Topics:

• Notation for limits and one-sided limits

• Properties of limits

A slightly improved version of the notes I handed out in class is here:

MAT1475LimitsAnalytically

The substitution property says that for certain types of functions, as long as c is a number in the domain of that function, it will be true that $\displaystyle \lim{x\rightarrow c}f(x) = f(c)$

There are some other functions which have the substitution property in addition to polynomials and rational functions. Here are some that you know from previous math classes:

The trig functions $\sin(x)$, $\cos(x)$, $\tan(x)$, $\cot(x)$, $\sec(x)$, $\csc(x)$

The exponential function $b^{x}$ for any positive base $b$ ($b\neq 1)

The logarithmic function $\log_b(x)$ for any positive base $b$ ($b\neq 1)

In particular, the functions $e^x$ and $\ln(x)$ have the substitution property.

We use the properties of limits to shortcut the process of finding a limit whenever possible from now on. We will have to resort to other means when these properties cannot be used, though.

Homework:

• If you have not already joined Piazza: Look for the invitation to join our Piazza discussion board, and join by following the link. The invitation has been sent to whatever email address you have in Blackboard, and it will be from “The Piazza Team”. Or you can go directly to Piazza here and join using your City Tech email address. We will be using the discussion board as a way to ask questions about the homework or the material discussed in class.

• Last call: If you have not already done so, log in to WeBWorK following the instructions here (which were also handed out in class) and make sure to enter your City Tech email! (If you do not yet have a City Tech email, please enter it as soon as possible.) You must have your City Tech email address in your User information in WeBWorK in order to be able to use the “email your instructor” feature, and also it will be used to send out your midterm grades.

• Do the WeBWorK assignment  “Limits-Continuity1”.  For people who joined this class this week, you still have some time to work on the previous assignment as well. Make sure that you follow the instructions given in the homework assignment last time.

• Also do the following problems from the textbook: p. 28 #7-13 odd, 19-25 odd. I may ask for volunteers to put some of these on the board.

• Monday 12 February is a holiday (no classes meet).

• There will be a Quiz on Wednesday. The question will be a randomly selected homework problem from today’s assignments.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Note on my schedule: for the time being, I will only be able to be online to read and reply to emails at certain times of the day. (It is possible that I may be online at other times but I cannot guarantee it.) The times are roughly:

Monday – Friday early morning

Monday-Thursday around 2:30-3:00 PM

Sunday-Thursday evenings around 9-10 PM

Please be aware of this if you need to contact me by email. Thanks.

Topics:

• Review of the concept of limits (for the benefit of several new students) and using tables in Desmos to guess the value of the limit (or whether or not there is a limit)

Here are links to the Desmos graphs and tables that I saved from class:

Showing that $\displaystyle \lim_{x\rightarrow 0}\frac{\sin(x)}{x} = 1$: notice that $\frac{\sin(x)}{x}$ is undefined when x=0, but the limit as x approaches 0 does exist and is equal to 1. The idea is that we are trying to guess what value the function “should” have when x=0, based on the values nearby.

Limit of a piecewise-defined function: This shows how to enter a piecewise-defined function in Desmos. Since the function is defined differently on each side of the target value of x, we have to use two tables, one for x approaching from the left and one for x approaching from the right.

Limit of a polynomial function: This is a table I made for problem 7, $\displaystyle \lim_{x\rightarrow 1}\left(x^2+3x-5)$ from the homework. Notice that in this case, the limit as x approaches 1 is -1, which is the same as f(1). This is a special situation!

It is very important that you understand what we are doing when we find a limit as x approaches some number c and how that is different from finding the value of the function when x=c. There are times when $\displaystyle \lim_{x\rightarrow c}f(x)$ will be equal to $f(c)$, but that does not mean that they are the same thing conceptually! In fact, that is a special situation that we will be discussing later when we get to the idea of continuity.

• Review of the three ways that a limit may fail to exist.

The limit fails to exist in these situations:

* when the function approaches different values from the left and from the right. (This tells us that in order for the limit to exist, there must be two “one-sided” limits and they must equal each other.)

* when the values of the function either grow without bound (“go to infinity”) or decrease without bound (“go to negative infinity”) as x approaches c. This happens when there is a vertical asymptote to the graph of the function. We will go into this in more detail later on.

* when the values of the function oscillate in such a way that they do not keep getting closer and closer to any fixed number, as x approaches c. (The example is $\sin\left(\frac{1}{x}\right)$)

• Average rate of change over an interval and instantaneous rate of change: I worked through my version of problem 1 from the WeBWorK except that I did not have time to complete the last part, where we are to find the limit of the $\frac{\Delta s}{\Delta t}$. If you have questions about how to compete this problem you can discuss on Piazza! See below.

Homework:

• If you have not already joined Piazza: Look for the invitation to join our Piazza discussion board, and join by following the link. Or you can go directly to Piazza here and join using your City Tech email address. We will be using the discussion board as a way to ask questions about the homework or the material discussed in class. (New students: I will send you your invitations sometime on Tuesday.)

• If you have not already done so, log in to WeBWorK following the instructions here (which were also handed out in class) and make sure to enter your City Tech email! (If you do not yet have a City Tech email, please enter it as soon as possible.) You must have your City Tech email address in your User information in WeBWorK in order to be able to use the “email your instructor” feature, and also it will be used to send out your midterm grades.

• Make sure that you have done all of the things that are in the First Day post and the post from last time!

• Do the WeBWorK assignment  “LimitsIntroShort”. Make sure that you are using  tables of values to find the limits in problems 9, 11, and 12! (If you go to a friend or tutor, they may try to show you some other way of finding these limits. Be insistent that you can only use the methods we have already learned in class!)

• There will be a Quiz on Wednesday. The question will be what was announced last time.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Note on my schedule: for the time being, I will only be able to be online to read and reply to emails at certain times of the day. (It is possible that I may be online at other times but I cannot guarantee it.) The times are roughly:

Monday – Friday early morning

Monday-Thursday around 2:30-3:00 PM

Sunday-Thursday evenings around 9-10 PM

Please be aware of this if you need to contact me by email. Thanks.

Topics:

• Discussion of problems 1, 3, and 4 from the homework last time. These are important for making sure that you understand the concept of a limit. You may also want to view one or more videos from the resources for the concept of a limit.
• Using Desmos graphing calculator to graph functions, in particular piecewise-defined functions like in the homework problems 13 and 15. (We will look at problem 15 next time.) You can type in the pieces of the definition, followed by the inequality restrictions on x enclosed in braces, like this: \[y = x^2 – x + 1 \{x\le 3\}\]\[y= 2x+1 \{x>3\}\]Another method for piecewise-defined functions is given here. Don’t forget that even if you use Desmos, you need to consider the domain of your function, because the graphing calculator cannot show you if there are holes in the graph!

Homework:

• Log in to WeBWorK, following the instructions given on this page. Start reading and working in the Orientation assignment to learn about how WeBWorK works. You should also then start the assignment “LimitsIntroShort”: do at least problems 6 and 7.
• View some of the videos in the post Some Resources for the concept of a limit for more thorough understanding, if you like.
• Look for your invitation to join the Piazza discussion board – it will have been sent by “the Piazza Team”, not by me, and it will have been sent to the email address that is listed in Blackboard for you. Or you can go directly to the Piazza board and join using your City Tech email address.
• Don’t forget about the homework from the text book from last time!
• There will be a one-question quiz at the start of class next time. The quiz will be on the question of whether it is always true that \[\displaystyle \lim_{x\rightarrow 5}f(x)\] is equal to \[f(5)\]. Be prepared to give an example where this is not true!

Here is a video introduction to limits from Khan Academy.

If you watch through the whole video, he does some examples by making use of the graph, and also numerically using a calculator.

More details and examples are in these videos and practice problems:

Also from Khan Academy: limits using tables

and here are practice problems for limits using tables.

Also from Khan Academy: limits from graphs

There are several videos in that sequence, followed by practice problems.

Background:

There is a very nice discussion of the intuitive meaning of a limit, using examples “from real life”, at betterexaplained. You may want to read through this to see why we need limits and other uses of them that will appear later in calculus.

Welcome to Calculus I!

Take a look around this website to see what resources are here. I am still adding things!

Before and after every class meeting, there will be a post of the day that contains notes, links to related materials, and details of what your homework assignment is for that day. Always check the blog post before heading off to WeBWorK, for example. (There will be a link to the WeBWorK when it is assigned, next time.

My course policies are on a separate page here.

In the future there will be separate pages for solutions to the Quizzes and for the Test Review problems sets.

Topics for today’s class:

• The idea of a limit: estimating limits by using a graph or tabulating values of the function.

• Three ways that a limit may fail to exist.

• Difference quotients and limits of difference quotients:

The idea is that the difference quotient represents the slope of a secant line through two nearby points – physically it represents the average rate of change of the function between those two x-values. We will be trying to find the slope of a tangent line to the graph at one point (an instantaneous rate of change) by making the points get closer and closer together, and see if the slopes of the secant lines approach a limit as we do so.

I hope to give more detailed notes (and using math notation) once I have a bit more time.

Homework:

• Download the textbook (or anyway Volume 1)  from this site:

http://www.apexcalculus.com/downloads

• Review the material we discussed in class, which is basically Section 1.1 in the textbook. Make sure that you understand the concept of a limit, the three ways a limit may fail to exist, and how to compute a difference quotient. (I prefer to compute the difference quotient as I did in class, by first computing f(x+h), then subtracting f(x) from that, and finally dividing by h. Note that we do not cancel out the common factor of h, because we are looking to see if there is a limit as h approaches 0.)

• Do the following problems from the textbook: p. 8 #1, 3, 4, and 7-23 odd. I may request that some of these be done on the board at the start of class next time.

For problems 7-23, where you need to approximate the limit graphically and numerically, you may want to try using Desmos as I did in class. There is also a Desmos app that you can use on your phone if you like. See the information on this page.

• Watch for the invitation to join the Piazza discussion board, which will be sent out over the next day or so to the email address that is listed for you in CUNYFirst. (Most likely it is your City Tech email address.)

Note also:
• Find and deal with your City Tech email: you must use this email address in WeBWorK and to join the Piazza discussion board. Also, City Tech is already sending you emails here!