Quiz 2.1-4 Topics

Here’s a list of topics that will be covered on this Friday’s 2.1-4 Quiz. I’ve tried to indicate where a similar homework problem would be helpful.

Calculate the slope of a secant line to a given level of accuracy. (2.1, #9)
Find an average rate of change from a data table. (2.1, #1)
Determine limits from a graph (2.2, #5,7)
Determine infinite limits given a function (2.2, #25)
Determine a value “a” to create a limit in a rational function (2.3, #59)
Apply the limit laws to determine a limit. (2.3, #1)
Determine delta given x, f(x), L and a (algebraically) (2.4, #1)
Determine delta given x, f(x), L and a (graphically) (2.4, #5)

That's it! I'll be in early on Friday. Don't forget the makeup test for chapter 1 has to be completed before the weekend.

2.4 The Precise Definition of a Limit

The intuitive definition of a limit gives us estimates that are as close to the exact answer as we might need. However, a more precise definition of the limit gives us the ability to get good estimates.

The precise definition of a limit says that if f is a function on an open interval that contains the number a, except possibly at a itself, then the limit of f(x) as x approaches a is L. We can write this as

Verbally, we can say that f(x) is close to L when x is close to a because we can make the values of f(x) within any distance from L by taking the values of x within a distance from a, but not equal to a. This is illustrated in the graph directly below.

One sided limits can also be precisely defined.

Infinite limits can also be precisely define by saying that if f is a function is on an open interval that contains the numbers around a, but excluding a, then
* Functions that become largely negative as x gets close to a have this definition: Let f be a function defined on an open interval that contains the number a, except possibly at a itself. Then

*For limits, one-sided limits, and infinite limits, we can use the precise definitions to prove the limits convincingly instead of vaguely as we do with intuitive definitions.

Additional Links:

Oregon State Limits

NCTU Limits

Reminder: Crystal, your turn next.

Today's lesson focused on "Limit Laws" and their uses:

There are eleven limit laws:

1.) lim x--> a[f(x)+g(x)] = lim x-->af(x) + limx-->ag(x)
*Verbally, this means that the limit of a sum is equal to the sum of the limits

2.) limx-->a[f(x)-g(x)] = limx-->af(x) - limx-->ag(x)
*Verbally, this means that the limit of a difference is equal to the diffenences of the limits

3.) limx-->a[cf(x)] = c limx-->af(x), where c is a constant
*Verbally, this means that the limit of a constant times a function is equal to the constant times the limit of that particular function.

4.) limx-->a[f(x)g(x)] = (limx-->af(x)) (lim x-->ag(x))
*Verbally, this means that the limit of a product is equal to the products of the limits

5.) limx-->af(x)/g(x) = (limx-->af(x))/(limx-->ag(x)) -if limx-->ag(x) is not equal to zero.
*Verbally, this means that limit of a quotient is the quotient of the limits(when and if the limit of the denominator is not zero)

6.) limx-->a[f(x)]^n = [limx-->af(x)]^n where n is a positive integer
*Verbally, this means that the limit of a function raised to the power n is equal to the entire limit raised to the nth degree

7.) limx-->ac = c
*Verbally, this means that the limit of a constant is equal to the constant

8.) limx-->ax = a
*Verbally, this means that the limit of x as x approaches a is equal to a,

9.) limx-->ax^n = a^n
*Verbally, this means that the limit of x to the nth degree as x approaches a is equal to a raised to the nth degree.

10.) This law is difficult to type of the computer, so I am going to verbally explain it. Basically this law states that the limit of the nth root of x as x approaches a is equal to the nth root of a.

11.) I will also explain this law verbally. It states that the nth root of f(x) as x approaches a is equal to the nth root of the entire limit of f(x) as x approaches a where n is a positive integer since you cannot take the root of a negative number.

Example Problems: What is another way to express the following limits using the laws listed above?
a.) limx-->2[f(x) + 5g(x)]
b.) limx-->1[f(x)g(x)]
c.) limx-->75
*Answers and Solutions:
a.) Using law #1 above, we can see that the limit of a sum is equal to the sum of the limits. Therefore, limx-->2[f(x) + 5g(x)] is equal to limx-->2f(x) + limx-->2 [5g(x)]. Then, according to law #3, we can see that that the limx-->2[5g(x)] is equal to 5 limx-->2g(x). Therefore, the answer is limx-->2f(x) + 5 limx-->2g(x).
b.) By using law #4, we can see that limx-->1[f(x)g(x)] is equal to the products of the limits of both f(x) and g(x) as x approaches zero. So, the answer is (limx-->1f(x))(limx-->1g(x)).
c.) According to law #7, the limit of a constant as x approaches a is the constant. Therefore, the solution is the constant 5.

Direct Substitution Property:
limx-->af(x) = f(a)
-functions with this property are continuous at a.

The Squeeze/Sandwich/Pinching Theorem:
-If f(x)<g(x)<h(x) when x is near a (except possibly when x is at a) and the limx-->af(x) =
limx-->ah(x) = L, then limx-->ag(x) = L.

*If there is a function whose limit is difficult to find, and you know of two functions that surround the original function around the limit, then you can find the limits of the two surrounding functions, and if they are equal, you can assume that the limit of the original function is the same as well.

Example Problem:
Find the limit of (x^2) sin(1/x) as x approaches 0.

Solution and Answer:
Since the graph of limx-->0 (x^2)sin(1/x) does not clearly show a limit, it is necessary to find two graphs whose limits would equal that of the original problem. So, we know that -1<sin1/x<1, and by multiplying each side by x^2, we get (-x^)2<x^2sin(1/x)<(x^2). From this we can find the limits of the graphs of (-x^2) and (x^2) as x approaches zero. We find that the limits of these two graphs as x approaches zero is equal to zero, then limx-->0(x^2)sin(1/x) is also equal to zero.

Other Important Theorems:
1.) limx-->af(x) = L if and only if limx-->a-f(x) = L = limx-->a+f(x)
*Verbally, this means that the limit of f(x) as x approaches a is the limit if an only if the limits of f(x) as x approaches a from both the negative and positive sides is equal to the limit.

2.) If f(x)<g(x) when x is near a(except possibly at a) and the limits of f and g both exist as x approaches a, then limx-->af(x)<limx-->ag(x).

Websites to help you understand the concepts of 2.3 more!

http://www.calculus.net/ci2/search/?request=category&code=1243&off=0&tag=9200438920658

http://www.geocities.com/mathdepot/squeeze.htm

Reminder to Jessica to write tomorrow's blog!

A couple jokes to keep you guys on your feet!

Q: What's the integral of (1/cabin)d(cabin)?

A: A natural log cabin!

Q: How does a mathematician induce good behavior in her children?A: "I've told you n times, I've told you n+1 times..."

Section 2.2

The Limit of a Function

- limits occur when the x value approaches a certain value, f(x) also approaches a certain value
- the general notation for limits: lim x->a f(x) = L,"the limit of f(x), as x approaches a, equals L"
- f(x) can be arbitrarily close to a but will never equal a
- calculators can often make mistakes! (refer to example 2 on pages 94-95) on a calculator, if t is sufficiently small, you will get the value 0.
- when a graph shoes infintely many values for x that approach 0, like on the graph of lim x->0 sin (pi/x). the limit does not exist.
- when guessing a limit using a calculator, you cannot be 100% sure because if you can persevere using smaller and smaller values of x, results may be different.

One-Sided Limits
- Definition: written as lim x-> a- f(x) = L, the left hand limit of f(x) as x appraoches a [or the limit of f(x) as x approaches a from the left] is equal to L if f(x) is arbitrarily close to L, and x sufficiently close but LESS THAN a.
- "the right hand limit of f(x) as x approaches a is equal to L" is written lim x-> a+ f(x) = L, and in this case we are only considering when x>a. this leads to the following definition:
lim x->a f(x) = L if and only if lim x->a- f(x) = L and lim x-> a+ f(x) = L

Infinite Limits
- occurs when y values become infinitely large when x approaches a.
- do not regard infinity as a number!!!!
- the limit of f(x) as x approaches a is negative infinity when the values of x decrease without bound.
- the line x = a is a vertical asymptote of the curve y= f(x) if at least one of the following is true:
*can't insert infinity symbol, so I will represent using the word*

• lim x->a f(x) = infinity
• lim x->a- f(x) = infinity
• lim x->a+ f(x) = infinity
• lim x->a f(x) = negative infinity
• lim x->a- f(x) = negative infinity
• lim x->a+ f(x) = negative infinity

4 steps in determining the limit of a function:

1. try plugging in values
2. simplify algebraically
3. look at a graph
4. if all of the above don't work, then the limit D.N.E. (does not exist)

Sample Problem: (a simple, yet important concept)

(my internet lags so i apologize for not being able to post pictures)

what is the limit as x approaches 2 from the left? the right? does limx->2 exist? (refer to graph in link)

http://www.math.uncc.edu/~bjwichno/spring2005_math1241_002/Review_Calc_I/Images/1_over_x_2.gif

answer: no, because as x gets arbitrarily close to 2, the values become infinitely small (as x approaches from left) and infinitely large (as x approaches from right). there is, however, an asymptote at x=2.

links:

your ol' classic wiki favorite:
http://en.wikipedia.org/wiki/Limit_of_a_function

and this one, which most likely sums it up better than i do!http://www.math.hmc.edu/calculus/tutorials/limits/

and THIS ONE, is a cool simulator graph for limits. it's quite addicting... o.O http://www.math.psu.edu/courses/maserick/limit/limit.html

I hope that helped...and I believe it is the Lauren's turn?

Ending with a quote today...but I've always liked this quote from the "Father of Modern Mathematics" (pretty sure we'll deal with his concepts later on)...

"I think, therefore I am..." - Rene Descartes.

well, that's it. jaa ne...mata ne! (cya, and until next time!)

Section 2.1

There are lots ways to say "derivative" - Instantaneous Slope, Velocity, etc. The first section in chapter two focuses on finding it by means of a tangent line to a curve. As we know, the slope of a line is equal to the "rise" divided by the "run". or the change in Y divided by the change in X. But while this system works for finding the slope of a straight line, we have to use it in a very specific way to find the slope of a given point on a curve. For instance, to find the slope at X=1 on the line Y=X^2...

• We must have two points to find the slope of any line. Since we are looking for the slope at X=1, one of our points will be (1,1)
• To find the second point, we should try some different points. First, let's use (2,4)
• The slope between (1,1) and (2,4) is 3/1. Let's see how it changes as our second point gets closer to (1,1)
• If our second point is (1.5, 2.25), the slope is 1.25/.5, or 2.5
• If we get ridiculously close to (1,1) then our point could be (1.0001, 1.00020001), in which case our slope would be .00020001/.0001, or almost exactly 2.
• As we can see, the closer our second point gets to 1, the closer the calculated slope is to 2
• This can be expressed in the equation: Slope f(x) x->a = (f(x)-f(a))/(x-a) , where a is the point at which you want to find the slope, and x is a point that is (ideally) infinitely close to a.

Two interesting websites that I found that illustrate the concept of tangent lines in relation to instantaneous slope (both require the Java plugin).
http://www.ies.co.jp/math/java/calc/doukan/doukan.html
, and a much more technical version,
http://www-math.mit.edu/18.013A/HTML/tools/tools04.html

And now for the first of my series of the greatest and most meaningful quotes of all time...
“Then after World War Two, it got kinda quiet, ‘til Superman challenged FDR to a race around the world! FDR beat him by a furlong, or so the comic books would have you believe. The truth lies somewhere in between…”
—Abe Simpson (Better known as Grandpa Simpson)
And on that note, Lauren has the next post...

Test 1 Topics

Here’s a list of topics that will be covered on this Thursday’s Chapter 1 Test.  I’ve tried to indicate where a similar homework problem would be helpful.

• Interpolation – given data or a graph, estimate/determine new points in the relationship/function. (1.1 - #17, 1.2 - #11,19)


• Extrapolation – given data or a graph, estimate/determine new points in the relationship/function. (1.1 - #17, 1.2 - #11,19)


• How does a graph change when certain values within the function vary? (1.4 - #31,34)


• Given f(x) and g(x) (algebraically, numerically or graphically), determine values for f(g(x)) given x. (1.3 - #55)


• Determine f(x) given a graph. (1.5 - #17)


• Determine the range of a function. (1.1 - #23)


• Determine the domain of a function. (1.1 - #23)


• Determine an appropriate viewing window for a given function. (1.4 - #7,13)


• Determine g(x) as a result of transformations of f(x). (1.3 - #3)


• Work with inverse functions. (1.6 - #19)


• Sketch a graph illustrating a given functional relationship. (1.1 - #11)


• Sketch a graph given a transformed function. (1.3 - #3)

Prep AB Calculus C 2006-07

no, god is not a mathematician. To be a mathematician requires black and white rationality. To create the world we live in requires quite the opposite. Things don't always make sense in today's world and not everything can be answered with a formula.

God: Mathematician or wizard?

As far as the premise goes, that a omniscent, omnipotent, omnipresent being called "God" created the entire universe as we know it, yes he was in a sense a mathematician (in fact the first mathematician). Math reflects the systematic and quantitative structure of the universe as well as the consistent natural laws; so, the creation of the universe created the original basic postulates of math. In this sense "God" created the universe in the structure of what we now describe as math. Therefore, "God" was a mathematician and mathematicians could be described as the discoverers of "God's mysterious ways".

not really....

God isn't a mathematician because if he were, humans wouldn't have to learn math since it would already be engrained in humans. What would be the point of making humans learn math if he could've given that knowledge to us. He's not that cruel.

not a mathematician

I don't think God is a mathematician because although he is omniscient (or whatever omni- all knowing is), Math is mankind's way to quantify how things are, so God probably doesn't deal in numbers. He gave us the ability to use numbers to explain to each other how large, heavy, or what path things are, but he didn't use math to create these things (thus, since he doesn't work in mathematical terms, he cannot, by definition, be a mathematician).

"In the beginning God created the heavens and the earth." - Genesis 1:1

Is God a mathematician?

In the original sense of the word god, being a super powerful or omnipotent being, a god would have no use for numbers. As it is natural for our bodies to regulate body temperature without knowing the exact temperature outside in Kelvin or degrees Celcius, I imagine it would be just as simple for the being of "God" to automatically know the specifics on an unconscience level in order to produce a desired result. God has no use for numbers, why need math?

Lockeman

yay math!

Is God a Mathematician?

God must be a mathematician because the laws that govern the way things work on his world are all based on math. Populations and species grow at rates that can be computed mathematically. The laws of physics are mostly based on math. Math occurs naturally, such as in the fibonacci series, and the numbers pi and e are present in the world. The only way this could be is if God is a perfect mathematician who applied his logic to the creation of his world.

Is God a mathmetician?

I think that God must be a mathematician simply because of the incredible mathematical details that accompany the world he has created. It takes a mathematicians skill to craft such an amazing world full of mathematical wonder. For example, the classic fibonacci shell spirals, as seen below. With any ordinary spiraled shell copied on paper, one can continue adding squares around the spiral and each new square will have a side which is as long as the sum of the latest two square's sides. Not only is this seen on sea shells, but on the arrangement of seeds on flowering plants also. This kind of mathematical perfection and harmony could have only been created by a God who is a mathematician.

Blog Postings

As a reminder, the requirements for each blog posting will consist of:

• A review of the main point of the class lecture/demonstration.  This summary should highlight any relevant formulas and/or graphs and communicate your interpretation of the concept covered in class. (15 pts)  For this part of the posting, I am looking for quality, not quantity.


• An example problem, including a statement of the problem, the answer, and the solution method.  (For your first post, using an example covered in class is acceptable, for additional postings, original examples will be required.) (10 pts)


• A link to an additional Internet resource supporting the Topic of The Day. (5 pts)


• A reminder to the next BlogMaster of their responsibility to post. (5 pts)


• A “personalization” of your posting.  This personalization can be a comment about the day’s class, an image, a quotation, a question posed for discussion, a joke, or something else that reflects you as a student.  These personalizations must be in good taste!  (5 pts)

In addition to your posting, you will be expected to comment on a minimum of two (2) of your classmates postings during each quarter.  These comments must either further enhance your classmates’ understanding of the posted topic or further a discussion question posed in the original posting.

Additional Notes:

• Postings will be due within 24 hours of class.  I will post a schedule of class scribes for the first quarter once everyone has joined the class blog.


• For help with posting equations and graphs, please feel free to come ask me for assistance.


• Initially, the blogs will be hosted on blogger.com.  As the year progresses, we hope to migrate to an internal website.


• While we are on blogger.com, there is some software available through the website that allows creation/editing of posts via Microsoft Word.

Blog Policies

There are some things I want you to remember about blogging. Many of things have been discussed by other teachers and classes, so I will paraphrase them here and try to give them proper credit:

First of all, our class will not be the only people to view our postings. The Internet is accessible almost everywhere these days, and even if a post is deleted, there’s no guarantee that the posting hasn’t been copied and propagated to other sites or linked to from those sites. This has a couple of implications:

First, privacy. We will only be using first names on the site. If I post pictures or video, no one will be identified, other than “Mr. French’s class”. Do not use pictures of yourself for your profile here. If you want a graphic image associated with your profile, use an “avatar” – a picture of something that represents you but is not you. Here’s a link to a fun image creator.

Second, etiquette, appearance and common sense. Bud the Teacher has these suggestions, among others:

1. Students using blogs are expected to treat blogspaces as classroom spaces. Speech that is inappropriate for class is not appropriate for our blog. While we encourage you to engage in debate and conversation with other bloggers, we also expect that you will conduct yourself in a manner reflective of a representative of this school.


2. Never EVER EVER give out or record personal information on our blog. Our blog exists as a public space on the Internet. Don’t share anything that you don’t want the world to know. For your safety, be careful what you say, too. Don’t give out your phone number or home address. This is particularly important to remember if you have a personal online journal or blog elsewhere.


3. Again, your blog is a public space. And if you put it on the Internet, odds are really good that it will stay on the Internet. Always. That means ten years from now when you are looking for a job, it might be possible for an employer to discover some really hateful and immature things you said when you were younger and more prone to foolish things. Be sure that anything you write you are proud of. It can come back to haunt you if you don’t.


4. Never link to something you haven’t read. While it isn’t your job to police the Internet, when you link to something, you should make sure it is something that you really want to be associated with. If a link contains material that might be creepy or make some people uncomfortable, you should probably try a different source.

Are there other considerations we should take into account? Use the comment feature to add any others or to clarify/expand on one of the above.

Welcome!

Congratulations! You found our class blog! This is where we as a team will hopefully create a resource to help us conquer any issues that arise during our class this year. This is the place to talk about what’s happening in class; to ask a question you didn’t get to ask in class; to share your knowledge with fellow classmates and any other Internet users who choose to read our notes;…and most importantly it’s a place to reflect on what we’re learning.

A large part of retaining knowledge requires reviewing and discussing new information on a regular basis. This blog is intended to help each of you do just that. Between creating your own posts and commenting on your classmates’ posts, you will have the opportunity to explore each of the topics we cover this year in greater depth. I hope you will use this forum to help yourself and your classmates in whatever ways you can think of.

Blogging Prompt

Occasionally I will include a posting of my own, either to clarify a concept or to generate some further discussion. These postings will have a title similar to the one above this paragraph.

To get things rolling, here’s a question for you to think about and respond: Is God a mathematician? Why or why not?

Don’t forget to email me with the information I requested in class so I can include you on the team!