Here are the notes on logistic details such as the structure of the final exam and picking up Exam 5. In preparing for the final, you might want to look at the list of objectives for the course. The objectives most relevant to Part A of the final exam are highlighted in green.
My schedule for the next few days is
| Thursday, May 6 | 8:00-10:00 am | Available for appointments |
| 11:00-noon | Office hour | |
| 1:00-2:00 pm | Available for appointments | |
| Friday, May 7 | 8:00-noon | Available for appointments |
| 3:00-4:00 pm | Office hour | |
| Monday, May 10 | 10:30-11:30 am | Available for appointments |
| 2:30-3:30 pm | Office hour | |
| 3:30-4:30 pm | Available for appointments | |
| Tuesday, May 11 | 8:00-11:00 am | Available for appointments |
| Section | Problems to do | Submit | Due date |
|---|---|---|---|
| 1.1 | #1-45 odd, 51-59 odd | None | |
| Real numbers | #1-6 from handout | None | |
| 1.3 | #5,7,11,15,19,35,37,43,47,53 | #20,42 | Monday, January 26 |
| 1.4 | #13,17,19,21,25,41,43 | #42,48 | Wednesday, January 28 |
| 2.1 | #1-9 odd,23,27,35,39 | #36,40 | Monday, February 2 |
| 2.2 | #1,5,9,11,13,17,19,21,23 | None | |
| 2.2 | #25,27,29,37,41,43,53,55 | #24,52 | Wednesday, February 4 |
| 2.3 | #1-4,7,11,17,19,21,39,41 | #12,40 | Friday, February 6 |
| 2.4 | #1-4,13,19,21,27,37,61,63,69 | None | |
| 1.2 | #3,5,9,11,13,17,19,21 | #12,32 | Monday, February 16 |
| 3.1 | #5,7,9,15,23,27,31,45,51,63,65 | #26,50 | Tuesday, February 17 |
| 3.2 | #1-27 odd, 31,33 | #24,42 | Monday, February 23 |
| 3.3 | #1-41 odd, 47,51 | #26,56 | Wednesday, February 25 |
| 3.5 | #9-47 odd | #38,44 | Friday, February 27 |
| 3.6 | #3-13 odd,19,25,27,35,41 | #14,42 | Monday, March 8 |
| 3.7 | #7,13,19,27,31,37,42 | #14,44 | Wednesday, March 10 |
| 3.8 | #1,5,19,23,25,33,37,39,55 | #38,40 | Friday, March 12 |
| 3.8 | #49-51 | None | |
| 4.1 | #1-13 odd, 19,23,25,29,37,43,53,57,59 | 26,52 | Friday, March 26 |
| 4.6 | #5,7,9,15,17,18,21,22,26 | None | |
| 4.3 | #13,15,17,21,29,37,41,43,45,47,51,60,61 | #32,48 | Tuesday, April 5 |
| 4.4 | #5-23 odd, 25,29,37,39,47,53 | 44,52 | Friday, April 9 |
| 4.5 | #7-21 odd, 23,27,31,33,35,41,51,53,55 | #24,36 | Tuesday, April 13 |
| 4.2 | #9,19,31,33,41,43,45,51 | None | |
| 5.2 | #1-11 odd, 15,17,19,23,25 | #18,24 | Friday, April 23 |
| 5.3 | #3,7,17,19,23,27,38,39 | None | |
| 5.1 | #1-29 odd, 45,47 | None | |
| 5.4 | #1-21 odd, 25,53 | None |
You will need a PDF viewer to read the files posted here. Visit the Adobe website to obtain a free reader (all major platforms are supported).
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.
Here's a movie for Section 3.7 Problem 37.
Section 1.1 of the text is review of specific ideas from algebra and trigonometry that we will use in this course. I will not cover this material in class except in addressing questions from the homework I have assigned from this section. You will need to judge for yourself how well you understand this material and come see me outside of class if you need some help with it.
You should also look at the problems on real numbers handout. In class, I meant to mention the following 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.
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.
Tomorrow, we will discuss questions you have from the Section 1.4 homework. After that, we will start in on the first ideas of calculus. If you feel that you need to continue reviewing ideas from geometry, algebra, and trigonometry, come talk with me.
For Problem #48 from Section 1.3, it would be good to think about how you expect the angle θ to change as x changes. What is the value of θ for x=0? (Note that this corresponds to the person looking straight up the wall at the picture.) What happens to the values of θ as x increases? What is the value of θ as x goes to infinity? Sketch a graph of what you think θ vs. x should look like. After you complete the problem and get a formula for θ as a function of x, plot this on your graphing calculator and compare it to the first plot you made.
In Section 2.1 of the text, you can skip the subsection entitled "Formal Definition of a Limit" for now. We will come back and talk about the ideas in this subsection later.
For the Section 2.2 problems, you should focus on computing limits by using the algebraic methods given in the text and class examples. If you get stuck or are not sure about your results, you can fall back on making an input/output table or a graph on your calculator. In general, an algebraic computation is better because it eliminates guesswork.
In class today, we looked at the limit of (1+x)1/x as x approaches 0. From the numerical evidence, it appears that this limit exists and has a value of approximately 2.71. (You can get more digits in an approximation by using inputs closer to 0 than what we got to in class). This limit turns out to be exactly the special number usually labeled e. The exponential function with base e is the most natural exponential function to use in calculus. We will see the reason for this later. So, the function f(x)=ex is called the exponential function.
We will have an exam next Tuesday, February 10 covering Chapters 1 and 2 except Section 1.2. The exam will be from 8:00 to 9:20 am. You need to let me know as soon as possible if you are not available for the extended period of time on this day.
In preparing for the exam, you should make sure you have done and understood all of the assigned homework problems. Come talk with me or get help from a tutor if you have questions on any of these problems. You should also read the relevant sections of the textbook and review class notes. As you do so, make a list of questions and get answers for them. It is useful to write down your questions. Trying to phrase a question often helps you pinpoint exactly what it is that you are working to understand. If you have time, you can do some of the review problems at the end of each chapter. This can be useful because these problems are not in the context of a specific section and so are more like an exam situation.
In class today, you worked on some limit problems. These differ from other limits we have looked at only in having more than one variable in the problem. We will see limits like this when we study derivatives after the exam. It is possible that a limit of this type will be on the exam. You should try to finish these problems as homework and ask any questions you have on them in class on Monday.
Here is a copy of an old first exam from the last time I taught Math 121. You can use this to get some idea of how I write exams. Don't assume I am going to write our exam by just making small changes to the old exam. You should also note that the old exam did not cover continuity. 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.
Last week, we defined a tangent line as "the line we see if we zoom in on a curve at a point." We then looked at getting the slope of a tangent line in the case that the curve is the graph of a function. The "zooming in" process corresponds to a limit. We define the derivative of a function as the limit of a difference quotient. We interpret the derivative as giving the slope of the tangent line (at the point on the graph corresponding to the input we are using). This week, we will focus on rules and techniques that allow us to compute derivatives without doing a limit explicitly. These rules and techniques are based on the limit definition of derivative.
As we discussed in class, you can redo one problem from Exam #1 for additional credit. This is due in class on Friday with no exceptions.
By now, you have noticed that I use h in difference quotients while the text uses Δx. That is, I use x+h as the second input and the text uses x+Δx. You can choose either one (or any symbol you want). I like to use h because I find it easier to see the algebra. On the other hand, the notation Δx tells you what the symbol means if you use the convention that Δ means "change in".
In class, I gave the following tentative dates for exams: Exam #2: Tuesday, March 2; Exam #3: Tuesday, March 30; Exam #4: ?; Exam #5: Tuesday, May 4 (in the last week of classes).
If you want your scores included in the web version of the gradebook, send me an e-mail with a code name. If you want to see what this will look like, go to the web page for some of my recent courses.
We will have our second exam next Tuesday, March 2 from 8:00 to 9:20 am. The exam will cover Sections 1.2 and 3.1 to 3.5.
Project #2 is due on Friday, February 27. Be sure to note that you are asked only to analyze differentiability of the given functions at x=0. You will find it much easier if you substitute x=0 into the difference quotient before you analyze the limit. Do not try to analyze the limit of the difference quotient with x left as a variable.
For the second exam, a well prepared student should be able to
As you work on the problems on implicit differentiation in Section 3.6, try to use the more compact notation. You might want to try the first few problems by substituting f(x) for y, but should work toward being able to implicitly differentiate directly without this substitution. Here's the handout we worked on in class.
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.)
Problems 1-9 of Section 3.6 involving calculating the derivative dy/dx from an implicit equation. Here are some plots related to some of these problems.
Problem 37 of Section 3.7 is the one involving a truck and a car moving along perpendicular roads. Here's a movie that might help you visualize the situation.
In class today, we talked about the tangent line (or linear) approximation. We went through this twice in two different notations. Each notation style is useful depending on the context.
I also handed out Project 3 which will be due on the Friday after Spring Break.
In class today, we looked at two methods for estimating a root of a function. The first was the bisection method. This does not use calculus. It is an easy idea that will always work. On the other hand, it can take many iterations to get an estimate within a desired tolerance. The second method is called the Newton-Raphson method. This method uses calculus because it involves following a tangent line to its x-intercept. The Newton-Raphson method is more complicated than bisection and it doesn't always result in good estimate. On the other hand, it often takes relatively few iterations to produce a good estimate. The Newton-Raphson method is discussed at the end of Section 3.8 in the textbook. I have assigned a few problems from Section 3.8 that involve using the Newton-Raphson method.
In class today, we talked about Problem 51 from Section 11.8. Part of this problem asks you to use the Intermediate Value Theorem. We went over this in class today. This theorem is discussed on pages 76-77 of the text. I also had a handout showing two plots of the graph of the function given in this problem.
For the essay that constitutes Project 3, I want you to focus on ideas and concepts rather than techniques and applications. What are the big ideas and how are they related? You can mention techniques and applications, but these are not the main focus. For example, you might mention the chain rule (a technique for computing derivatives) by name, but probably should not explain how the chain rule works. Similarly, you might mention related rate problems but probably should not give a detailed example. In thinking about your essay, you might find it useful to look at the Table of Contents for the text. Each chapter also begins with a more detailed list of contents.
I will be out of town for the first part of next week. I will be back and will check e-mail Thursday and Friday. You can send an e-mail if you have any questions. Have a good Spring Break.
I've made a change to this course web page so that you can see the scores I have recorded for you on assignments and exams along with a course total based on the weights given in the syllabus. In the contents list above, you will see the item "Check your scores." This link will take you to a form where you input your last name and UPS student ID number. If all goes correctly, when you submit this, you should get back a table with your score, the possible score, and the class average for each assignment and exam. If something goes wrong when you try this, send me an e-mail letting me know.
We will have our third exam next Tuesday, March 30 from 8:00 am to 9:20 am. This will cover Sections 3.6 through 3.8 and the first part of Chapter 4.
We are temporarily skipping over Sections 4.2-4.5 and going to Section 4.6. This section is about applied optimization problems. For now, we will focus only on problems that involve optimizing some geometric quanitity. Section 4.6 consists of a series of examples. The ones that involve geometric optimization problems are Examples 1, 2, 5, and 6.
We will have our third exam next Tuesday, March 30 from 8:00 am to 9:20 am. This will cover Sections 3.6, 3.7, 3.8, 4.1, and the relevant parts of 4.6. I will post a list of objectives for this exam by the end of the week. To prepare for the exam, you should read the relevant sections of the text and work on the assigned problems (both those to be submitted and those not submitted). Keep a list of questions that come up in your reading and work on the problems. Get answers to these questions from the many sources available to you:
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 #3 will cover Sections 3.6, 3.7, 3.8, 4.1, and the parts of 4.6 that concern geometric optimization problems. In Section 3.6, you can skip the subsection entitled "Logarithmic differentiation" and in Section 3.8, you can skip the subsection entitled "Marginal analysis in economics." For this exam, a well-prepared student should be able to
The tutors who work in the Center for Writing and Learning have arranged a study session for Monday evening at 8 pm in Howarth 109. This is open to everyone in the class.
Here's a rough plan for the remainder of the course:
In working on the problems from Section 4.3, you will often need to solve an equation to find critical points. You should first look for the exact solutions (typically by factoring). If you cannot find exact solutions, use your calculator to get approximate solutions. On a TI-86, you can use the POLY feature for polynomial equations. If the equation is not polynomial, you can use the SOLVER feature. The TI-83 also has the SOLVER feature. On a TI-83, you can find the SOLVER under the MATH menu if you scroll down to the bottom. If you need the relevant part of the manual for your calculator, go to this TI web page.
I've given out Project #4. If you need a copy, go down to the Projects section of this web page.
In Section 4.4, the text gives a formal definition of what it calls "limits to infinity." In class, we will not work with this formal definition. Instead, we will adopt the following informal definitions.
Definition: The number L is the limit of the function f as x increases without bound (written x→+∞) if the outputs f(x) are close to L for all positive inputs far from 0.
Definition: The number L is the limit of the function f as x decreases without bound (written x→−∞) if the outputs f(x) are close to L for all negative inputs far from 0.
If you look back, you will see how these informal definitions are similar to the informal definition we used for "regular" limits at the beginning of the course. You can think of the symbols x→+∞ as shorthand for the phrase x increases without bound and the symbols x→−∞ as shorthand for the phrase x decreases without bound
In the informal definitions above, the phrases "f(x) are close to L" and "positive inputs x far from 0" are ambiguous. One could ask "How close to L?" and "How far from 0?" The formal definition (in the box on page 217 of the text) resolves any ambiguity. In the formal definition, the variable ε measures how close f(x) is to L and the variable N measures how far x is from 0.
On Problem 32 in Section 4.3, many people had difficulty in solving the equations to get zeroes and critical points. These equations involve trig functions. I've written a handout that explains how to solve these equations. You might want to read this if you had questions about how to solve these equations. You might also want to read it as an example of writing in mathematics.
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.
In grading the essays for Project #3, I used a rubric that you can look at in you want more detail on your score.
Exam #4 will be Friday, April 16 from 8:00 to 8:50. It will cover the material from Sections 4.2, 4.3, 4.4, and 4.5. I will post a list of objectives for this material later this week.
Exam #4 will cover Sections 4.2, 4.3, 4.4, and 4.5. In Section 4.2, you can skip the subsection entitled "Rolle's theorem." For this exam, a well-prepared student should be able to
Problem 52 of Section 4.4 has two parts (a) and (b). In part (a), you show that functions of a certain form have horizontal asymptote y=a/r and that the graph of a function of this form crosses this asymptote for the input x=(at-cr)/(br-as) provided br≠as. In part (b), you are asked to plot the graphs of two functions g and h. Here are the graphs (g on the left and h on the right):
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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.
I've moved the due date for the Section 5.2 homework back to Friday. We can go over one or two more problems from this section in class tomorrow. You should also work on the problems on the handout "Some accumulation problems."
You should continue working on the problems from Section 5.2 and the handout "Some accumulation problems." As you do, think about the similarities in computing the area of a region bounded by the graph of a function and computing the accumulation for a quantity from information about the rate of change in the quantity. On Friday, we will define integral. You will see that the definition is just formalizing what we have been doing in computing area and accumulation. The problems for Section 5.3 are listed below but you need not look at them until after class on Friday.
You should work on the problems from Section 5.3. Be sure to note that the first few problems are asking you to approximate the value of a given definite integral using a given number of terms in a Riemann sum. You are not asked to compute the exact value.
We began looking at antiderivatives today. The basic idea is to "undo" differentiation. Everything we will know about antidifferentiation comes from what we know about differentiation.
Exam #5 will be on Tuesday, May 4 from 8:00 to 9:20. It will cover Sections 5.1, 5.2, 5.3, 5.4, and possibly 5.5.
I have assigned Project #5 as extra credit. It is due on Wednesday, May 5.
In class, we looked at a definite integral for the function f(x)=exp(-x2). The plot on the left below shows the graph of f(x). The plot on the right shows the graph of an antiderivative F(x) for f(x). (This particular antiderivative satisfies the condition F(0)=0. The graph of any other antiderivative is a vertical shift of this graph corresponding to adding a constant C to F(x).) It is not possible to give a formula for F(x) using only elementary functions (roughly speaking, this means algebraic functions, trigonometric and inverse trigonometric functions, exponential and logarithmic functions.) The First Fundamental Theorem of Calculus applies to any definite integral for f(x)=exp(-x2). However, without a formula that allows us to calculate exact outputs for F(x), the First FTC is not useful. To get any information about the value of a definite integral for f(x)=exp(-x2), we need to use a finite number of terms in a Riemann sum. This is the topic of Project 5.
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Exam #5 will cover Sections 5.1, 5.2, 5.3, and 5.4. 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