuming that R(x) ≥ r(x) ≥ 0 whenever a ≤ x ≤ b, where r and R are continuous functions and a ≤ b are constants, 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. Assuming that H(x) ≥ h(x) whenever a ≤ x ≤ b, where h and H are continuous functions and 0 ≤ a ≤ b are constants, 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)?
38. Surface area of revolution:
    • Date assigned: December 8 Friday.
    • Date due: December 11 Monday.
    • Reading from the textbook: Pages 372–375 (§6.4).
    • Problems due:
      1. Assuming that f(x) ≥ 0 whenever a ≤ x ≤ b, where f is a continuously differentiable function, 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. Assuming that g(y) ≥ 0 whenever c ≤ y ≤ d, where g is a continuously differentiable function, 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?

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