## Math 1510 Calculus I

### Homework and In-class Assignments

Assigned work is listed here with most recent work listed first.
Definitions
• "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 and WeBWorK (Reading and Homework). These assignments contribute to your "Homework" grade.
Assigned Work
• Chapter 4 Review Problems
• In class: (The numbers refer to the numbering in the online WileyPlus "Review Exercise and Problems for Chapter 4"): 50, 68, 78, 80, 96, 100
• 50. Find the dimensions of the closed rectangular box having a square base $$x$$ by $$x$$ cm and height $$h$$ cm of maximum volume, given that the surface area is $$8$$ $$\hbox{cm}^2$$
• 68. The hypotenuse of a right triangle has one end at the origin and one end on the curve $$y=x^2e^{-3x}$$, with $$x \geq 0$$. One of the other two sides is on the $$x$$-axis, the other side is parallel to the $$y$$-axis. Fnd the maximum area of such a triangle. At what $$x$$-value does it occur?
• 78. A polystyrene cup is in the shape of a frustrum (the part of a cone between two parallel planes cutting the cone), has top radius $$2r$$, base radius $$r$$ and height $$h$$. The surface area $$S$$ of such a cup is given by $$S = 3\pi r\sqrt{r^2+h^2}$$ and its volume $$V$$ by $$V=\frac{7\pi r^2 h}{3}$$. If the cup is to hold $$200$$ ml, use a calculator or a computer to estimate the value of $$r$$ that minimizes its surface area
• 80. For $$a > 0$$, the following line forms a triangle in the first quadrant with the $$x$$- and $$y$$-axes: $a(a^2+1)y = a-x.$
• (a) In terms of $$a$$, find the $$x$$- and $$y$$-intercepts of the line.
• (b) Find the area of the triangle, as a function of $$a$$.
• (c) Find the value of $$a$$ making the area a maximum.
• (d) What is this greatest area?
• (e) If you want the triangle to have area $$\frac{1}{5}$$, what choices do you have for $$a$$?
• 96. Ice is being formed in the shape of a circular cylinder with inner radius $$1$$ cm and height $$3$$ cm. The outer radius of the ice is increasing at $$0.03$$ cm per hour when the outer radius is $$1.5$$ cm. How fast is the volume of the ice increasing at this time?
• 100. A voltage, $$V$$ volts, applied to resistor of $$r$$ ohms produces an electric current of $$I$$ amps, where $$V = IR$$. As the current flows the resistor heats up and its resistance falls. If $$100$$ volts is applied to a resistor of $$1000$$ ohms the current is initially $$0.1$$ amps but rises by $$0.001$$ amps/minute. At what rate is the resistance falling if the voltage remains constant?
• Also work on these problems (worked solutions in WileyPlus): 67, 77, 101
• 67. A rectangle has one side on the $$x$$-axis and two corners on the top half of the circle of radius $$1$$ centered at the origin. Find the maximum area of such a rectangle. What are the coordinates of its vertices?
• 77. A ship is steaming due north at $$12$$ knots ($$1$$ knot $$= 1.85$$ kilometers/hour) and sights a large tanker $$3$$ kilometers away northwest steaming at $$15$$ knots due east. For reasons of safety, the ships want to maintain a distance of at least $$100$$ meters between them. Use a calculator or computer to determine the shortest distance between them if they remain on their current headings, and decide if they need to change course.
• 101. A train is heading due west from St. Louis. At noon, a plane flying horizontally due north at a fixed altitude of $$4$$ miles passes directly over the train. When the train has traveled another mile, it is going $$80$$ miles per hour, and the plane has traveled another $$5$$ miles and is going $$500$$ miles per hour. At that moment, how fast is the distance between the train and the plane increasing?
• Chapter 6
• Section 6.4
• In class: 12, 22, 26, 36
• Then do: 17-45 odd
• Section 6.3
• In class: 16, 20, 26, 32
• Then do: 15, 14, 19, 21-25, 29, 33, 34
• Section 6.2
• In class: 82, 92, and the following problem from the 6th edition:
• 113. In this problem we examine the expression $$x^x$$
• (a) Explain why you can rewrite $$x^x$$ as $$x^x = e^{x\ln x}$$ for $$x>0$$.
• (b) Use your answer from part (a) to find $$\frac{d}{dx}\left(x^x\right)$$.
• (c) Find $$\int x^x(1+ \ln x)\ dx.$$
• (d) Find $$\int_1^2 x^x(1+ \ln x)\ dx.$$ exactly using part (c). Check your answer numerically (that is, using your calculator).
• Then do: 76, 77, 79, 81, 82, 85, 86, 88, 91, 93, 94, 100, 101, and the following problem from the 6th edition:
• 114. The origin and the point $$(a,a)$$ are at opposite corners of a square. Calculate the ratio of the areas of the two parts into which the curve $$\sqrt{x}+\sqrt{y} = \sqrt{a}$$ divides the square.
• Section 6.1
• In class: 20, 22, 32, 36
• Then do: 15-17, 21-35 odd
• Chapter 5
• Section 5.4
• 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
• 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
• 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)?
• Section 5.1
• In class: 12, 16, 38
• Then do: 25-27, 31, 32, 35-37, 39, 41, 43
• 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 $$R$$.
• 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
• In class: 34, 44, 52
• Then do: 17, 18, 23, 25, 30, 35, 41, 42, 45, 47, 48, 50, 51, 55
• Section 4.4
• In class: 38, 42, 62
• Then do: 19, 28, 41, 47, 52, 60, 63, 67
• Section 4.3
• 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
• 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
• 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).)
At board: 2-40 even
• 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)
• 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.
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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.
• Then do: 34, 36, 43, 47, 52, 57, 66
• 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
• 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
• 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
• 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
• 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
• 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
• 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?

• 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.