1. General review:
• Date due: August 27 Tuesday.
• Skim: Everything through Section 1.2 (everything through page 18);
• Skim: Section 1.6 through "Finding Inverses" (pages 38–41).
• Everything through Section 1.4 (everything through page 6);
• Optional: Section 1.5 (page 7).
• Exercises 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?
2. Limits informally:
• Date due: August 28 Wednesday.
• 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);
• 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:
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?
3. Continuity informally:
• Date due: August 29 Thursday.
• 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:
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?
4. Defining continuity:
• Date due: September 3 Tuesday.
• 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 (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.
5. Defining limits:
• Date due: September 4 Wednesday.
• 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:
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) = _____.)
6. Evaluating limits and checking continuity:
• Date due: September 5 Thursday.
• Reading from my notes: Sections 2.6&2.7 (pages 14–17).
• 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);
• 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).
• Reading from my notes: the very bottom of page 12 through page 14 (§2.5).
• Exercises due:
1. If you're taking the limit of a rational expression as x → c, and you get 0/0 when you evaluate the expression at x = c, then what factor can you cancel from the numerator and denominator to simplify your expression (and then evaluate the limit)?
2. 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?
3. What is the limit of (sin x)/x as x → 0?
• Date due: September 9 Monday.
• 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:
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, ∞).
8. Differences and difference quotients:
• Date due: September 10 Tuesday.
• 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: 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.
9. Derivatives as limits:
• Date due: September 11 Wednesday.
• Reading from the textbook: Chapter 3 through Section 3.1 (pages 116–118).
• Exercises 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.
10. Derivative functions:
• Date due: September 17 Tuesday.
• Section 3.2 (pages 120–125);
• Section 3.3: "Second- and Higher-Order Derivatives" (page 136).
• Exercises 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.
11. Differentiating polynomials:
• Date due: September 18 Wednesday.
• Reading from the textbook: Section 3.3 through "Powers, Multiples, Sums, and Differences" (pages 129–132);
• Exercises due:
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)?
12. Rules for differentiation:
• Date due: September 19 Thursday.
• Sections 3.3&3.4 (pages 23&24);
• Optional: Section 3.5 (page 25).
• Section 3.3: "Products and Quotients" (pages 133–136), skipping Examples 6.b and 7.b;
• Skim: Section 3.6 (pages 154–158), focussing on Examples 1, 6.a, 6.b, and 7;
• Optional: Section 3.8 through "Derivatives of Inverses of Differentiable Functions" (pages 167–169).
• Exercises due: Write answers using prime notation, not d/dx.
1. 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)?
2. 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 _____.
3. If f and g are differentiable everywhere and h(x) = f(g(x)) for all x, then what is h⁠′⁠(x)?
13. Differentials:
• Date due: September 23 Monday.
• Section 3.6 (pages 26&27);
• Optional: Section 3.7 (pages 27&28).
• Reading from the textbook: Optional: Section 3.11: "Differentials" (pages 196&197).
• Exercises 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.
14. Implicit differentiation:
• Date due: September 24 Tuesday.
• Reading from my notes: Section 3.6 (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: 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 = _____.
15. Exponential functions:
• Date due: September 25 Wednesday.
• Skim: Section 1.5 (pages 33–37);
• 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:
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.
16. Logarithmic functions:
• Date due: September 26 Thursday.
• Skim: Section 1.6: "Logarithmic Functions", "Properties of Logarithms", "Applications" (pages 41–44);
• Section 3.8: "Derivative of the Natural Logarithm Function" (pages 170&171);
• Section 3.8: the rest of "The Derivatives of au and logau", "Logarithmic Differentiation" (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:
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.
17. Trigonometric operations:
• Date due: September 30 Monday.
• Skim: Section 1.3 (pages 21–27);
• 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:
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.
18. Inverse trigonometric operations:
• Date due: October 1 Tuesday.
• 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);
• The rest of Section 3.9 (pages 179–182).
• Exercises due:
1. Simplify asin x + acos x (where asin may also be written as arcsin, sin−1, and other ways, and similarly for acos);
2. Write the differential of atan u (where atan may also be written as tan−1 and other ways) using u, du, and algebraic (not trigonometric) operations.
19. Using derivatives with respect to time:
• Date due: October 2 Wednesday.
• Reading from my notes: Section 3.9 (pages 30&31).
• Reading from the textbook: Section 3.4: everything through "Motion Along a Line" (pages 139–143).
• Exercises 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. 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?
20. Harmonic motion:
• Date due: October 8 Tuesday.
• Reading from the textbook: Section 3.5: "Simple Harmonic Motion" (pages 150&151).
• Additional reading: Simple harmonic motion on the English-language Wikipedia (version last edited by me).
• Exercises due: 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.
21. Related rates:
• Date due: October 9 Wednesday.
• Reading from my notes: Optional: Review the full paragraph on the top of page 30.
• Reading from the textbook: Section 3.10 (pages 184–188).
• Exercise due: 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, state whether or not, in the context of that example, 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.)
22. Sensitivity and linear approximation:
• Date due: October 10 Thursday.
• Reading from my notes: Section 3.10 (pages 31&32).
• Section 3.11 through "Linearization" (pages 192–195);
• Section 3.11 "Estimating with Differentials", "Error in Differential Approximation" (pages 197–199);
• Section 3.11: "Sensitivity to Change" (page 200).
• Exercises 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.
23. Mean-value theorems:
• Date due: October 14 Monday.
• Reading from my notes: Section 3.12 (pages 33&34).
• 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: 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 _____.
24. Increasing and decreasing functions:
• Date due: October 15 Tuesday.
• Reading from the textbook: Section 4.3 through "Increasing Functions and Decreasing Functions" (pages 228&229).
• Exercises due: 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.
25. Constant functions:
• Date due: October 16 Wednesday.
• Section 4.2: "Mathematical Consequences", "Finding Velocity and Position from Acceleration" (pages 223–224);
• Optional (but we'll come back to this in Calculus 2): The rest of Section 4.2 (pages 224–226).
• Exercises due: 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.
26. L'Hôpital's Rule:
• Date due: October 17 Thursday.
• Reading from my notes: Section 3.13 (page 34).
• Section 4.5 through "Indeterminate Powers" (pages 246–251);
• Optional: Section 4.5: the rest "Proof of L'Hôpital's Rule" (page 251, page 252).
• Exercises due: If D is any direction in the variable x, 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 infinite while limDg(x) is zero;
4. limDf(x) is infinite while limDg(x) is a nonzero real number;
5. limDf(x) and limDg(x) are both infinite.
27. Absolute extrema:
• Date due: October 23 Wednesday.
• Reading from the textbook: Section 4.1: "Finding extrema" (pages 215–217).
• Exercises 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?
28. Local extrema:
• Date due: October 29 Tuesday.
• Reading from the textbook: The rest of Section 4.3 (pages 229–231).
• Exercises due: Suppose that a, b, and c are real numbers with a < c < b, and f is a function defined on the interval (a, b). Also suppose that f is continuous on (a, b), differentiable on (a, c), and differentiable on (c, b). (In particular, f must be continuous at c, but f may or may not be differentiable at c.) For each of the following circumstances, state whether f has a strict local maximum at c, a strict local minimum at c, both, or neither.
1. If f⁠′⁠(x) < 0 when a < x < c, while f⁠′⁠(x) < 0 when c < x < b.
2. If f⁠′⁠(x) < 0 when a < x < c, while f⁠′⁠(x) > 0 when c < x < b.
3. If f⁠′⁠(x) > 0 when a < x < c, while f⁠′⁠(x) < 0 when c < x < b.
4. If f⁠′⁠(x) > 0 when a < x < c, while f⁠′⁠(x) > 0 when c < x < b.
29. The second-derivative test:
• Date due: October 30 Wednesday.
• 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: Suppose that f is twice continuously differentiable on an open interval and that c is a number in that interval with f⁠′⁠(c) = 0. For each of the following circumstances, state whether f must have a strict local maximum at c, f must have a strict local minimum at c, or the given information is not enough to tell.
1. If f⁠″⁠(c) < 0;
2. If f⁠″⁠(c) = 0;
3. If f⁠″⁠(c) > 0.
30. Concavity:
• Date due: October 31 Thursday.
• Reading from the textbook: Section 4.4 through "Points of Inflection" (pages 233–237).
• Reading from my notes: Section 3.14 (page 35).
• Exercises 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.
31. Graphing:
• Date due: November 4 Monday.
• Optional: Section 1.4 (pages 29–32);
• The rest of Section 4.4 (pages 237–242).
• Exercise due: Suppose that a function f is continuous everywhere; has critical points at x = −20, 0, 7, and 12; inflection points at −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?
32. Graphing asymptotes:
• Date due: November 5 Tuesday.
• Reading from my notes: Section 3.15 (pages 35&36).
• Reading from the textbook: Optional: Section 2.6: "Horizontal Asymptotes" and "Oblique Asymptotes" (pages 99–102).
• Exercises 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.
33. Applied optimization:
• Date due: November 6 Wednesday.
• Reading from my notes: Section 3.16 (pages 36&37).
• Reading from the textbook: Section 4.6 through "Examples from Mathematics and Physics" (pages 255–258).
• Exercises 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 the limit of u, as x approaches 0, 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?
34. Optimization in economics and finance:
• Date due: November 7 Thursday.
• Reading from my notes: Section 3.17 (pages 37&38).
• Section 3.4: "Derivatives in Economics and Biology" (pages 143–145);
• Section 4.6 "Examples from Economics" (pages 258&259).
• Exercises due:
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)?
35. Newton's Method:
• Date due: November 11 Monday.
• Reading from my notes: Section 3.11 (pages 32&33).
• Reading from the textbook: Section 4.7 (pages 266–269).
• Exercise due: 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′⁠.
36. Riemann sums:
• Date due: November 12 Tuesday.
• 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: 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.
37. Riemann integrals:
• Date due: November 13 Wednesday.
• Reading from the textbook: Section 5.3 (pages 307–316).
• Reading from my notes: Chapter 4 through Section 4.1 (page 39).
• Exercises due: 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?)
38. Antidifferentiation:
• Date due: November 19 Tuesday.
• Section 4.8 through "Finding Antiderivatives" (pages 271–274);
• Section 4.8 "Indefinite Integrals" (pages 276&277).
• Reading from my notes: Section 4.2 (pages 40&41).
• Exercises due:
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).
39. The Fundamental Theorem of Calculus:
• Date due: November 20 Wednesday.
• Reading from the textbook: Section 5.4 through "The Relationship Between Integration and Differentiation" (pages 320–327).
• Reading from my notes: Section 4.3 (pages 41&42).
• Exercises due:
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); what is ∫abf(t) dt?
40. Integration by substitution:
• Date due: November 21 Thursday.
• Section 5.5 (pages 332–337);
• Section 5.6 through "Definite Integrals of Symmetric Functions" (pages 339–342).
• Exercises 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 ∫abf(g(x)) g⁠′⁠(x) dx?
41. Differential equations:
• Date due: November 25 Monday.
• Reading from the textbook: Section 4.8: "Initial Value Problems and Differential Equations", "Antiderivatives and Motion" (pages 274&275).
• Section 4.4 (page 42);
• Chapter 5 through Section 5.3 (pages 45–47), especially Section 5.3 (page 47).
• Exercises due: 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.
42. Planar area:
• Date due: November 26 Tuesday.
• Section 5.4 "Total Area" (pages 327&328);
• The rest of Section 5.6 (pages 342–345).
• Exercises due:
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?
43. Arclength:
• Date due: December 2 Monday.
• Reading from the textbook: Section 6.3 (pages 375–379).
• Exercises due:
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?
44. Volume of revolution:
• Date due: December 3 Tuesday.
• Chapter 6 through Section 6.1 (pages 356–363);
• Section 6.2 (pages 367–372).
• Exercises due:
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)?
45. Surface area of revolution:
• Date due: December 4 Wednesday.
• Section 6.4 (pages 381–384).
• Reading from my notes: Section 4.6 (pages 43&44).
• Exercises due:
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.)
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