| Section | Problems to do | Submit | Due date |
|---|---|---|---|
| Real numbers | #1-6 from handout | #2,6 | Thursday, September 2 |
| 1.3 | #5,7,11,15,19,35,37,43,47,53,67 | #20,42 | Friday, September 3 |
| 1.4 | #13,17,19,21,25,41,43 | #42,48 | Wednesday, September 8 |
| 2.1 | #1-9 odd, 23,27,35,39,41 | #36,40 | Friday, September 10 |
| 2.2 | #1,5,9,11,13,17-29 odd,37,41,43,53,55 | #24,52 | Monday, September 13 |
| 2.3 | #1-4,7,11,17,19,21,39,41 | None | |
| 2.4 | #1-4,13,21,27,37,39,61,69 | #38,40 | Thursday, September 23 |
| 3.1 | #5,7,9,15,23,27,31,45,51,65 | #26,50 | Wednesday, September 29 |
| 3.2 | #1-27 odd, 31,33 | #24,42 | Thursday, September 30 |
| 3.3 | #1-41 odd, 47,51 | #26,56 | Friday, October 1 |
| 3.5 | #9-47 odd | #36,38 | Monday, October 4 |
| 3.4 | #5,9,17,19,23,40,53,55,66 | None | |
| Limits 1 | #1-4 from first handout | #2,4 | Friday, October 15 |
| Limits 2 | #1 from second handout | #1 | Friday, October 15 |
| 3.6 | #3-13 odd,19,25,27,35,41 | #14,42 | Thursday, October 21 |
| 3.7 | #7,13,19,25,27,31,42,43 | #14,44 | Monday, October 25 |
| 3.8 | #1,5,19,23,25,33,37,39,55 | #38,40 | Thursday, October 28 |
| 4.1 | #1-13 odd | None | |
| 4.1 | #19,23,25,29,37,43,53,57,59 | #26,52 | Friday, November 5 |
| 4.6 | #5,7,9,15,17,18,21,26 | #12,24 | Wednesday, November 10 |
| 4.3 | #13,15,17,21,29,32,37,41,43,45,47,55,58,61 | #30,48,60 | Thursday, November 11 |
| 4.4 | #5-23 odd, 25,29,37,39,47,53 | #44,50 | Monday, November 15 |
| 4.5 | #7-23 odd, 27,31,33,35,41,51,53 | #24,36 | Wednesday, November 17 |
| 4.2 | #9,19,31,33,41,43,45,51 | None | |
| 5.2 | #1-11 odd, 15,17,19,23,25 | None | |
| 5.3 | #3,7,17,19,27,38,39 | None | |
| 5.1 | #1-29 odd, 45,47 | None | |
| 5.4 | #1-21 odd, 25,53 | None |
You can look at exams from last time I taught Math 121. 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 should also note that we were using a different textbook so some of the notation is different. There are also differences in the material covered on each exam. 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.
Nothing fun yet. If you're bored, check out the spong monkeys here.
There are two supplements to the text that you might find useful. One is the Student Mathematics Handbook and Integral Table for Calculus. This reviews topics from geometry, algebra, and trigonometry. It can serve as a useful reference. The Student Mathematics Handbook is included on a CD that comes with the text when it is purchased new. You might want to check if you already have this. The other is the Student Survival and Solutions Manual. This has solutions for the odd-numbered problems in the text.
Here is an important fact about real numbers: Between any two real numbers, there is at least one rational number and at least one irrational number. You can use this fact for Problem 6 on the real numbers handout.
The material in Sections 1.1 and 1.2 is review of selected topics from high school mathematics. We will not cover these sections at the beginning of the course. Instead, we will look at each topic as we need it. For now, you should look through these two sections to get some idea of what topics are covered.
When you have a question on homework or reading, you should fill out a question form and bring it to class. This will help me determine what homework problems and questions to look at in class. It might also help you since formulating a specific question oftens produces an answer that question. For this, try to make your questions detailed and specific.
In class, we started on the first topic of calculus, the limit of function. So far, we've just looked at examples. You should read the beginning of Section 2.1 and work on the problems I've assigned so far from that section. You should also work on the first project.
If you feel a need to review some trigonometry, look for a handout I will post later this weekend.
I have posted a first draft of a trigonometry review. If you are feeling rusty on trigonometry, you might try working through this to remind yourself of some important details. I haven't tested this out before so I don't know how effective it is. If you try it, I welcome your feedback with suggestions for improvement. Check back later for updated versions.
In our initial look at limits, we relied on numerical and graphical evidence to conjecture a limit. For the problems from Section 2.1, numerical and graphical evidence are the only tools you had. We are now moving into algebraic ways of analyzing limits. For the problems from Section 2.2, you should use algebraic tools as much as possible. If you get stuck, you can always fall back on numerical or graphical evidence.
In Section 2.1, you can skip the last subsection "Formal Definition of a Limit" for now. We will come back to this later.
We will have our first exam next Friday, September 17. This will cover Sections 1.3, 1.4, 2.1, 2.2, and 2.3. In order to give you a little more time, I would like to start the exam at 7:30 am. Please check your schedule to see if you a commitment that would prevent you from starting at 7:30 am. We will make a decision on this in class Monday.
For the first exam, a well prepared student should be able to
Here are some general comments in regards to grading Project #1:
Section 2.4 concerns exponential and logarithmic functions. Much of this is review of high school mathematics. The new idea comes in being careful about the definition of exponentiation with an irrational exponent. The definition used in our text is based on requiring continuity. The problems from this section focus on what should be familiar properties of exponential and logarithmic functions. For Problems 13 and 21, you should use these properties to give an exact result rather than using your calculator to produce a decimal approximation.
We are going to be pursuing two tracks simultaneously. The first is to develop a precise definition of limit. Initially, this track will consist of assignments developing the pieces we need to put together a precise definition. (The first one concerns intervals and inequalities.) The second track is to begin our study of derivative. This is the main idea of the semester. The definition of derivative uses limit. Our time in class will mostly be devoted to this second track, although we will take time to address questions on the limit definition assignments. Working on two tracks simultaneously might cause some confusion about what to be doing when. If so, just ask me for clarification.
We have defined derivative as the limit of a difference quotient, provided the limit exists. For the problems in Section 3.1, use this definition and the interpretation of derivative as slope of a tangent line. Next week, we will develop technniques that allow us to calculate derivatives more efficiently. These techniques will allow us to avoid explicitly computing the limit of a difference quotient.
The second assignment on the path to a precise definition of limit concerns meeting a specified tolerance.
The goal for this week is to learn results and rules that allow us to compute derivatives without going back to the definition (as limit of a difference quotient). For the problems from Section 3.2, you should use the results and rules we have so far: the power rule, the constant factor rule, the sum rule, the product rule, and the quotient rule. It is important to keep in mind that all of these come from the definition of derivative as limit of a difference quotient.
In class, we did not talk about higher derivatives. The idea here is very simple: Start with a function f(x). Compute its derivative f '(x). This derivative is itself a function so we can compute the derivative of the derivative. This is called the second derivative of the original function f(x). It is denoted f ''(x). For example, with f(x)=x3, we get f '(x)=3x2 and f ''(x)=6x. See pages 116-117 of the text for more details.
Exam #2 will be next week. Check you schedule to see if you can do the exam on Thursday from 8:00 to 9:20.
We now have all of the tools in place to compute derivatives without going back to the definition of derivative as the limit of a difference quotient. These tools incude the derivatives of specific basic functions (power, trig, exp, log) and general rules for combinations of functions (sum, product, quotient, composition). You should be working at computing derivatives quickly and correctly. This takes lots of practise so it is very important that you do all of the assigned homework. If you feel this is not enough, find more problems to do. Try the even problems from the text or ask me for more.
Today we talked about a second interpretation of derivative. This interpretation is as rate of change in one variable with respect to changes in another variable. One example is the rate of change in position with respect to change in time. This is given by the derivative of the position with respect to time and defined as the velocity.
We will have Exam #2 on Friday from 7:30 to 8:50. This will cover Sections 2.4 and 3.1-3.5. I will post a list of objectives later this week.
We will have Exam #2 on Friday from 7:30 to 8:50. This will cover Sections 2.4 and 3.1-3.5. For this exam, a well prepared student should be able to
We are now looking at a precise definition of limit. The wording in the definition we will use is slightly than that given in Section 2.1 of the text. You can read the wording we will use and some examples on this handout. The handout also has four problems that are assigned as homework.
In class, I distributed a handout giving a proof of the sum property for limits. After class, I rewrote part of this because the argument we went through in class is a bit nicer than what I wrote up in the original version. This handout also has a problem that is due on Friday along with the problems from the previous handout.
In working on the problems for Section 3.7, you should start by drawing a relevant diagram and labeling relevant quantities with symbols you choose. Don't put in given values yet. You should then determine which quantities are constant and which are changing (typically changing in time). Then, you should use your imagination to make your diagram into a movie. In your head, you should see the varying quantities changing and you should look at the relations between them. Ask yourself questions such as "If quantity A increases, does quantity B increase, stay the same, or decrease?" When you finish the problem, go back and check if you answers are consistent with the thinking you did at this stage. If not, something is wrong and you should reconcile the difference. Check if your initial thinking was wrong or if you made some error in setting up relations or calculating.
For these problems, you might need various geometry formulas (for areas and volumes of specific shapes). If there is one you don't remember, you can look it up in the Student Mathematics Handbook. Also note that for Problem 31, there is a picture at the top of the second column on page 163. (The wording of the problem is given at the bottom of the first column so it is easy to miss the picture.)
We spent the entire class period looking at problems from Section 3.7 and did not get to the ideas from Section 3.8. We will do this on Monday so I've pushed back the due date for the Section 3.8 problems. This weekend, you should focus on the problems from Section 3.7.
Exam #3 is on Friday from 7:30 to 8:50. It will cover the precise definition of limit, Sections 3.6 to 3.8, and the parts of Section 4.1 on local extrema (but not the parts on global extrema).
We have moved Exam #3 to Monday from 7:30 to 8:50.. It will cover the precise definition of limit, Sections 3.6 to 3.8, and the parts of Section 4.1 on local extrema (but not the parts on global extrema). In Section 3.6, you can skip the subsection entitled "Logarithmic differentiation" and in Section 3.8, you can skip the subsections entitled "Marginal Analysis in Economics" and "The Newton-Raphson Method for Approximating Roots."
For this exam, a well-prepared student should be able to
In working on the optimization problems of Section 4.6, you might find it useful to read the box on page 238. This describes a series of steps that are typical for these problems. (Not all problems fall into this pattern, but many do.) I also find it useful to "make a movie" in my head after I have drawn the relevant figures. To do this, I have to identify which quantities are constant and which are variable. Then I picture the variable quantites changing and watch to see what happens to the objective quantity (i.e., the quantity we want to minimize or maximize). Sometimes, you can see the solution to the problem, particularly if there is a lot of symmetry. You can also see any restrictions on the variables in the problem. (This is the "practical domain" in Step 4 of the box on page 238.)
Exam #4 will be next Friday, November 19 from 7:30 to 8:50. It will cover Sections 4.1-4.6. I'll post a list of objectives for this exam some time next week.
You should be working on problems from Sections 4.4 and 4.5. In addition, you should be working on Project #3.
Section 4.2 is about the Mean Value Theorem. The first topic in this section of the text is Rolle's Theorem. Rolle's Theorem is a special case of the Mean Value Theorem because it includes the additional hypothesis that f(a)=f(b). The text includes this special case as a stepping stone in proving the Mean Value Theorem. We are not going to focus on the proofs in this section so you can go lightly over Rolle's Theorem. You should read and understand the Mean Value Theorem itself. You will also need to read and understand the Zero-Derivative Theorem and the Constant Difference Theorem. These results will help with the problems I have assigned and will play a role in Chapter 5.
Exam #4 will cover Sections 4.1-4.6. In Section 4.2, you can skip the subsection entitled "Rolle's theorem." The relevant examples in Section 4.6 are Examples 1, 2, 5, and 6. For this exam, a well-prepared student should be able to
Today, we began our final big topic of the course, namely integrals and integration. These pair with derivatives and differentiation as the big, applicable ideas in calculus. (Limits are the foundation for each of these big ideas.) The goal in class today was to understand the problem of computing area. This problem motivates the definition of integral in the same way that the problem of computing slope motivates the definition of derivative.
In class, we discussed a general process for computing area in the case of a region that is bounded above by the graph of a function f, below by the x-axis, on the left by the vertical line x=a, and on the right by the vertical line x=b. It takes a lot of detail and notation to describe this general process. The summation notation is useful here because it is more compact. If you have not worked much with summation notation in the past, you will need to carefully read the subsection "Summation Notation" on page 286 of the text. Problems 1-11 give you practice with the summation notation and some useful summation formulas given in the box on page 287 of the text.
Your last project is to write an essay on some aspect of calculus we have covered in this course. You should choose a feature of calculus that you find to be interesting or useful. Here's some ideas:
Exam 5 is on Wednesday, December 8 from 7:30 to 8:50. It will cover material from Sections 5.1 through 5.4. In Section 5.4, you can skip the subsection entitled "The Second Fundamental Theorem of Calculus."
I will give you the usual half-sheets of trigonometric identities and derivatives. I will also give you the summation formulas shown in the box on page 287 of the text. For this exam, a well-prepared student should be able to