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!