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### Homework and In-class Assignments

- "With reading": Do these problems after you read a section, in preparation for class. Do not turn your solutions in.
- "In class": These problems will be assigned to be done in class in groups. Each group will turn in one solution set to be graded for credit.
- "Then do": Do these problems after the section has been covered in class. These problems are generally a little more challenging than the ones assigned with the reading. Do not turn your solutions in.
- Questions in italics are NOT available in the Wiley Plus system.
- You must also do the assignments posted in Wiley Plus (Reading and Homework). These assignments contribute to your "Homework" grade.

**Definitions**

- Chapter 5
- Section 5.4
- With reading: 1-23 odd
- In class: 30, 42, 44, 54
- Then do: 25, 26, 29, 31, 32, 35, 37-39, 41, 53, 55, 64-66, 68

- Section 5.3
- With reading: 1-13 odd
- In class: 18, 30, 34, 44
- Then do: 15, 17, 19, 21, 23, 25, 26, 29, 33, 35, 39, 42, 43, 47, 53, 55, 63

- Section 5.2
- With reading: 1-21 odd
- In class: 34, 58, 60, 64
- Then do: 23, 24, 35, 37, 39, 44, 45, 49, 53, 55, 61, 63, 65, 70-73, 75, and the following problem from the 6th edition:
- 87. Consider the integral \(\int_1^2 \frac{1}{t} dt\) in Example 1. By dividing the interval \(1 \leq t \leq 2\) into \(10\) equal parts, we can show that
\[0.1\left(\frac{1}{1.1}+\frac{1}{1.2}+ \ldots + \frac{1}{2}\right) \leq\int_1^2\frac{1}{t} dt\] and
\[\int_1^2\frac{1}{t} dt \leq 0.1\left(\frac{1}{1}+\frac{1}{1.2}+ \ldots + \frac{1}{1.9}\right).\]
- (a) Now divide the interval \(1 \leq t \leq 2\) into \(n\) equal parts to show that \[\sum_{r=1}^n \frac{1}{n+r} < \int_1^2\frac{1}{t} dt < \sum_{r=0}^{n-1} \frac{1}{n+r}.\]
- (b) Show that the difference between the upper and lower sums in part (a) is \(\frac{1}{2n}\).
- (c) The exact value of \(\int_1^2\frac{1}{t} dt\) is \(\ln(2)\). How large should \(n\) be to approximate \(\ln(2)\) with an error of at most \(5\cdot 10^{-6}\), using one of the sums in part (a)?

- 87. Consider the integral \(\int_1^2 \frac{1}{t} dt\) in Example 1. By dividing the interval \(1 \leq t \leq 2\) into \(10\) equal parts, we can show that
\[0.1\left(\frac{1}{1.1}+\frac{1}{1.2}+ \ldots + \frac{1}{2}\right) \leq\int_1^2\frac{1}{t} dt\] and
\[\int_1^2\frac{1}{t} dt \leq 0.1\left(\frac{1}{1}+\frac{1}{1.2}+ \ldots + \frac{1}{1.9}\right).\]

- Section 5.1
- With reading: 1-23 odd
- In class: 12, 16, 38
- Then do: 25-27, 31, 32, 35-37, 39, 41, 43

- Section 5.4
- Chapter 4
- Section 4.8
- With reading: 1-19 odd, 23-33 odd
- In class: 38, 42, and the following (slightly edited) problem from the 6th edition:
- 68. A hypothetical moon orbits a planet, which in turn orbits a star. Suppose that the orbits are circular (the moon around the planet, and the planet around the star), and that the moon orbits the planet 12 times in the time it takes the planet to orbit the star once. In this problem we will investigate whether the moon could come to a stop at some instant (relative to the star).
- (a) Suppose that the radius of the moon's orbit around the planet is 1 unit and that the radius of the planet's orbit around the start is \(R\) units. Explain why the motion of the moon relative to the star can be described by the parametric equations \[x=R\cos(t)+\cos(12t), y = R\sin(t)+\sin(12t).\]
- (b) Find values for \(R\) and \(t\) so that the moon stops relative to the star at time \(t\). (Note: you are not asked to find ALL possible values for \(R\) and \(t\), just one pair that works.)
- (c) On a graphing calculator or computer, plot the path of the moon for the value of \(R\) that you found in (b) and make a sketch of this plot to include in your solution. Experiment with other values for \(R\).

- 68. A hypothetical moon orbits a planet, which in turn orbits a star. Suppose that the orbits are circular (the moon around the planet, and the planet around the star), and that the moon orbits the planet 12 times in the time it takes the planet to orbit the star once. In this problem we will investigate whether the moon could come to a stop at some instant (relative to the star).
- Then do: 35, 37, 39, 40, 41, 43, 44, 49, 50, 52, 53, 55, 57, 59, 60 and the following problem from the 6th edition:
- 69. Derive the general formula for the second derivative \(\frac{d^2y}{dx^2}\) of a parametrically defined curve: \[\frac{d^2y}{dx^2} = \frac{\left(\frac{dx}{dt}\right)\left(\frac{d^2y}{dt^2}\right)-\left(\frac{dy}{dt}\right)\left(\frac{d^2x}{dt^2}\right)}{\left(\frac{dx}{dt}\right)^3}.\]

- Section 4.7
- With reading: 1-41 odd, 44-47
- In class: 44, 46, 54, 64, 72, 88
- Then do: 49-77 odd, 83-89 odd

- Section 4.6
- With reading: 1-15 odd
- In class: 34, 44, 52
- Then do: 17, 18, 23, 25, 30, 35, 41, 42, 45, 47, 48, 50, 51, 55

- Section 4.4
- With reading: 1-15 odd
- In class: 38, 42, 62
- Then do: 19, 28, 41, 47, 52, 60, 63, 67

- Section 4.3
- With reading: 1-15 odd
- In class: 24(a), 30, and the following problem from the 6th edition:
- 65. A rectangular swimming pool is to be built with an area of 1800 square feet. The owner wants a deck of width 5 feet along each side of the pool and of width 10 feet at each end. Find the dimensions of the smallest piece of (rectangular) property on which the pool can be built satisfying these conditions.

- Then do: 17-23, 25, 27, 29, 34, 35, 37, 38, 40, 42, 46, 47, 49, 51, 59, 60

- Section 4.2
- With reading: 1, 2, 3-19 odd
- In class: 36, 42, 44
- Then do: 28, 31, 32, 34, 35, 37, 39, 41, 45, 47, 48, 53

- Section 4.1
- With reading: 1-23 odd, 25-28
- In class: 42, 46, 50, 58
- Then do: 31, 41, 43-45, 47, 52-54, 59-63

- Section 4.8
- Chapter 3
- Section 3.10
- With reading: 1, 3, 5-9
- In class: 10, 15, 26
- Then do: 11, 14, 18, 19, 20, 22, 23, 27

- Section 3.9
- With reading: 1-9 odd
- In class: 14, 31-32, 46
- Then do: 15, 12, 17, 23, 34, 41, 43, 44, 48, 50

- Section 3.8
- With reading: 1-15 odd, 16 (in Wiley Plus: 1, 3, 5, 11, 15, 16)
- In class: 30, 32
- Then do:
*19, 20, 21*, 29,*31, 35*

- Section 3.7
- With reading: 1-29 odd (in Wiley Plus: 1, 5, 9, 13, 15, 17, 25, 27)
- In class: 16, 28, 36
- Then do: 32,
*33, 35, 39, 40, 41*, "AP47" (online only)

- Section 3.6
- With reading: 1-41 odd (in Wiley Plus: 1, 9, 13, 17, 25, 35, 39, 41)
- In class: 64, 66 and the following problem (from the 6th edition)
- 84.(a) Calculate \[\lim_{h\to 0}\frac{\ln(1+h)}{h}\] by identifying the limit as the derivative of \(\ln(1+x)\) at \(x=0\). (Method: use derivative formulas to find \(f'(0)\) for \(f(x) = \ln(1+x)\). Then instead write the limit definition expression for \(f'(0)\). Compare the two.)
- 84. (b) Use the result of part (a) to show that \[\lim_{h\to 0}(1+h)^{\frac{1}{h}}=e.\](Method: let \[C = \lim_{h\to 0}(1+h)^{\frac{1}{h}},\] then take the natural logarithm of both sides and recognize one side from (a). Solve the resulting equation for \(C\).)
- 84. (c) Use the result of part (b) to calculate the related limit \[\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n.\] (Method: make an appropriate substitution in the left side of the expression in (b).)

- Then do: 43, 50,
*52, 53*, 55, 62,*63, 65, 75, 76 and the following problem (from the 6th edition)*- 85. (a) For \(x>0\) find and simplify the derivative of \(f(x) = \arctan(x) + \arctan\left(\frac{1}{x}\right)\).
- 85. (b) What does the result of (a) tell you about \(f\)?

- Section 3.5
- With reading: 1-49 odd (in Wiley Plus: 1, 5, 9, 13, 17, 19, 25, 29, 31)
- In class: At board: 2-46 even
- Then do:
*51*, 53,*57*, 58, 61,*63, 65, 67, 71*, 75, 77, "81" (online only)

- Section 3.4
- With reading: 1-57 odd (in Wiley Plus: 1, 7-11 odd, 17, 19, 31, 33, 43, 45)
- In class: 70, 86; At board: 2-56 even
- Then do: 58, 61,
*63, 65*, 67,*69*, 71, 73,*75*, 77,*83, 85, 87, 89-92, 94, 97*

- Section 3.3
- With reading:
*1, 2*, 3,*4*, 5, 7,*9, 11*, 13,*15, 17*, 19, 21,*23*, 25,*27, 29* - In class: At board: 4-30 even
- Then do:
*31-39 odd, 40, 41, 43, 45*, 49,*51-53, 55, 56, 59, 67*, "90" (online only)

- With reading:
- Section 3.2
- With reading: 1,
*3*, 5, 7, 9,*11*, 13,*15*, 17,*19*, 21,*23-39 odd* - In class: 40, 44, 46; at board: 2-38 even
- Then do: 42,
*43, 48, 50, 52, and the following problems (from the 6th edition)*:- 57. Find the value of \(c\) so that the point \((c,0)\) lies on the tangent line to the graph of \(y= 2^x\) at the point \((0,1)\).
- 58. Use the equation of the tangent line to the graph of \(e^x\) at \(x=0\) to show that \(e^x \ge 1+x\) for all values of \(x\) (Hint: draw a graph that shows both \(y=e^x\) and the tangent line)
- 59. For what value(s) of \(a\) are \(y=a^x\) and \(y=1+x\) tangent at \(x=0\)? Explain.

- With reading: 1,
- Section 3.1
- With reading:
*1, 3-5*, 7,*9*, 11,*13-21 odd*, 23, 25,*27-33 odd*, 35,*37-41 odd*, 43,*45-49 odd* - In class: 72, 76, 84, 96; at board: 6-48 even
- Then do:
*59, 61, 62-67*, 69,*75, 85, 94, 95, 99, 103, 105*

- With reading:

- Section 3.10
- Chapter 2
- Section 2.6
- With reading:
*1*, 2,*3,4* - In class: 3, 8, 12, 16
- Then do:
*7*, 9,*11*, 13, 15, 18,*19, 21, 23*

- With reading:
- Section 2.5
- With reading:
*1*, 2, 4, 8-14 - In class: 28, 38, 40, 48
- Then do:
*16*, 21, 25,*27, 29*, 37, 41,*45, 46*

- With reading:
- Section 2.4
- With reading: 1, 2, 3, 6, 10,
*11-15 odd*, 16 - In class: 42, 50, 52, 54
- Then do:
*23*, 27, 33,*32, 39*, 43,*44*, 51,*53*

- With reading: 1, 2, 3, 6, 10,
- Section 2.3
- With reading: 1,
*3*, 5, 7, 11, 13, 15, 17,*18*, 21, 23,*25, 27* - In class: 32, 42, 48, 50, 52
- Then do:
*31, 36, 38, 41, 49, 51*, 54,*55*, 56,*60, 62*

- With reading: 1,
- Section 2.2
- With reading: 1, 5, 6, 7, 9, 12,
*13-18* - In class: 20, 24, 36, 42
- Then do:
*10, 21-23, 27, 33*, 35, 45-49 odd,*51-53 odd*, 57-65 odd

- With reading: 1, 5, 6, 7, 9, 12,
- Section 2.1
- With reading: 1, 3, 5,
*9*, 12, 13,*14*, 15-18,*19*,*20*, 21 - In class: 22, 24, 28, 30
- Then do: 25, 31, 33,
*34, 40, 42*

- With reading: 1, 3, 5,

- Section 2.6
- Chapter 1
- Section 1.9
- With reading: 1-3, 5-23 odd, 24-26, 27-33 odd, 35-37
- In class: 40, 52, 60
- Then do:
*38*, 39, 41, 45,*46*, 49,*50*, 51, 53, 55,*58*, 59, 63, 65

- Section 1.8
- With reading:
*1*,*3*,*30*, 32, 36, 38, also*12*, 17 from Section 1.10 (Available in Wiley Plus) - In class: Section 1.10 #18, then 50, 54
- Then do:
*6-8*, 9,*11-27 odd*, 39, 42,*43-51 odd*,*52*,*55*,*57*,*60*,*61, 65, 67*, Section 1.10:*1, 10, 11*

- With reading:
- Section 1.7
- With reading:
*1-5*,*7*, 11-16, 17, 21 - In class: 38 and the following problem (from the 6th edition):
- 84. (a) Sketch the graph of a continuous function \(f\) with
*all*of the following properties:- \(f(0) = 2\)
- \(f(x)\) is decreasing for \(0 < x \leq 3\)
- \(f(x)\) is increasing for \(3 < x \leq 5\)
- \(f(x)\) is decreasing for \(x > 5\)
- \(f(x) \to 9\) as \(x \to \infty\)

- 84. (b) Is it possible that the graph of \(f\) is concave down for all \(x > 6\)? Explain.

- 84. (a) Sketch the graph of a continuous function \(f\) with
- Then do: 34,
*36*,*43*,*47*,*52*, 57,*66*

- With reading:
- Section 1.6
- With reading: 1,
*2*, 3,*5*,*7*, 9,*11*, 13,*16*, 17, 19, 23,*24-26*,*33-44* - In class: 50, 54, 56, 68
- Then do:
*18*,*45*,*48*,*51*, 55,*67*,*69*,*70*,*82*

- With reading: 1,
- Section 1.5
- With reading: 1,
*3-9 odd*,*11*, 13, 15,*19*, 21, 23, 24-26,*27-29*, 41 - In class: 62, 69
- Then do:
*37*,*42*,*50-53*, 59, 61, 70, 73,*78*,*80*

- With reading: 1,
- Section 1.4
- With reading: 1-9 odd, 10,
*14*, 16, 21, 27, 30, 31, 61 - In class: 38, 43
- Then do:
*36*,*37*,*45*, 47, 53, 57

- With reading: 1-9 odd, 10,
- Section 1.3
- With reading:
*5*, 7,*10*,*12*, 13, 15,*17*,*19*, 21, 33, 35, 37, 57 - In class: 43, 44, 53, 56
- Then do:
*36*, 42, 45, 50, 51, 54, 58,*75*,*79*

- With reading:
- Section 1.2
- With reading: 1-5, 7, 15, 17,
*34*,*35* - In class: 14, 19, 22
- Then do: 11, 16, 25, 27,
*28*, 29,*30*, 31, 32,*41*,*46*, 48,*49*,*53*,*70*

- With reading: 1-5, 7, 15, 17,
- Section 1.1
- With reading: 1,
*3*, 5, 9, 13, 19, 23,*26*, 29, 31 - In class: Two parts, first 34 and the "overhead"; then 33, 35, 45, 55
- Then do: 27, 33,
*38*, 43, 53, 58,*65*, 67,*71*,*85*

- With reading: 1,

- Section 1.9
- Due on Friday September 1:

Please write a paragraph or two about yourself. What is your mathematical background? Why are you taking this class? What are you hoping to get out of this class? Are you comfortable with using a graphing calculator and/or Desmos? How comfortable do you feel with the WileyPLUS system? How do you intend to ensure a successful experience in this class? Visit the web site for this class - tell me something about it that you like, and maybe something that could be improved. Write anything else you wish.

Finally: Where is New Zealand?

**Assigned Work**

**Important: send an email to your family, telling them when your tests and final exam are in this class, advising them NOT to book travel that conflicts with these events.**