Containment and Equality
If A and B are sets, then A is said to be contained in B iff (if and only if) every element of A is contained in B. So A⊆B means that A is a subset of B.
Example:
All squares ⊆ all rectangles
All right triangles ⊆ all triangles
Important! This implies the idea of forwards and backwards logic: If Joe has three million dollars, he is a millionaire. If Joe is a millionaire it doesn’t necessarily mean he has three million dollars, he could have one million dollars and
still be a millionaire. Likewise, all squares are rectangles but not all rectangles are squares.
A=B iff A⊆B and B⊆A
Example:
{x:x^2=4}={-2,2}
{x:x^2<4}={x:-2<x<2}

Sets and Other Elementary Subjects
Sets are a collection of things called objects. Objects are all unambiguously defined. In other words, objects have unmistakably clear definitions with one meaning and one interpretation that leads to one conclusion. This may seem convoluted because we are
so used to words and phrases having different meanings and whatnot, but not in this case. Look at some examples to get a better idea what it means for objects to be unambiguously defined.
Objects Not Objects
Cars Cool cars
Children Nice children
Temperature Comfortable temperature
Baseball players ...
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Hello Students!
Start this year off strong with good organizational and note taking skills. Make sure you understand the material and are not just taking notes aimlessly. Try to take in what your teacher is saying and don't be afraid to ask questions!! If you start taking
the initiative to learn and understand now, college will be a much more pleasant experience for you. Trust me!
Stay organized and plan your homework and study schedule!
Quiz yourself!
Study with friends!
READ YOUR TEXTBOOK! :)
Remember, homework isn't busy work and a chance to copy down your notes, it is part of the learning process. This is especially important with math, as it builds on itself and understanding the basics will make the other subjects easier!
Have a fantastic and fun year!

Today, the future depends on you as much as it does on me. The future also depends on educating the masses in Science, Technology, Engineering, and Math, otherwise known as STEM. As a new tutor to WyzAnt, I hope to instill the importance of these subjects
in student's lives, as well as, the lives around them.
Besides the fact that, "the average U.S. salary is $43,460, compared with the average STEM salary of $77,880," (Careerbuilder) these subjects are interesting and applicable to topics well beyond the classroom. Success first starts with you; I am only
there to help you succeed along the way. STEM are difficult subjects. Yet when you seek out help from a tutor, like myself, you have what it takes to master them.
Please enlighten me on students looking to achieve and succeed rather than live in the past and think I can't as opposed to I can. We can take the trip to the future together, one question at a time

All too often, calculus textbooks misrepresent the proof this formula:
\frac{d}{dx} e^{x}= e^{x}
The texts by Finney, Demana, et al. usually introduce, without explanation or proof, the limit below:
\lim_{h \rightarrow 0} \frac{ (e^{h}-1) }{h}=1
The problem with this approach is that it deprives the student of key concepts regarding the exponential function, ex . The student often thinks of e as the number that is 2.71828...... because Precalculus and Calculus teachers define it as
such. However, that definition is not the logical definition, but a mere incidental byproduct. The logical definition of e is the exponential base in the function whose tangent line has a slope of 1 at x=0 in the function f(x)=ex . The calculation
of the numerical value of this base to be 2.71828... is a later development that results from the definition. It is not the logical definition.
Khan Academy sidesteps this...
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There's no such thing as the square root of a negative number. Right?
Since squaring a number is defined as multiplying it by itself, and multiplying a negative times a negative gives a positive, all squares should be positive. Right?
So any number you want to take the square root of should be positive to begin with. Right?
So what if it's not?
What do you do if you're chugging through a problem and suddenly find yourself confronted with
x = √(-9)
It seems like to finish this problem we'll need to take the square root of a negative number – but we can't, so what do we do? Drop the sign and hope nobody notices? Mark it as 'undefined' like dividing by zero? Give up? Cry?
Well, actually, we don't have to do any of that, because we've got an imaginary friend to help us.
Meet i.
i is a mathematical constant, whose sole definition is that i2 = -1. Or, in other words,
i = √(-1). i is an imaginary number...
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When it comes to using a legitimate online resource to help with tutoring mathematics, or answering mathematical questions I use Wolfram.com.
This website is very diverse and allows the user to input any mathematical equation, formula etc.
With subject areas of mathematics, such as calculus, Wolfram.com has proved to be extremely beneficial, especially when working with difficult integrals and derivatives.
With the Pro version of this website, which is well worth its value, you will be provided step-by-step instructions on how to solve the particular problem that you have inputted.
Check out this website and explore the countless benefits it has to offer.
Keith

I'm so glad my GMAT student improved upon his score!! He's in for BIG things...
Now, I'm looking for college students to tutor! I want to see even more successes this year!!
There is is a reason the student-to-teacher ratio was small in ancient times. IT WORKS!!! :)

I am a University of Utah mathematics major and I love the word FREE. (cheap is good too)
I don't have a lot of money so any Free resources to help me study are worth it to me. Since I know a lot about mathematics that is what I will be posting here.
The key to Mathematics is Learning, Practicing, Learning, Practicing, and sometimes it goes in the opposite order: Practicing, Learning, Practicing, Learning. But either way a good resource to me has a bit of both: they teach you how and why you do something
and they make you do it as well. A really good resource will teach you how and why, make you try it, and then will show you why you got it wrong and what you should have done, and then make you do more problems of the same type. So then, without further ado,
here are the resources:
Paul's online notes (type it in google it will be one of the first to pop up)
http://tutorial.math.lamar.edu/
his notes are free, come with worked out...
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I've started brushing up on Calculus. I studied Calculus in high school and took two semesters in college, bu that was forty years ago. It's really interesting how persponal memories pack themselves in along with Diffrential Equations and Integrals.

Hello everyone,
One of my Calculus students had an interesting Related Rates problem that I had to go home and think about for a while in order to figure out. The problem was set up as such:
A 25 inch piece of rope needs to be cut into 2 pieces to form a square and a circle. How should the rope be cut so that the combined surface area of the circle and square is as small as possible?
Here's what we'll need to do:
1. We will have to form equations that relate the length of the perimeter and circumference to the combined surface area.
2. We will then differentiate to create an equation with the derivative of the surface area with respect to lengths of rope.
3. Wherever this derivative equals 0 there will be a maxima or minima, and so we will set the derivative = to 0 and determine which critical points are minima...
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You can find some really good resources for math test prep in the used bookstores in a college town. Some examples that I like are: (1) Humongous Book of ______________ Problems (fill in the blank with your math topic); (2) the REA Problem Solvers series;
and (3) the Schaum's Outlines. If you don't live near a college town it might be worth a Saturday trip just to buy books. Alternately, all of these are available (used) through the Amazon Marketplace sellers at really low prices.
You should preview each title of these book series that you might be considering to be sure you like the authors style. Each one is different. You may like one series' treatment of Pre-Calc but prefer a different series for Calculus.
So how do you use these books ?
They are an alternate resource for explanations of basic concepts and problem solving techniques. You should use them as 'hint mills' and sources of problems to...
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You'd think that, "If I'm paying for tutoring, he should be answering MY questions. Not the other way around."
While I can sympathize with the general sentiment, I'd say,"you're way off base there!"
I think that the tutor/teacher/coach should never ask the student directly,"Do you understand __________ ?" Not knowing the subject matter, how would the student know/evaluate/determine if they understood or not ? Generally they can't, that's why the need
a tutor. Rather than ask about specific content, directly, I ask questions to determine if the student understands the material and how the pieces fit together. Sometimes that's five or six questions.
Here's my general GAME PLAN: Find out where they are. Tell them, show them, then see what they heard and saw.
When your tutor's asking you questions, he/she is probably working the same kind of plan. You can help them help you by always providing the syllabus...
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Hi All!
In the spirit of giving, starting on 11/29/2013, I will be offering a few brainteasers/ trivia questions where the first 3 people to email me the correct answer will receive a free, one hour, tutoring session in any subject that I offer tutoring for (via
the online platform)! That's right free! Get your thinking hats on everyone!
Merry Christmas!!
Andrew L. Profile

Another assignment meant another stressful evening. I was 12 weeks into AP Calculus and I was so worried I wasn't going to be able to understand the class. So far I had completed the assignments, but I never felt I understood what I was doing. Our assignment
was on rates of change. Hours went by and I was still trying to figure out the problem. How do I even start?
It was like a door opened and light flooded in. I knew how to do it! I wrote my steps down, checked the answer in the back of the book, and there it was. My answer matched! It was one of those moments when your confidence soars. It seems silly now
that I got so excited about solving that one problem, but I consider that moment a defining moment when I knew I could be good at math.
The rest of the year was still challenging, but I felt like I knew how to get better at solving math problems: do as many problems as I could from the...
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I have many students tell me that they are afraid to ask a stupid question in class. I tell them that there are NO stupid questions, only stupid mistakes because you didn't ask the question! Too many smart students in Calculus think that by asking a
question they will appear weak. What most students don't know is that probably there are many other students with the same question in their head that they are afraid to ask! The look of relief on other students' faces when some else asks a question is amazing.
Another thing smart students have a problem with is writing down each step. Newsflash: by the time you get to Calculus you can no longer do the problems in your head. Calculus problems generally are difficult because it is not just a matter of memorizing
a formula and applying it. In Calculus you are expected to extrapolate the knowledge you have learned to problems you have never seen before. This is...
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I used to do this and I see a lot of students who do this common mistake when studying. Maybe you are working through old homework problems to prepare for an exam in math or physics and you have the solutions in front of you. You get to a certain point
and you get stuck, so you check the solution, see what the next action you have to take is, and then continue working through the problem. Eventually you get an answer that may (or may not) be right and check the solution again. If it is, you feel great and
move on. If it isn't you compare the work and see what you did wrong and understand the mistake so you move on. All this is a fine way to start studying, but the major mistake is that most students don't go back to that problem and try to do it again. Even
if you were able to understand the solution or the mistake you made, you never actually got through the problem completely without aid. So now if you come to this problem on your test, this will be the first time you actually...
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Naturally, Anything new can be challenging. For example, Calculus, now my favorite math topic, once was something somewhat confusing. How did I master Calculus? By asking people around me to explain it. The trick, at least for me, isn't how you explain
it, it's how you define it. When someone finally stated "the change in y with respect to x" I finally understood. It was an immediate understanding of all concepts of calculus.
So what my secret? It's learning everyone elses secret until I find one to make mine!

I've heard this sentiment over and over--sometimes from students, and sometimes, I'll admit, in my own head.
Last night, I was working on my own math homework, and there was one problem I just couldn't get my head around. I read the book, looked back at my class notes, and even sat down with a tutor for a while, and still, when I tried a new problem of the same type
on my own, it just didn't work!
"Maybe I'm not as good at math as I thought," I told myself. "Am I REALLY smart enough for bioengineering?"
It was hard, but I told myself "YES!" And I kept working. I laid the assigned problems aside and started doing other problems of the same type from the book. I checked my work every time. Each problem took at least ten minutes to solve, and the first three
were ALL wrong! I kept going. I got one right, and it made sense! I did another, and it was half right, but there was still a problem. I did another, and it was right...
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Developing a grounded understanding of numbers, and number operations provides the firmest foundation for learning math. Touching, seeing, and manipulating physical objects are perhaps the surest way to accomplish that in the beginning.
Developing the practice of drawing pictures to reflect an arithmetic or story problem is the next step and soon becomes a central tool for thinking through a math problem whether represented in math and science, or encountered in life.
Finally, talking about, through, and around math, arithmetic, problems, and solutions is equally important to proficiency in math and any other area of education, socialization, and life.
It is important to recognize the preferred learning style of each student in order to achieve the best opportunity to that student’s learning and performance. Yet, excellent teaching includes multiple approaches and learning styles on the way to each student’s
full facility, proficiency, and confidence. This necessary...
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