| Section | Problems to do | Submit | Due date | Comments |
|---|---|---|---|---|
| 1.1 | 1,3,7,9,11,13,15,21,23,33,39,49,51,55,57 | 26,62 | Friday, September 5 | |
| 1.2 | 1,3,5,9,15,17,19,27,31,33,35,41,63 | 8,18 | Monday, September 8 | |
| 1.3 | 1,5,9,13,15,17,31,41,51,61,63 | 42,52 | Thursday, September 11 | |
| 1.4 | 1,3,5,9,11,13,15,21,25,29,33 | 24,36 | Friday, September 12 | |
| 1.5 | 1,3,5,7,9,15,25,27,29,31,37,39,43,51,53 | 10,40 | Tuesday, September 16 | |
| 2.1 | 1,7,19 | None | - | For Problem 19, focus on (a), (b), and (c). |
| 2.2 | 1,5,9,11,17,29,33,37 | None | - | |
| 2.2 | 25,27,29,35,45,55,57,61,65 | 44,64 | Monday, September 22 | |
| 2.2 | 69,81,82 | None | - | |
| 2.4 | 1,3,5,7,9 | None | - | |
| 2.4 | 17,23,25,29,31,37,39,41,47,51,53,55 | None | - | |
| 2.6 | 5,7,9,11,13,19,23,27,35,39,41 | 16,36 | Tuesday, September 30 | |
| 2.7 | 9,15,21,27,29,33 | 8 | Friday, October 3 | |
| 3.1 | 1,3,5,7,13,19,27-30,31,35,39,41 | 22 | Friday, October 3 | |
| 3.2 | 1-11 odd, 17-35 odd, 47,51,63 | 12,24 | Tuesday, October 7 | |
| 3.4 | 1,3,7,9,11,13,15,17,21,25,31,33,37 | 24,34 | Thursday, October 9 | |
| 3.3 | 3,9,13,15,21,25,27 | None | ||
| 3.5 | 1,3,5,7,9,13,17,19,23,25,29,31,33,35,37,39,41,47,61,65 | None | ||
| 3.6 | 1,3,9,11,17,27,41,43,53 | 14,44 | Monday, October 26 | |
| 3.7 | 9,11,13,15,21,25,27,29,57,59 | None | ||
| 3.8 | 21,23,33,41 | None | ||
| 3.7 | 67,71,75,89,93 | None | ||
| 3.9 | 3,13,15,17,23,25,31 | None | ||
| 3.10 | 5,6,11,15,17,19,25,39,47,49,61 | None | ||
| 4.1 | 1-6,7,9,11-14,23,31,33,41,47,51,53,69 | None | ||
| 4.3 | 3,7,11,17,21,23,25 | None | ||
| 4.4 | 1,3,5,17,25,41,43,65,69,77 | None | ||
| 4.5 | 7,9,11,15,19,25,29,32 | None | ||
| 4.2 | 1,5,7,45 | None | ||
| 4.2 | 12,15,19 | None | ||
| 4.6 | 1,15,19,21,23,25,33,43,45,63 | None | ||
| 4.6 | 38,40,46,47,49,51,53 | None | ||
| 2.3 | 37,43,45 | None | ||
| 4.8 | 1-19 odd, 23 | None | ||
| 4.8 | 29,31,35,43,51,61,81,121,123 | None | ||
| 5.1 | 1,3,5,7,9,13,19 | None | ||
| 5.2 | 1,7,13,19,39 | None | ||
| 5.3 | 9,11,13 | None | ||
| 5.4 | 1,3,5,7,15,17,19,21,27,29,33,57,69 | None | We'll look at these problems in class on Monday. Last homework assignment! |
This morning, I'll be around my office until about noon. This afternoon, I'll be here from about 1:30 until at least 3:00.
This morning, I will be in or around my office until 11:45. This afternoon, I'll be here from 2:00 until at least 4:00.
I've put together a master list of objectives for the course. The objectives that are most relevant to Part A of the final exam are highlighted.
Some of you have told me that you will be having a study session on Sunday at 2 pm on our usual classroom. Everyone from the class is welcome to join in.
I've finished grading Exam #5. You can pick it up whenever you can find me in my office. Feel free to e-mail or call if you want to check that I'm here before heading over.
Details for logistics of Exam #5, grading, and the final exam are on this handout.
Exam #5 will be on Tuesday December 9 from 9:30 to 10:50 am. It will cover material from Sections 2.3, 4.8, and 5.1-5.4. (You can use the "equal-size subintervals" definition of definite integral from class rather than the more general definition from the text. The First Fundamental Theorem of Calculus will not be included.) For this exam, a well-prepared student should be able to
For this exam, I will provide the sum formulas given on p. 327 of the text.
I'll have an evening office hour from 7:30 to 8:30 tonight in our usual classroom.
In class, we looked at the Second Fundamental Theorem of Calculus and how to use this to efficiently evaluate definite integrals. The Second Fundamental Theorem of Calculus also tells us that integration undoes differentiation in a certain sense. The First Fundamental Theorem of Calculus (which we'll take a quick look at on Monday if we have time) tells us that differentiation undoes integration. Together, the two theorems tell us that differentiation and integration are inverse operations.
Exam #5 will be on Tuesday December 9 from 9:30 to 10:50 am. It will cover material from Sections 2.3, 4.8, and 5.1-5.4. (You can use the "equal-size subintervals" definition of definite integral from class rather than the more general definition from the text.) For this exam, a well-prepared student should be able to
For this exam, I will provide the sum formulas given on p. 327 of the text.
In class, we'll work with a simplified definition of definite integral that uses equal-size subintervals. Details are on this handout. The handout also includes a connection to the more general definition given in Section 5.3 of the text in case you are interested.
Project #3 is due tomorrow.
Exam #5 will be on Tuesday December 9.
We are building up to defining the definite integral for a function over an interval. Our story here will parallel the way we developed the idea of derivative. In both cases, we start with a definition. Using the definition to compute a specific result is generally hard work. So, we develop results and rules that allow us to compute much more efficiently. So, why start with a definition if it's hard to use? Here's two reasons:
I've assigned a few problems from Section 5.2. Most of these deal with using the "sigma" notation for sums. The last deals with computing an exact area as we did in class today.
In class, we used this worksheet to introduce a new idea: Given information about rate of change over an interval, deduce something about the accumulated change. If we think in terms of graphs, this is equivalent to deducing something about the area between the graph of a function and the horizontal axis for a given interval on the axis.
In class, I distributed Project #3. This will be due on Friday December 5.
I've assigned additional problems from Section 4.8
Have a great break. Get lots of sleep.
As you now know, mastering the precise definition of limit is not easy. The main point I want you to take away is that there is a deeper layer of precision underpinning everything we've done in calculus. Precision is an essential feature of mathematics. You can often use mathematics without mastering the precise level but you should always be aware that a precise level exists in case you encounter a problem that requires more precision.
For the specific example of limits, here's how the precise level works:
After you have used the precise definition to establish a few basic rules and results, you no longer need to go back to the precise definition.
We've seen a similar pattern with derivatives. With derivatives, we
Mathematicians care about precision not to be perverse or sadistic, but rather in order to prove precise statements about what is true. Users of mathematics can sometimes get away with not understanding the precise mathematics. But, it can't hurt (well, other than the pain of learning) to know the precise details or at least that the precise details are available. The more you know about the precise mathematical details, the more likely you are to be able to correctly handle new or unusual situations that might arise in applying mathematics. An analogy is often made to driving a car. You can drive a car well without knowing much about the mechanics of a car. But, the more you know about how a car works, the more likely you will be able to correctly handle any new or unusual situation you might encounter.
In any case, we've moved on to a new idea, namely antiderivative. I've assigned problems from Section 4.8. (We will be omitting the material in Section 4.7.)
When we first talked about limits early in the semester, we did not stop to formulate a precise definition. We did so today. Here's the handouts I distributed in class:
Getting comfortable with the precise definiton of limit takes some work and time. We will probably leave this topic before you become truly comfortable with the definition. I've assigned a few problems from Section 2.3 to help you gain some understanding and comfort.
My office hour for today is moved to 1:00-2:00 (from the usual 1:30-2:30) because I have another commitment. I'll also be available from 3:00-4:00.
Exam #4 will be on Thursday from 9:30 to 10:50 am. It will cover material from Sections 4.2 through 4.6. For this exam, a well-prepared student should be able to
I'll have an evening office hour from 7:00 to 8:30ish on Wednesday in our regular classroom.
I've assigned a few more problems from Section 4.6.
Project #2 is due tomorrow.
Exam #4 will be on Thursday, November 20 from 9:30 to 10:50 am.
I've changed the due date for Project #2 to Tuesday but I encourage you to finish it earlier so you have time to focus on preparing for Exam #4.
Exam #4 will be on Thursday, November 20 from 9:30 to 10:50 am
I've assigned a few more problems from Section 4.2
In class, we looked at the first problem from this handout. You should look at the other problems before class on Thursday. I've also assigned a few problems from Section 4.2. I'll assign more after Thursday's class when we've talked about a few uses of the Mean Value Theorem.
I haven't assigned additional problems. Keep working on anything from Section 4.5 you don't yet understand.
In class, I distributed Project #2. This will be due on Monday November 17.
Many applications involve determining what value of one variable should be used to get the biggest (or smallest) possible value of a a related variable. To turn this into a mathematical problem, you need to
Once you've done this, you have a mathematical problem: Find the global maximum (or global minimum) for the function on the relevant domain. You can use the tools of calculus to analyze this problem. In practice, the most difficult step is often building a formula to relate the variables.
Our next exam will come up quickly. It's tentatively scheduled for Thursday, November 20.
Exam #3 will be on Tuesday, November 4 from 9:30 to 10:50. It will cover material from Sections 3.6-3.10, 4.1, and 4.3 in the text. For this exam, a well-prepared student should be able to
For the exam, I will provide you with the derivatives of the inverse trigonometric functions. I could ask you to derive one of those results as an exam problem.
In class, we talked about how to find the global minimum and the global maximum for a continuous function defined on closed interval. I've assigned problems from Section 4.1 related to this idea. We also looked at how to classify a critical value as a local minimum, a local maximum, or neither. I've assigned problems from Section 4.3 related to this idea. In class, we looked at two methods for doing this classification:
We've skipped over the ideas in Section 4.2 for now. We'll come back to these after the exam.
I will have an office hour on Sunday from 4:00 until whenever everyone is happy or I am tired (whichever comes first). I'll be in our regular classroom. On Monday, I'll be free to meet from 2:00 until 5:00. I have a regularly scheduled office hour from 2:30 to 3:30 when you can drop by and I'll definitely be there. Otherwise, I'll probably be there. If you want to lock in a specific time, send me a note to make an appointment.
Exam #3 will be on Tuesday, November 4 from 9:30 to 10:50. It will cover material from Sections 3.6-3.10, 4.1, and 4.3 in the text. For this exam, a well-prepared student should be able to
Most of what we did in class today is learning new language. You should work to make sure you understand precisely what is meant by phrases such as "global maximum" or "local minimum". Note that extreme means "minimum or maximum" so the plural extrema means "minima or maxima". I've assigned a few problems from Section 4.1.
Exam #3 will be next Tuesday. It will cover material we have done since the previous exam, including ideas we discuss in class tomorrow.
I've changed the due date for Project #1 to Thursday, November 6
Here's a map of Mt. Rainier mudflow regions. And here's a map of Pierce County volcano evacuation routes. Merriam-Webster says lahar is a Javanese word (note this is not Japanese).
Linearization is one example of trading in a function for an approximation that is easier to work with. If you carry on in mathematics, you'll see this idea used in many places. A classic example of using linearization is the small-angle approximation often used in physics, namely sinθ ≈ θ for θ near 0 (that is, for small angles). If you know any physics majors, ask them about the physics club t-shirt from last year.
Your first project is due next Monday, November 3.
We will have Exam #2 next Tuesday, November 4 (Election Day!).
As you have discovered, most related rates problems can involve lots of good thinking. Here's a few things to keep in mind:
Keep working on the assigned problems from Section 3.9. We'll go over at least one more of these quickly at the beginning of class tomorrow.
As I mentioned last week, I will de-emphasize how much homework you submit. In place, we will do some projects. Your first project will center on Problem 24 in Sectin 3.9. For projects, I will expect carefully written reports on your work. In class tomorrow, I'll distribute a handout with specific requirements and expectations.
We'll discuss the exam schedule in class tomorrow. One option is to have Exam #3 on Tuesday, November 4 and Exam #4 on either Thursday, November 20 or Tuesday, November 25. You can vote by email in advance of class if you have preferences to express.
Some of you missed the fact that a few problems from Section 3.6 were due today. I've changed the due date for these to Monday. I've also assigned problems from Section 3.9 to look at.
I've assigned a few more problems from Section 3.7
Remember the Chemistry Magic Show today at 4:00 and 7:30 pm in Schneebeck Concert Hall.
In class today, we looked at the main ideas from Sections 3.7 and 3.8 so I've assigned problems from these sections. There are a few additional ideas in these sections that we'll look at after break. I've also assigned two problems from Section 3.6 to be submitted.
We will be skipping over subsections on parametric equations. This includes the subsection on pages 170-171 in Section 3.5 and the small subsection on page 186 in Section 3.7.
Have a great break!
Getting comfortable with implicit differentiation takes some thought and practice. To begin with, you can substitute f(x) for y (assuming we are computing dy/dx). This should help make clear when it is that you need to use the chain rule. If you do a problem by substituting f(x) for y, try redoing the problem without the substitution. You'll need to keep in mind that, behind the scenes, we are thinking of y as a function of x without denoting it explicitly.
Exam #2 will be on Tuesday from 9:30 to 10:50 am. It will cover material from Sections 2.6, 2.7, 3.1, 3.2, 3.3, 3.4, and 3.5. There's a list of specific objectives below.
I'll be available this afternoon from 2:00 until about 5:00. I'll return to our usual classroom (TH 283) for an evening office hour from 7:00 until after 8:00. Stop by this afternoon or evening if you have questions.
Exam #2 will be on Tuesday from 9:30 to 10:50 am. It will cover material from Sections 2.6, 2.7, 3.1, 3.2, 3.3, 3.4, and 3.5. For Exam #2, a well-prepared student should be able to
My goal for today's class was to make the chain rule (for differentiating a composition) seem obvious by starting with an applied situation. Looking at a situation with units motivates the correct expression. We did not have time to pull everything together in our discussion today. This wasn't helped by the fact that what I got up on the board wasn't quite as organized as I had hoped. I've put together this handout to help clarify what went on today. (Note that I changed the specific values a bit from what we used in class since 0.04 was showing up in two different by coincidence and I wanted to eliminate that coincidence). Please read through this handout before class tomorrow. If you notice any typos, please send me a note so I can correct them.
I've assigned problems from Section 3.5 but don't expect you to look at these until after tomorrow's class. We'll address questions from these in class on Monday. This material will be on Tuesday's exam so the turn-around time for mastering the chain rule will be fairly short.
I've assigned problems from Section 3.3. The first of these deal with position, velocity, and acceleration for an object moving on a line (as opposed to moving in a plane or in space). The ideas here are straightfoward:
Exam #2 will be Tuesday, October 14 from 9:30-10:50 am.
I've assigned additional problems from Section 3.4, including two problems to submit.
Exam #2 will be Tuesday, October 14 from 9:30-10:50 am.
We've now established derivatives for some specific functions (constant, power, the exponential, sine, cosine) and rules for differentiation combinations (sum, difference, product, and quotient) of functions. You should do enough practice to become very proficient on calculating with the results and rules. I've assigned problems from Sections 3.2 and 3.4. I'll assign more problems from Section 3.4 next week.
A few of the problems I've assigned ask you to compute a second derivative. The idea is easy. Start with a function f. Compute the derivative to get a new function f′. Now compute the derivative of f′ to get yet another function which we denote f″ (that is, f with a double prime). We call f&Prime the second derivative of f. For example, if we start with f(x)=x3, we differentiate to get the first derivative f′(x)=3x2 and then differentiate again to get the second derivative f″(x)=3(2x)=6x.
Next week, we'll add one more rule to our collection. This rule will allow us to differentiate a composition of two functions. The rule will allows us to differentiate a function such as f(x)=cos(x2).
After looking at your preferences (which broke evenly between the two options) and where we are in the course material, I think it would be best to have Exam #2 on Tuesday, October 14. So, we'll have Exam #2 on Tuesday, October 14.
In class, we developed results for some specific functions, namely power functions and the exponential function. We did not develop results that will allow us to compute derivatives for combinations (sum, difference, product, quotient, composition) of functions. We'll do this in class tomorrow. I'll then assign problems from Section 3.2.
In the meantime, you should do the following:
In class tomorrow, we'll talk about the timing of Exam #2. Two options are Thursday, October 9 and Tuesday, October 14. Think about whether or not you have a strong preference.
I've assigned additional problems from Section 3.1. I've also designated a problem from Section 2.7 and a problem from Section 3.1 to be submitted on Friday.
So far, we've been computing derivatives by starting with the definition (as the limit of a difference quotient). We'll next turn our attention to developing rules that will allow us to compute derivatives much more efficiently. We'll have two kinds of rule:
In class today, we hit the main topic of this course head-on: derivative. We've previously hinted at the ideas that motivate defining derivative. Most generally, a derivative gives us a way of quantifying rate of change in one variable with respect to change in another variable. Derivatives give us a way of quantifying the rate of change for each specific value of the second variable (as opposed to an average rate of change for an interval of values). You can think of slope as a special case of this. Slope is the rate at which y changes with respect to changes in x for points along the curve given by y=f(x).
Give a function f(x) and a specific input, say a, we can think about the tangent line (presuming one exists) at (a, f(a)) in two ways:
I've assigned problems from Sections 2.7 and 3.1. I'll assign more problems from Section 3.1 after class tomorrow.
We well skip over material in Section 2.5 for now. In class today, we talked about the main idea of Section 2.6, namely continuity. I've assigned some problems from Section 2.6.
Exam #1 will be on Thursday from 9:30 to 10:50 am. It will cover material from Sections 1.1 to 1.5, 2.1, 2.2, and 2.4. For Exam #1, a well-prepared student should be able to
* As an example of "build a function that models a given real-world situation", think about the cost function you came up with for Problem 62 in Section 1.1.
Here's my standard advice on how to prepare for an exam:
I will be available this afternoon (Tuesday) for office hour from 1:30 to 2:30 and for appointments from 2:30 to 4:30. On Wednesday, I'm available for appointments most of the morning and after 2:00 in the afternoon. E-mail or call if you want to set up a time to meet.
The last topic before the exam is "limits at infinity". These are a straightforward extension of limits at "regular" points. For a limit as x→∞, think of a list of inputs x that get large without bound (so something like 10, 100, 1000, ...). Think of these inputs as getting close to &infin. For each of these, compute the output f(x) to get a list of outputs. If there is an obvious number to put in the output column opposite &infin, that number is the limit of f(x) as x→∞.
Exam #1 will be Thursday from 9:30 to 10:50 am. It will cover material from Sections 1.1 to 1.5, 2.1, 2.2, and 2.4. We'll talk in class tomorrow about what you can expect.
In class, we talked about the Sandwich Theorem. This appears in Section 2.2. For some reason, an important example of the Sandwich Theorem is in Section 2.4 in the subsection "Limits Involving sin(θ)/θ" on pages 88-89. So, you should include that subsection in your reading about the Sandwich Theorem. I've assigned a few problems from Section 2.2 that deal with the Sandwich Theorem.
We are going to skip over the material in Section 2.3 for now. We'll come back to this later. The idea in Section 2.3 is to give a precise definition of the limit of f(x) at x=0 is L. The precise definition takes some time to understand so we'll continue to work with a somewhat vague definition for now. As you read Sections 2.4 and beyond, you will occasionally see references to this precise definition of limit. You can ignore these for now.
Section 2.4 has two main ideas. The first of these is one-sided limits. This is staightforward so I have assigned some problems from Section 2.4 to look at over the weekend. You'll need to read the first few pages of Section 2.4 in order to understand the ideas and notation before attempting the assigned problems.
In class, we are now developing rules and techniques that allow us to compute a limit rather than merely conjecture a limit based on looking at a table of inputs and outputs. I've assigned additional problems from Section 2.2.
There are some limit statements that cannot be computed algebraically. For example, there is nothing we can do to re-express sin(x)/x in a way that lets us cancel common factors that are creating issues with division by zero. To have a convincing argument that the limit of sin(x)/x at x=0 is 1, as we have conjectured, we'll need a new tool. That tool is the Sandwich Theorem and is what we'll talk about in class tomorrow.
In class, our first look at limits has been by using tables of inputs and outputs. This view gives us a good idea of what limits mean but is not very efficient. Over the next few days, we'll develop techniques for computing limits.
I've assigned a few problems from Section 2.2. In class, we've talked about some of the ideas in Section 2.2 but not all of them. I'll assign more problems from Section 2.2 after class on Thursday.
We'll skip over Section 1.6 for now and pick up these ideas later.
In class today, we talked about how to define rate of change for a specific value of time as opposed to an average rate of change for an interval of times. The essential idea is that defining rate of change requires a limit. So, we'll next talk in more detail about limits (the main topic of Chapter 2). After we better understand limits, we'll return to rate of change (Chapter 3 and beyond).
I've assigned a few problems from Section 2.1
Exam #1 will be Thursday September 25 from 9:30-10:50. Let me know as soon as possible if your schedule would prevent you from working on the exam until 10:50. Note that class normally ends at 10:20 on Thursdays.
I've added a few problems to the assignment for Section 1.5, including two problems to submit next Tuesday.
Exam #1 will be on either Tuesday September 23 or Thursday September 25. I'll make a final decision on this later today or tomorrow and then post details here.
If you want more fundamentals on trigonometry, you might try working through the exercises on this handout.
I've assigned some problems from Section 1.5. We'll start class tomorrow with addressing questions you bring on these problems. I'll assign some additional problems from Section 1.5 (including ones to submit) after class tomorrow.
Exam #1 will be on Tuesday September 23. (Note that this differs from the tentative schedule included on the course syllabus.) I would like to see if we can use the 80-minute period from 9:30-10:50 for the exam. I've just sent out an email asking each of you to reply letting me know whether or not your will be available from 9:30-10:20 am on Tuesday September 23 for this exam. Note that class normally ends at 10:20 on Tuesdays.
Feel free to come ask whenever you have questions on class, reading, or homework. Another source of help is tutors at the Center for Writing, Learning, and Teaching. Alison Paradise is the math faculty member who has hours at the CWLT this year. In addition, peer tutors are available at the CWLT for math and many other subjects. Details on the availability of peer tutors are on this page.
In addition the CWLT tutors, the math department has its own tutors available to help on a drop-in basis. These tutors are stationed in Thompson 390 (the math lounge) at the following times:
I've assigned problems from Section 1.4 on exponential functions. Some of this involve applications in which some quantity changes with exponential growth or decay. Come talk with me if you are not familiar with this type of application
If you are a science major, considering a science major, or are interested in science, you should try to attend talks in the Thompson Hall Science and Mathematics Seminar, held most Thursdays at 4 pm in Thompson 175.
We'll look at a few more problems from Section 1.3 in class tomorrow and then turn our attention to exponential functions. Note that I've changed the due date from the problems to be submitted from Section 1.3.
I will post an assignment from Section 1.4 later this afternoon for those who want to get ahead a bit.
When we talked in class about composing functions, I pointed out a distinction to make in notation. A similiar distinction holds for other ways of combining functions. For example, you should try to distinguish between the meanings of (f+g)(x) and f(x)+g(x). The distinction is subtle but worth some effort to understand.
We'll continue reviewing prerequisite ideas in class primarily by addressing questions you bring on reading and homework problems. As you read the text and work on problems, make note of any unfamiliar language and terminology. Ask questions about these by e-mail, during class, or in person outside of class.
As I mentioned in class, the material in Chapter 1 is mostly review of topics from high school mathematics. Everyone has a different background so there may be some topics with which you are not familiar or comfortable. We'll spend most of our class sessions this week addressing the questions you have on assigned homework. If you are feeling unsure about your mathematical background, come talk with me in the next or two.
In class on Thursday, we'll look at questions you have on assigned problems from Section 1.1. If we have time, we'll talk a bit more about real numbers.
Check out the Astronomy Picture of the Day.
You can look at exams from last time I taught Math 180. You can use these to get some idea of how I write exams. Don't assume I am going to write our exams by just making small changes to the old exams. You can use this old exam to prepare for our exam in whatever way you want. I will not provide solutions for the old exam problems.
