Problem sets and quizzes

Here are the quizzes and their associated problem sets (Quiz 1, Quiz 2, Quiz 3, Quiz 4, Quiz 5, Quiz 6, Quiz 7, Quiz 8, Quiz 9):

1. Continuity and limits:
   • Date taken: October 11 Thursday.
   • Problems from Section 1.1 (pages 11–13): 7, 8, 13, 23, 25, 26, 71.
   • Problems from Section 1.2 (pages 18–21): 5, 23, 24.
   • Problems from Section 2.2 (pages 66–69): 1, 3, 9, 10, 15, 19, 25, 29, 35, 37, 43, 53, 57, 65.a.
   • Problems from Section 2.3 (pages 75–78): 7, 9, 11, 13, 15, 17, 23, 27, 49.
   • Problems from Section 2.4 (pages 83–85): 1, 3, 5, 11, 17, 23.
   • Problems from Section 2.5 (pages 94–96): 1, 3, 5.d, 7, 9, 13, 15, 19, 21, 25, 27, 29, 39, 43, 77.
   • Problems from Section 2.6 (pages 107–109): 1, 7, 9, 11, 15, 21, 25, 27, 29, 35, 41, 45, 49, 53, 57.
   • Problems from Section 4.1 (pages 214–217): 1, 3, 5, 7, 9, 15, 17, 20.
   • Extra credit (essay): Explain your background in mathematics and what you are going to use this course for.
2. Derivatives:
   • Date taken: October 18 Thursday.
   • Problems from Section 2.1 (pages 56–58): 1, 3, 5.
   • Problems from Section 3.1 (pages 117–119): 1, 2, 5, 11, 19, 25, 35, 36.
   • Problems from Section 3.2 (pages 124–127): 1, 3, 5, 27, 28, 29, 30, 31, 34, 35, 37, 41, 43, 45, 47.
   • Problems from Section 3.3 (pages 135–137):
     • Where the textbook writes ‘d/dx’ before an expression in parentheses, it really should write the prime symbol after the expression instead, since these are expressions for functions: 53;
     • 61, 69;
     • Extra credit: Show at least one intermediate step for each part: 74, and also show (as part c) how the Reciprocal Rule is a special case of the Power Chain Rule.
   • Problems from Section 3.6 (pages 158–161): 9, 11, 13, 87.a–e, 104.
   • Problem from Section 3.8 (pages 174–176): 3.
3. Differentials and exponentials:
   • Date taken: October 25 Thursday.
   • Problems from Section 3.11 (pages 198–201): 19, 21, 23, 31, 33.
   • Problems from Section 3.3 (pages 135–137): 1, 3, 5, 17, 19, 20, 23, 25, 29, 31, 35, 41, 51.
   • Problems from Section 3.6 (pages 158–161): 23, 31, 33, 35, 37, 83.
   • Problems from Section 3.7 (pages 164&166): 1, 3, 7, 21, 27, 29.
   • Problems from Section 1.5 (pages 37&38): 11, 15, 19.
   • Problems from Section 1.6 (pages 48–50): 43, 45, 47, 49, 51, 55, 57, 61, 63, 65.
   • Problems from Section 3.8 (pages 174–176): 21, 27, 39, 47, 51, 57, 65, 75, 89.
   • Additional extra-credit problem: Many of the rules for differentiation are similar to rules for logarithms: taking the differential or logarithm of the product of two expressions results in two terms, with one derivative or logarithm in each; starting with division instead of multiplication changes addition into subtraction in the differential or logarithm; taking the differential or logarithm of a power involves multiplying by the exponent, etc. The connection here involves the simple form of a logarithmic function's derivative (which is a multiple of the reciprocal function). So, suppose that c is a real number, and suppose that f and g are two functions that are differentiable at c. For simplicity, suppose further that f(c) = 1 and g(c) = 1. (This isn't really necessary, but it will keep extra factors of f(c) and g(c) from cluttering up your equations.) Notice that ln f(c) and ln g(c) are both 0. Also notice that fg, f/g, and fⁿ (meaning f raised to the power of n), where n is any constant, have the same properties as f and g above (they are differentiable at c and their values at c are all 1). Finally, notice that the Logarithm Rule, in functional form, states that the derivatives of ln f and ln g at c are f′(c) and g′(c) respectively (because f(c) and g(c) are 1).
     1. Work out the derivative at c of ln(fg), using the Logarithm Rule and the Product Rule but not any other properties of logarithms; then compare this to the derivative of ln f + ln g at c. (Show at least one intermediate step for each of these.)
     2. Now work out the derivative at c of ln(f/g), using the Logarithm Rule and the Quotient Rule; then compare this to the derivative of ln f − ln g at c. (Also show at least one intermediate step for each of these.)
     3. Finally, work out the derivative at c of ln(fⁿ), where n is any constant, using the Logarithm Rule and the Power Chain Rule; then compare this to the derivative of n ln f at c. (Show at least one intermediate step for each of these too.)
     This shows that ln(fg), ln(f/g), and ln(fⁿ) have the same derivatives as ln f + ln g, ln f − ln g, and n ln f, respectively. (You can do the same for a logarithm with any other base b, if you keep track of the extra factors of ln b; again, I used base e to keep it simpler.) These also all have the value 0 at c. There is a general principle that if two functions have the same derivative everywhere and the same value at some particular c, then they must be the same functions entirely (at least on any interval containing c where they are defined). So in this way, the algebraic properties of logarithms follow from the facts that the derivative of the natural logarithm is the reciprocal and that the logarithm of 1 is 0, with each additional property of logarithms being a consequence of the corresponding rule of differentiation. (The reason that there is no rule for the logarithm of a sum is that (f + g)(c) is not 1, which makes the argument break down.)
4. Trigonometry and time:
   • Date taken: November 1 Thursday.
   • Problems from Section 1.3 (pages 27–29):
     • 5, 6, 9, 11, 31, 33;
     • These really use the Half-Angle Formulas: 47, 49.
   • Problems from Section 3.11 (pages 198–201): 25, 27, 29, 37.
   • Problems from Section 3.5 (pages 151–153): 1, 3, 5, 9, 11, 13, 15, 19, 23, 31, 35, 61.
   • Problems from Section 3.6 (pages 158–161): 27, 29, 39, 43, 47, 65, 97, 98.
   • Problem from Section 3.7 (pages 164&166): 11.
   • Problems from Section 3.8 (pages 174–176): 63, 93.
   • Problems from Section 1.6 (pages 48–50): 67, 69.
   • Problems from Section 3.9 (pages 180&182):
     • 1, 7, 9, 11, 13–20, 21, 23, 25, 31, 35, 37, 39;
     • Extra credit: Show at least two intermediate steps: 50.
   • Problems from Section 3.4 (pages 144–147): 1, 3, 5, 7, 9, 13, 17, 18, 21, 25.
   • Problems from Section 3.10 (pages 187–190): 1, 2, 3, 6, 12, 13, 15, 23, 27, 29, 31, 41.
5. Linearization and L'Hôpital:
   • Date taken: November 8 Thursday.
   • Problems from Section 3.11 (pages 198–201): 1, 2, 3, 4, 5, 6, 7, 10, 11, 13, 14, 15, 16, 51, 52, 53, 57.
   • Problem from Section 3.4 (pages 144–147): 30.
   • Problems from Section 4.2 (pages 222–224):
     • 1, 5, 9, 11, 13, 15, 21, 25, 29, 31, 39, 43, 48;
     • Extra credit: 62.
   • Problems from Section 4.3 (page 228): Assume that the graphs include all unmarked endpoints but do not extend beyond the portion shown (so the functions' domains are [−4, 4]): 15, 16, 17.
   • Problems from Section 4.5 (pages 248&249): 1, 3, 5, 11, 13, 15, 21, 27, 37, 43, 51, 55, 59, 60, 78.a.
6. Graphs of functions:
   • Date taken: November 15 Thursday.
   • Problems from Section 4.1 (pages 214–217): 11, 12, 13, 14, 23, 27, 37, 39, 41, 55, 61, 71.
   • Problems from Section 4.3 (pages 228–230): 1, 3, 5, 7, 13, 19, 23, 29, 33, 43, 47.a&b, 49.a&b, 55.a&b, 67.
   • Problems from Section 4.4 (pages 238–241):
     • 1, 3, 5, 7, 11, 19, 23, 27, 43, 53, 81, 82, 83, 84, 85, 87, 101, 103, 104, 105, 106, 107, 108.
     • Extra credit: Assume that the second derivative is also defined everywhere, and give reasons for your conclusion: 114.
   • Problems from Section 1.4 (page 33): Optional: 1, 2, 3, 4.
7. Transitional topics:
   • Date taken: November 28 Wednesday.
   • Problem from Section 3.4 (page 146): 23.
   • Problems from Section 4.6 (pages 255–260): 1, 3.c, 7, 9, 11, 13, 15, 29, 31, 37, 39, 43.a, 51, 53.a.
   • Problems from Section 4.7 (pages 264&265): 1, 3, 5, 9, 10, 13, 21, 23.
   • Additional extra-credit problem: Newton's Method is guaranteed to work under certain conditions given by the Newton–Kantorovich Theorem, which you can find at the end of §3 on page 3 of my handout on applications of differentiation (PDF). Of the 4 applications of Newton's Method in Exercises 4.7.1, 4.7.9, and 4.7.10, to which of these does this theorem apply, and why?
   • Problems from Section 5.2 (pages 299&300): 1–6, 7, 9, 11–16, 17, 19, 23, 29, 31, 33, 35, 39, 43.
   • Problems from Section 5.1 (pages 291–293): 1–19 odd.
   • Problems from Section 5.3 (pages 308–315): 9, 11, 13, 15, 17, 27, 35, 43, 49, 71.
8. Integration:
   • Date taken: December 5 Wednesday.
   • Problems from Section 4.8 (pages 271–275):
     • 1–24, 27, 29, 35, 39, 41, 45, 49, 51, 55, 61, 65, 75, 77, 81, 83;
     • Note that the book is conflating the function y with its output y(x) at the input x: 95, 97, 105.
   • Problems from Section 5.4 (pages 320–323): 1–15 odd, 23, 29, 39, 43, 47, 51, 57, 59, 61, 77, 78, 79.
   • Problems from Section 5.5 (pages 329–330):
     • 1–7 odd, 15, 17, 21, 25, 27, 31, 35, 39, 47, 55, 57, 61;
     • Note that the book is conflating the function s with its output s(t) at the input t: 73, 75.
   • Problems from Section 5.6 (pages 337–340): 1–9 odd, 13, 19, 25, 31, 37, 41, 45.
   • Problems from Section 4.2 (pages 222–224): 40, 45, 47.
   • Additional extra-credit problem: A positive quantity undergoes exponential growth if its rate of change with time is also positive and proportional to the size of the quantity (so a constant multiple of that size). If t stands for time and A stands for a quantity undergoing exponential growth, then write down a differential equation involving A and t to say that the rate of change of A with time is a positive constant k times A. If the value of A is a when t = 0, then solve this initial-value problem by manipulating the equation algebraically so that only one variable (and its differentials) appear on either side of the equation (after which you can integrate both sides; this is called separation of variables). You should then be able to see why this situation is called ‘exponential’ growth. (The unspecified constants k and a should both appear in your answer.)
9. Geometric applications:
   • Date taken: December 12 Wednesday.
   • Problems from Section 5.6 (pages 337–340): 47, 51, 55, 57, 59, 67, 69, 75, 81, 87, 99.
   • Problems from Section 6.3 (pages 376&377):
     • 1, 3, 7, 9, 13, 15.a, 19.a;
     • Extra credit: Show at least two intermediate steps (hint: implicit differentiation): 30.
   • Problems from Section 6.1 (pages 361–364): 1, 5, 7, 13, 15, 17, 21, 23, 25, 35, 45, 51.
   • Problems from Section 6.2 (pages 369–371): 1, 3, 5, 9, 15, 21, 25, 27, 31, 39.
   • Problems from Section 6.4 (pages 381–383): 1–7.a odd, 13–21 odd.

The permanent URI of this web page is http://tobybartels.name/MATH-1600/2018FA/quizzes/.