The 14 things I wanted to remember for Exam 1

I love this kid’s facial expression! That is seriously what I look like when I feel time running out and I’m not finished with an exam… Anyway, I wanted to write up a quick review for Exam 1. Basically what I plan on putting here are all the important forumulas and notes I took that I think I might need to remember for the exam.

#1.

#2. is the difference quotient

#3. The difference quotient = the average rate of change, the slope of the secant line

#4. The Derivative = The instantaneous rate of change, the slope of the Tangent line, the limit of the difference quotient, limit of the average rate of change, and limit of the slopes of secant lines.

#5. The Precise Definition of the Derivative: The derivative of a function is the limit of the difference quotient of that function as h approaches zero.

#6. Differentiation Shortcuts: Power Rule: If where n is a number, then

Constant Multiple Rule: If where k is a constant, then

Sum and Difference Rule: If then

Product Rule: If then

Quotient Rule: If then

#7. Functions are not differentiable at sharp bends in the graph, at any discontinuity, or if a vertical tangent line is present at that point on the graph.

#8. Anyone confused about how to convert a square root function into exponential form, here’s how it works:

if you have something like this it is equal to if you ever have a fraction as an exponent, the denominator (or bottom number) is the root it is (for example if you have something like it’s the square root of x cubed, or if you had something like that would be the 3rd root of x squared)… and obviously the numerator is the real exponent of x under the root.

It works both ways so I don’t feel the need to explain how to put it back into root form.

#9. Continuity of Certain Functions:

Linear Functions always continuous everywhere

Quadratic Functions always continuous everywhere

Polynomial Functions always continuous everywhere

Radical Functions continuous everywhere they are defined

Rational Functions continuous everywhere they are defined

Exponential functions always continuous everywhere

#10. Rational Functions can have 2 types of discontinuities: Vertical Asymptotes, and Point Discontinuities. They will not have a limit at a vertical asymptote, but will have one at a point discontinuity.

#11. How to find a discontinuity of a function: Set the denominator equal to zero. Here’s an example:

now we must set the denominator to zero

are our discontinuities.

Next you can consider f(x) as an algebraic expression and simplify it into g(x) (which is not the same function).

cancel out (x-2) on top and bottom, and you’re left with

You can use this simpler function to find the limit of the more complicated function. So in other words, a more important equation to remember is:

#12. for any value

#13. If you solve for the limit of an equation and end up getting this means it is a discontinuous.This also means you have more work to do, so you should plug the numbers into the simplified equation g(x) and try to solve for it.

#14. If you solve for the limit of an equation and end up getting a number over a zero (like you have found a vertical asymptote and the limit here DOES NOT EXIST. DNE.

I worked backward, but that’s about it.  Good Luck to everyone on the test!

Attempting to Make Equations out of Chemistry Problems, New Derivative Rules, & Exam 1!

Exam 1 approaches!!! DUN DUN DUUUUUUUHHHHHH.

I just finished up the homework due by Sunday and found myself feeling more confident than ever about this exam. I really feel that I understand this whole derivatives concept, and I love the new easy way to find derivatives. I don’t think I have a favorite, but it’s certainly faster to find the derivative applying the new rules we got this week in class.

I’ve actually been enjoying it so much, I decided that my Chemistry class should have such clear equations for math we have to do in there, and decided to write my own equations for the test we have coming up in Chemistry on Wednesday. I just think I will do so much better if I can plug numbers into an equation rather than second guessing myself. Before I go crazy on my Chemistry math… to make sure I help out my fellow classmates, I want to post the newest rules for Derivatives.

When , Or in other words… when ,

When , Or in other words… when ,

So a workable example of this would be:

if you are wondering how I got 6x and 8… refer to my last post.

Example 2:

which can be simplified into

On to Chemistry Math!

For Combustion Reactions:

is the symbol for any combustion reaction with A being any compound reacting with the diatom of O.

represents the combustion of hydrocarbons (compounds made only of hydrogen and carbon)

represents a combustion of C,H, and O that also forms

Easy enough, but now we move on to more difficult concepts

The Mole:

1 Mole = the amount of matter that contains as many objects (atoms, molecules, ions, etc.) as the number of atoms in exactly 12 g of Carbon 12.

Avogadro’s Number helps us understand how many objects are in a mole.

objects in a mole

To better understand what a mole is, (and yes, this totally applies to math right now) 1 mole of anything is  objects. The mole helps us measure really really small things (i.e. atoms) in workable numbers.

Here’s how my chemistry notes put it:

(Molecular Mass) 1 molecule of = 18.02 amu

(Molar Mass) 1 mole of = 18.02 g

1 Mole of H2O = 6.022 x molecules of H2O

1 Mole of H2O contains atoms of Hydrogen

I find that to be useful but not as useful as if I had an equation that laid it out for me. So here’s what I came up with:

has atoms of x, and atoms of y.

To find mole using the number of atoms you can use the following equation:

To find grams in a mole:

where Y stands for an element, A stands for the subscript and C stands for the number in grams of that element.

To find molecules of a compound with a given number of grams:

For our purposes, we’re going to use an element of E_a.

To find atoms of a specific element in a compound in a specified amount in grams:

where E=element, a=subscript (number of elements in the compound), H=grams of the compound present, B=grams of the compound in 1 mole.

I now see how confusing this could be. Especially if you aren’t in Chemistry… but it makes a lot more sense to me now that I’ve written it all out, and I’m going to conclude with my final math equations for Chemistry.

Determining Empirical Formulas from Molecular Formulas and Vice Versa

This is pretty simple. It’s kind of like factoring actually. In order to determine an empirical formula from a molecular one, all you have to do is get the simplest whole number ratio.

For example, the molecular equation for Glucose is this: the subscripts are all divisible by 6, so the empirical formula would look like this:

It’s a little harder going from the Empirical formula to the Molecular one… but here’s how it works.

First you need the molecular weight. Then you need the Formula Weight.

For example, Formaldehyde has a molecular weight of 30.03. It’s molecular formula is look familiar? That’s because it’s molecular formula is also its empirical formula, which happens to also be the empirical formula of glucose… awkward, right? Cause I’d much rather eat some glucose than some formaldehyde… so it’s pretty important that we recognize that an empirical formula doesn’t usually give us enough info.

Next we have Acetic Acid. It’s Empirical formula is also , however its molecular weight is 60.06. In order to find out what it’s Molecular Formula is, we can do this simple equation:

Where MW= Molecular Weight, FW=Formula Weight Once you solve for x, you can use it to create the new Molecular Formula which would look like this

To show exactly what I mean, we can plug-in the info for Acetic Acid and get its molecular formula since we know it’s MW=60.06 and it’s FW = 30.03 (weight of formaldehyde) (you can also get this by adding the molecular weight of 1 carbon, 2 hydrogens, and 1 oxygen using the periodic table as a reference).  Let’s plug it in:

And we have our Molecular Formula!

We can also find it using the Molecular Mass, for example:

Cortisol’s Molecular Mass = 362.47 g. It’s made of 69.6% C, 8.34% H, and 22.1% O by mass. What is in its molecular formula?

I created an equation to help solve this and it goes a little something like this:

362.47x.696=C

362.47x.0834=H

362.47x.221=O

C=252g H=30.23g O=80.11g

Now that I have those numbers I can take the g/mole of each element and get the number of each element in the molecule.

Now we know that the molecular formula of Cortisol is

But what if we didn’t have the Molecular Mass? If we had just the percentages, we could solve for 100g. It could look something like this:

But we need to get to a whole number so first we have to divide by the lowest number:

That won’t work, because we still have a fraction, and we need a whole number, so we have to find a multiple that will make it whole, which in this case happens to be 5.

Multiply each by 5, and you get what we got before:

TADA!!!!

So I realize this was a lot of Chemistry, but I felt that the math really was connected to my calculus class, and I am certainly more adept at doing the algebra now that I’m working with more complex figures like Derivatives.

Good Luck to you ALL on the 1st Exam! In celebration of making it to our first Exam, I am going to change up my blog background.

Oh… and happy early Valentines to you all!

Word Problems help make connections

The calculus exam is coming up… Monday.  Looks like this will be a good opportunity to see if what I’ve learned is really sticking. So far I feel completely comfortable with what I’ve learned, and think I will do well. However there’s always the fear in the back of my mind that I’m missing something that I’m unaware of.

Derivatives, so far, are understandable. I realize it’s going to get harder and harder, but the homework assigned to us yesterday was a pretty easy thing to do, and I found that the application problems really helped me get a better grasp on why I would want to know the derivative.

For example, there was a problem asking how many a 3 year old ram’s horn diameter would be. So all I had to do was plug in 3 to the given function and I easily arrived at my answer. What was cool, was the minute I realized that the derivative of the original function was the answer to the next question, which was asking me what the average rate of grown per year for that 3 year old ram would be. Then you just have to find the derivative of the function, plug 3 into that and you’re golden.

So it’s finally coming together for me that a derivative helps you find the instant rate of change for a specific year or specific degree, while if you wanted to find the average rate for something a whole, that would be the difference quotient. What I mean is, had the question asked me what the average rate of growth for all the ram’s possible ages would be, that would be a whole different process than just finding the average rate of growth per year for the 3 year old ram’s horn.

I don’t know if that made any sense to you guys, but it totally clicked here.

I’m hoping we eventually learn how to read a word problem and then formulate it into a function on our own… I’d like to know where they got the functions I got to work with in our homework this weekend. I think that would be useful, and I do imagine we will get there by the end of the semester and probably earlier than that.

As a semi-unrelated side note, I found this awesome Iphone app that lets you make your own meme posters… so now I can make my own math comics… excellent.  I have to find ways to keep it interesting, right?

But until I find a way to make math funny I am going to borrow one to close out my post.

Derivatives… ARGH! And how to get a rational difference quotient for Square Roots.

I just spent 4 hours finishing a 14 problem set of homework. I’m so annoyed. It’s all because I kept getting confused about what I was being asked, so if you are feeling a bit confused, I am about to lay it all out for myself and whoever may benefit from this.

Derivative = Instantaneous Rate of Change = Slope of the Tangent Line = Limit of the Difference Quotient.

Difference Quotient = Slope of Secant Line = Average Rate of Change

So if given a function, such as and asked to find the slope of the tangent line, all you have to do is find the limit of the difference quotient. For example:

must be solved for first this can be simplified to now you have to solve for the limit.

f'(x)=4x which is also your tangent line slope.

Don’t make the mistake I made, where they asked me to find the secant line’s slope and I kept giving the stupid TANGENT line slope. I am so irritated at myself about this. 4 hours, people… 4.  It’s just ridiculous.

Thanks to komplexify.com for the image

If they ask you to solve for the slope of the secant line all you have to do is give the difference quotient which was (in the equation above)

The other issue I know a lot of people (including myself) have been struggling with is with square roots and the difference quotient.

So I’m going to go ahead and post how to solve for the square root as simply but detailed as I can.

So let’s say you have the function . And you have to solve for the derivative, which cannot have a zero on bottom, and when you make h=0, and you get 0 as your denominator (which is inevitable with this equation)… how do you work around this? Well, you have to rationalize the numerator. Rationalize the what now? I’ll show you…

First let’s remember our difference quotient:

Now we’ll plug in our stuff:

How do we rationalize it? We have to multiply it by itself, just like rationalizing the denominator.. sorta. It should look like so:

* 

It looks confusing but what I like to do from here is show you exactly how you would multiply the top guys to get an easy number:

* * *

which could be simplified into 4x+4h – 4x = 4h and then put it back on top of the denominator and it now looks like this:

 cancel out the h’s and you have your new solution

Now that it’s simplified if you need to find the Derivative/Slope of the Tangent Line/Instantaneous Rate of Change you can take the limit of the difference quotient:

and you get aka

My brain hurts.

My Brain on Math

Today I made the realization that I speak equations better than I speak math in sentences. Derivatives/Rationalizing square roots, etc always makes more sense when I see it laid out nice and pretty in mathematical symbols. Am I alone in this? I don’t know.

As our Prof. told us, we’re supposed to write at least one in-depth blog. As I’ve been trying to understand what “in-depth” might consist of, I seem to have calculus and this blog on my mind all the time. I am actually really enjoying calculus, and thus far have not found myself frustrated or lacking the ability to understand what the teacher’s explained. I really like her method of teaching; she puts things in words and in symbols and writes it out while she talks about it. Seeing it employs multiple senses, which is something I find to be valuable since we all learn in different ways. After many years of school and learning, I find that I learn best in a kinesthetic/visual way. I have to see it, then I have to do it, and then it’s in there. Even better if I see and do it at the same time (following someone). So this method is especially valuable to me. I can also look back at my notes anytime I get stuck on homework and find the solution I’ve already done, remember how I did it, and duplicate it for the new problem.

As I’ve been thinking about Calculus a few connections have started clicking for me. Firstly, I used to think I was a very right-brained thinker. I was quite emotional as a teenager, and found myself drawn to psychology. However, as time has gone by, I have realized I seem to be more left-brained after all. Math comes easily to me. I prefer logical arguments over emotional ones. I enjoy a + b = c. Some of my favorite hobbies have been connected to having a mathematically adept brain, such as playing the piano and oil painting. I do both of these. I’m not saying either of them is specifically left or right brained, but oil painting does require the ability to map out a canvas in a geometrical sort of way and make it look like it does in whatever you’re copying onto the canvas. Piano is absolutely mathematical. Piano is a combination of counting and hitting notes that work together musically, to play it requires that you only have the ability to read the music and time your fingers accordingly, but to write it requires much higher levels of mathematical abilities and intuition. I’ve tried writing a few songs and have to say it’s quite difficult, but thrilling to write music. Both also require creativity and soul to make your work BETTER than other’s. So I would have to say they use both sides of the brain, but you need some mathematical ability to do both and be good at them.

One thing that I’ve found difficult is going from Philosophy to Calculus class (they’re back to back). It’s like I’m having to switch from one side of the brain to the other and I find myself feeling a bit relieved to get out of Philosophy and get to some logical thinking and real conclusions. See, philosophy is like a math problem that never solves. It’s like you’re constantly factoring and never reaching a resolution, because I might see that a + b = c, but my fellow classmate might see that a + b does not equal c but instead equals LMNOP. In other words, you can logic out any argument on any side of the argument because opinions are infinitely variable. At least, that’s how it looks to me. So I sit in Philosophy arguing out whether or not homosexuality is moral, and then I run to calculus and learn how to find the derivative of a square root and I feel relieved about it.

Last, I go to Chemistry. Chemistry is a whole lot of math. I thought it would only be a little math (i.e. conversions), but even today we did more math. Sure, its simpler math than calculus, but being in calculus feeds my ability to do the math easily. Even today, we were learning to balance reactions. Basically what goes into a reaction has to come back out of a reaction with the same number of atoms. This is identical to algebra. So for example you have . You have to balance this equation here because at the moment it looks like you’ve got 2 Na on the left but only 1 Na atom on the right. So you have to rectify this by making it 2NaCl on the right. Then you have to make sure you have the same number of all the other elements on both sides. It looks like we have an extra Cl on the right now thanks to our 2 in front of NaCl, so now we have to even that out and put 2 in front of HCl. Now we have an even number of H and O so we’re fully balanced. So in the end you get this equation:

It makes beautiful mathematical sense. And I realize I probably would have figured this out without calculus, but I just really like that these classes tie in so well together.

Anyway, this is my brain, and my thoughts on math. I also found myself thinking of calculus when I heard this song… I want to somehow change the lyrics to make a calculus song soon… guess we’ll see if I can get creative enough:

Just in case you wanted to know the equations from her lyrics… I decided to write it out latex style. Hehe.

I couldn’t help myself. I promise I just posted this because she was talking about math… I’m actually not trying to tell anyone they’re dumb.

 

We’ve Finally Arrived at REAL Calculus

Here we go… finally into unknown waters. Wednesday was the first class day where I wasn’t going “Oh YEAH!” and instead was going “Oh… Ok, this is new.” Luckily, I followed the lesson well, and have it down (homework problems helped solidify it for me).

So derivatives combine the stuff we learned about limits (I knew this was going to tie in somehow) and the average rate of change of a function.

Things I found helpful from the lesson:

• The limit of the average rate of change of a function as h approaches 0 is the instantaneous rate of change.

This can be said in equation form like so: where is notation for instantaneous rate of change.

• The limit of the difference quotient of a function as ha approaches 0 is the derivative of that function.
• Instantaneous Rate of Change = Derivative
• The derivative of a function is another function.

Different Notations that all pretty much mean derivative/instantaneous rate of change:

  which is read “f prime of x”

which is read “dy dx”  (Remember here change in y over change in x? aka slope? yeah… same thing)

read “D sub x”

read “y prime”

read “ddx of f(x)”

So basically the average rate of change, slope of the secant line, and the difference quotient are all the same thing.

And the instantaneous rate of change is the same thing as the derivative.

Now that we got all of that out of the way, how the hell do we solve for a derivative? I’ll show you…

But 1st… a quick photo break…. I relate to this so much.

Anyway… I loved this because it spoke to my internal fear of not knowing what’s going on in class. On to me explaining how to find a derivative.

Let’s say you need to find the derivative for a function such as…

First you’re going to want to find the average rate of change for this equation. In this case finding the average rate should go something like this:

1.

2.

3.

4.

5. which =

Now that you have you Difference Quotient you may plug in the awesome new equation for instantaneous rate of change and solve like so:

1.  

2.  aka

TADA!!!!!

If any of you don’t understand this please comment and I’ll try to help you the best I can.

Limits of Rational Functions

I may end up writing more than 2 blog posts per week. I’m not trying to be an overachiever (Ok maybe I am). I think that by writing about what we learned in the last class, I can help solidify it in my mind and perhaps my viewpoint could help a classmate. So here’s an update from last class.

#1. I got my quiz back and got a 10 out of 10!  Woohoo!

#2. We learned about the Limits of Rational Functions and here’s what we learned:

Properties of Rational Functions:

• Rational Functions are continuous everywhere that they are defined.
• They can have Vertical Asymptote discontinuities OR
• They can have Point Discontinuities
• Rational Functions typically look something like this

What’s important to remember, is that rational functions are continuous everywhere EXCEPT where the denominator is equal to zero. Aka if you solve for x on the above equation, and you get you have a discontinuity.

But what does that mean for limits? Well, let’s look at the two types of discontinuities we are dealing with for a rational function.

Vertical Asymptote: A vertical asymptote occurs when a graph has two lines that never cross a certain point of x. It looks a lot like this:

Thank you http://www.analyzemath.com for the photo

Please note that this discontinuity of a function means the function has no existing limit.

Point Discontinuity: This is where a point on the function is not passed through by the line on the graph. For example:

Thanks to Wikipedia for the image

Note that at there is an unfilled circle, that is the point discontinuity. It is important to know that despite this discontinuity, a limit still exists for this equation, because limits do not care about the exact point, but only leading up to the point.

So what we’ve reached thus far is that when dealing with a vertical asymptote, a limit will not exist, but when dealing with a point discontinuity a limit will exist.

How to find the limit/discontinuities of an ungraphed equation:

Let’s take the equation

*In order to find the discontinuities, set the denominator to zero.

the discontinuities are at x=0 and x=2

you can also simplify the function as an algebraic expression to find g(x) like so:

= so now f(x)=g(x) everywhere they are DEFINED but they are not the same function.

Once you’ve simplified the function, you can use the simpler function to find the limit of the more complicated function.

Typically, if you do this, and can cancel out a factor, it’s a point discontinuity, but otherwise it is a vertical asymptote and no limit will exist. If you substitute your c into a rational function and you get where n is any non zero number, then there is a vertical asymptote at x=c and the limit of f(x) at x→c does not exist.

If you get factor it out into a g(x) and you should be able to find a limit.

If this seems like a lot of gibberish, the best way I can show you is by doing 2 problems and explaining it that way. Here we go:

which means it’s a point discontinuity and we can solve it using a factored version of the equation:

to which we can plug in x=3 and get

Problem #2:

this is a vertical asymptote, meaning there is no limit for this equation so

 

Chemistry & Calculus

I did not realize this when I signed up for the two classes this semester, but I get the feeling that being enrolled in both calculus and chemistry this semester was a great idea. I didn’t remember how much math was involved in chemistry until I was going through my first homework assignment and realized that conversions are doable through an algebraic equation. I know algebra is not quite as advanced as calculus… but I have a feeling that as I delve deeper into Chemistry (Chem 2, and then Organic Chem 1 & 2) I will need more and more calculus level math.

To show you what I’m talking about, I got a problem on my chemistry homework that went something like this:

Convert into .

Like a deer in headlights, I felt my heart sink. See, I have this little problem. I have the WORST ability to convert things. I cannot explain this phenomenon but basically what happens is when trying to convert say g to kg, I end up doing the opposite of what I was supposed to do… I think, ok 1 gram is 1000 kg, so to convert a kilogram into a gram, I simply multiply it by 1000. WRONG. I don’t know why I do this. Then the margin of error goes up when I try to convert the measurements separately, and end up with two wrong numbers on top of each other. I did at least 10 of the stupid chemistry problems about 20 times each until I finally got it right. But now, after my god-send of a chemistry teacher showed me the light, I find myself making no errors. All I have to do is line it up algebraically. So here is what it looks like algebra-style.

by cancelling out grams and centimeters cubed I was able to get to kg over cubed meters. This was much more coherent for me, lining up the numerator and denominator in a way that they continue to cancel each other out until you get the numbers you need. My chemistry teacher taught us this method and I think it’s the most genius method I’ve ever seen. No one has ever given me an algebraic way of converting… I’m thrilled!

Professor M. asked us to look at calculus in ways that it helped us in other areas of life, including other classes, and already I am seeing just the refresher we had at the beginning of the first week helping me out in chemistry. Perhaps having these two classes will help gain a deeper knowledge and retention of both. One can hope.

I get it…

When I first heard we were doing blogs for Calculus I was a bit skeptical. Why??? I tried to be neutral about it. I wanted to understand what our professor’s end goal was with this, and today it really hit me. I think it’s working. I love learning in unconventional ways. I used to hate how I would go to high school and be forced to follow a certain formula of learning and most of that was based on passing the SOL’s (Standard of Learning tests) instead of actually caring that we learned something worthwhile. I’ve also been disappointed in the amount of knowledge I never learned.

Example #1. I went to Germany when I graduated High School back in 2004 (don’t ask… I took a hiatus). I took an intensive learning class for 3 weeks and was the only American in my entire class. We had people in my class from Switzerland, Japan, China, Russia, Czechoslovakia, Italy, etc. and most of them did not speak English (surprising to me at the time). The teacher would ask me to tell him how tall I was, and I had no idea how to tell him in the metric system, so everyone laughed at me when I told him in inches how tall I was. Later, I was asked what the population of my home country was. I didn’t even know. I guessed at 1 million (still kicking myself over that one) and people laughed at me again. I just never learned this sort of thing in school. And this makes me incredibly angry.

I hate the way our public schools do things. I hate how everything is based on test scores and no one actually cares if we are LEARNING.  The “No Child Left Behind” BS only made it worse. Hear me out here. Some children need to be left behind. Some children are smarter and need to be pushed ahead. It may not be fair, but life is not fair. Some people are born beautiful, some people are born ugly. Some people are born smart, some people are born with only a brain stem. Life is not fair. So let’s not act like we’re all the same. Doing this sets us back as a society. We need to harness our strengths and work on our weaknesses even harder. I’m not saying weaknesses can’t be overcome, but if you pretend they don’t exist, or that a child has learned what they needed to learn when they have not, you do that child a disservice.

I had a point, I am trying to get back to it I promise. My point is, I think the professor has branched out, and taken a chance on us. I think that by doing these blogs we are able to learn in our own ways, and learn from each other. Instead of just handing us a book and telling us to read it, she’s making us read it and (if we do this the way it was intended) think about it, and apply it, and share the learning process with each other. Maybe someone else who blogs will come up with a better way for me to understand a certain concept, or link something that really drives the message home for me.

I also find that learning something using multiple senses/parts of the brain helps me to retain the knowledge. Not only can I visually learn math, I also get to write it out, and then type about it, while considering new ways to blog about it to keep this blog from being monotonous and make sure I get a good grade. Suddenly I realize the brilliance behind the plan, and I applaud my teacher for taking a risk, and not being a stereotypical math teacher. I feel lucky to be the first semester to do this project.

Anyway, I am really happy because the Quiz on Wednesday went really well for me, I feel like I have limits down and I remember functions a lot better. With that, I will leave you with a fun math-related photo I found, and I will try to write a little more about actual Calculus on my next post.

Learning Limits and Continuity

Today I finally got my Calculus book. Yes, it’s the one with the eagle on it… a majestic bird, but I’m not sure why it’s on a calculus book.

I also did the homework on Math XL today and found the website to be VERY helpful for me in solving the math problems. You never realize how much you have FORGOTTEN about math until you need all that basic knowledge to move on to higher knowledge. I’m irritated I have neglected math for so long… because deep down inside I find myself liking it (a lot). Math will never lie to you… not if you put the right numbers in and use the right equation. I like that about math. See, there are a lot of unknowns in this world, mysteries of life… I even read this fascinating article about 6 scientific discoveries that defy the laws of physics. It’s a fun article. But math.. you solve a problem and it’s SOLVED. You can rest easy at night and maybe even get the recommended full 8 hours of sleep (what’s that like? I can’t remember…).

Anyway – on to Limits and Continuity. Limits are pretty sweet. It’s defining the height of a graph at a particular point on the graph. The limit from the left enters the point at a certain height, and then enters from the right at the same or different point. Easy. Then it gets a little more complicated. Does the line enter the point from the same height on both sides? If so… it’s probably continuous, but not necessarily.

Continuity is also a relatively simple concept. Easiest way to define whether something is continuous is whether or not you can draw the line without picking up your pencil. If you have to pick up your pencil, it’s not continuous. So if the line passes through the point in question (x), and continues on… it is continuous. Easy.

But then you have to complicate it for me. How do we determine the limits without looking at the graph with an equation? What if we change the equation from a polynomial to an exponential function? What is a polynomial function? What’s an exponential function? HOW DID I FORGET ALL OF THIS???

It’s true… that’s where I was about 8 hours ago… but luckily I have a brain and was able to look it all up and thought I would share the little nuggets of information with you guys in case you had a massive brain fart like I did.

1st thing I kinda forgot. A function f(x) = y. So when you solve for x, you get y. Capiche? Also.. if a vertical line (notice I did not say horizontal here) intersects a graph in more than one point, the graph is not the graph of a function.

I also forgot the differences between types of functions so I listed them out here:

1. Linear Function:

2. A Quadratic Function: where a, c, and c are real numbers and .

3. Polynomial Function: where are real numbers called coefficients, with .

4. Rational Function: where p(x) and q(x) are polynomial functions and

5. Exponential Function: where and

These can all be found in Chapters 1 and 2 of our Calculus Book, but I find it useful to have these handy for solving problems, and I find things slowly coming back to me. I guess you have to force yourself into this full force or you won’t survive… at least me. I wish I were a genius with an amazing memory and a sponge-like brain, but I’m just not.