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# Blogs

## Calculus Blogs

Welcome back to the school everyone! I hope you all had a great summer. For all those whose summer was maybe a little too great, maybe those who’ve forgotten even the basics, we’re going to take it all the way back to arithmetic a.k.a “number theory”. A review of number theory is a perfect place to start for many levels. Calculus and a lot of what you learn in pre-calculus is based on the real number system. When we use the word “number” we are typically referring to all real number. But how can numbers be “real”? You can’t touch the number 6 or smell 1,063. You can’t boil 1/2 or stick it in a stew. So what’s so real about real numbers? The simple answer is this: a real number is a point on a number line (1).                               -2.5        -1      0 ... read more

When you have to take the derivative of 2 variables being multiplied, we use the product rule. If we have f(x)g(x), the derivative will be:   (f(x)g(x))' = f(x)g'(x) + g(x)f'(x).   Now, the primes can be a little confusing when you are learning how to apply this for the first time, so lets denote the number 1 to f(x) and the number 2 to g(x), where 1 refers to f(x) and 2 refers to g(x). The product rule becomes easy to memorize.    Sing it to yourself: 1d2 + 2d1   Whenever you have a "d" in front of the number we denoted for the first part of the function, you take the derivative, and if there is no "d", we simply copy the exact same function.    Example: (x+4)(x² - 1)   Here, your 1 = (x+4) and 2 = (x² - 1).   We compute 1d2 + 2d1 = (x+4)(x² - 1)' + (x² - 1)(x+4)'   Notice that the functions you need to take... read more

Calculus is easily the most complex of all the basic fields of math.   This blog is intended to help you recognize the basic calculus integration problem types, and how to approach them.   Types:   1) u-substitution: when one function is the derivative of the other function   2) integration by parts: when one function is being multiplied by another function   3) partial fractions: when you have a rational function and the denominator can be factored into linear or quadratic expressions   4) trig substitution: when you have a function that follows the pattern type of a trig identity, for example sin^2 + cos^2 = 1

I'm going to list what I believe are the key concepts that you need to master across different math subjects. These are the tools that I have to use most often in order to solve problems, so you should get very familiar with the theory behind them and very comfortable with applying them.   Algebra 1: order of operation (PEMDAS) solving equations slope-intercept form of linear equations point-slope form of linear equations systems of linear equations (elimination and substitution methods) inequalities domain and range undefined and imaginary expressions asymptotes (horizontal and vertical) discontinuities (removable and non-removable) rational expressions factoring quadratic formula radical properties exponent properties transformations and translations of functions   Algebra 2:   1. recognizing and factoring the three most common polynomial forms: quadratic equations common factor expressions... read more

Why it's important   You can use the quotient rule to answer questions like:   Find f'(x) when f(x) = (3 + x2)/(x4 + x).    What it is   I recite this rhyme to remember the quotient rule:   Low Dee High minus High Dee Low Draw the Bar and Square Below   Which means:   f'(x) = [low * dee high - high * dee low] / low2   Dee high means the derivative of the high function. You can guess which that is.   In our example, low = x4 + x and high = 3 + x2, so dee low = 4x3 + 1 and dee high = 2x.   f'(x) = [(x4 + x) * (2x) - (3 + x2) * (4x3 + 1)] / [(x4 + x)]2   That's it. That's the quotient rule.   Intuition   I like applying rules I just learned to cases where I know what the answer will be. This helps me build my confidence that I'm using the rule correctly.   x2 is... read more

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       ... read more

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 nicely,... read more

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... read more

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 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 problems,... read more

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... read more

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 make... read more

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... read more