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Everything That You Should Already Know  

Chat about Calculus

In this section Im going to outline all the simple little rules that you are supposed to know already, but may have forgotten.

Exponents

OK, this is simple enough. I will work this by example.

25 * 23 = 28 When multiplying two of the same number, with exponents, add exponents. Note that a more general form is x5 * x3 = x8. An even more general form is xy * xz = xy+z.

25 / 23 = 22  When dividing two of the same number, with exponents, subtract exponents. 

 
 

(25)3 = 25*3 = 215 When an exponent goes on a parenthesis containing one number or variable, multiply the exponent inside by the exponent on the outside. If a number does not have an exponent, then its exponent is actually 1, and 1 * a = a. Another example is:

(25)1/3 = 25/3

21/2 = 2     21/3 = 3√2     23/4 = 4√23 In this last case, it doesnt make a difference whether the exponent 3 is outside or inside the square root. All square roots can become the number to the exponent 1/2. All cube roots can be the number to the exponent 1/3. If an exponent is a fraction, then the upper number becomes an exponent, and the bottom is the root.

2-4 = 1/24 = 1/16

2-4 / 5 = 1 / (24 * 5) = 1/80  Any negative exponent means you can flip the number underneath a fraction, and make the exponent positive. If it is a whole number, as opposed to a fraction, place a denominator of 1 beneath it, because any number divided by 1 is equal to itself. Then multiply the 1 underneath by 24 and remove it from the top.

e-1 = 1/e        3-1 = 1/3        x-1 = 1/x        x-1/2 = 1/√x       x-2/3 = 1/3√(x2)

Some of these are really complicated to type out! If you don't understand it in this format it may be better for you to write the problem down on paper, in a normal fraction, not just a slash that this format constrains me to. 

 

Functions with e and ln

e is a number, equal to 2.71828, and is not to be confused with a variable, such as x.

ex refers to e to the power x for all values of x.
This is similar to 10x referring to 10 to the power x. 

Logarithms are the opposite of an exponent. For clarity, we will first deal with base 10, because you are naturally familiar with that. Log10 x is the reverse of 10x. When I say log10 100, I mean how many times to I multiply 10 (the base!) to get 100. The answer is 2, so log10 100 = 2. Log 10000 = 4, because 104 = 10000. Notice that I could have said log 104 = 4. It is not a coincidence that there are two 4's there. It doesn't always have to deal with whole numbers, though. !02.5 has a result of 316, so log 316 = 2.5. Notice that I didn't bother to write the base. (a subscript 10 in log10) All problems that you will deal with will most likely be base 10, or base e. When we say log, we usually refer to base 10; when we say natural log, or ln, we refer to base e.

Dealing with base e is not at all different, except the numbers are not in terms of 10. Loge x, otherwise known as natural logarithm, (ln) is the reverse of ex. ln e3 = 3. Since the two functions are the reverse, they can cancel each other out, as in the functions eln x and ln ex, which both are equal to x. Also, ln e is equal to 1, because you multiply e by itself once to get e. Thats all you really need for the moment. Just realize that e gets treated like its a number in all functions. In derivatives, learned in the beginning of calculus, e has a special quality which will be discussed.

Natural Log functions come up very frequently, so I use them for these handy rules, but these rules will work for base 10, or any other: 
ln 3 + ln 6 = ln 18 = ln 2 + ln 9 = ln 18 + ln 1 Multiplying on the inside of the log is the same as adding outside the log.
ln 3 - ln 6 = ln (3/6) = ln .5  When subtracting ln's, you divide.
ln ab = b ln a    When the number inside the ln has an exp. you can pull it out to the front. This will be crucial for taking some derivatives later on. ln e3 = 3(ln e)

 

Fractions

A fraction has two parts: the numerator (top half) and the denominator. (bottom half) 

When multiplying fractions, you must multiply the two numerators, and multiply the denominators. The resulting fraction is your answer. Note that in the following examples, once again, I am constrained by this format and I present all fractions with a slash. The denominator is surrounded by parentheses. 

3/x * y/7 = 3y/7x

When you are dividing by a fraction, you can flip the second fraction and multiply by it instead. See the following examples.

7/6 divided by 12/5 equals 7/6 * 5/12 = (7*5)/(6*12) = 35/72

4/(3/5) = (4/1)/(3/5) = 4 * 5/3 = 20/3

Every number can be thought of as a fraction, and this makes dealing with equations using both fractions and whole numbers much easier. Any number is equivalent to a fraction of that number over 1. The reason, of course, is that a fraction is another way of looking at division, without actually doing the division and coming out with a decimal. So 4 is equal to 4/1. 

This has several uses. Firstly, if you are ever asked to take the reciprocal, you now have a fraction to flip: The reciprocal of 2/3 is 3/2 and of 4/1 is 1/4. This also enables you to multiply and divide whole numbers with fractions. 

4*(9/8) = (4/1)*(9/8) = 36/8 = 4.5

 

Factoring

Factoring is usually very easy. You just have to know what to do before you do it. Ill try and fill in any blanks that you dont know, but don't think I'm going to explain the truly elementary basics.

Basically, you want to find what you can pull out of every number, and multiply them all by it.

(2x2 + 6x2 16x3)

To simplify the process, I'd like to point out that you can add the 2x2 and 6x2 to get 8x2.  So the equation is really (8x2 16x3). You can pull an 8 out of both of these, and an x2. When I say pull out an 8, I mean you can divide both terms by 8. You divide both terms by 8, and multiply the whole equation by 8. As long as you can divide all the groups inside the equation by any number, or power of x, you can pull it out. So it equals 8x2(1 2x). If you reverse this, you will see that 8x2 * 1 = 8x2 and 8x2 * 2x = 16x3, so 8x2(1 2x) = (8x2 16x3) = (2x2 + 6x2 16x3) which is our original equation. 

When you are asked to set an equation to 0, you often have to factor it out. If an equation is split into different parts which are multiplied against each other, than if any one of those parts is equal to 0 the whole equation is. This is why when we set an equation to 0 we factor it into parts, so that we can easily find the value(s) of x that make the whole equation equal to 0. 

x2 + 3x 4 = (x + 4)(x - 1) Im assuming you know how to do this, the reverse FOIL. If you have x3 + 3x2 4x then realize that you can factor an x out of every term, making x(x2 + 3x 4) which equals x(x + 4)(x - 1). If you have 4x2 + 12x 16 realize that a four can be factored out of every term, giving you a final answer of 4(x + 4)(x - 1). If you have both an extra x, and an extra number, then take out both of them.

For x(x + 4)(x - 1), if x = 0 then the equation equals 0, because the first of the three terms will be 0. If x = -4 or 1 or 0, one of the three segments of this equation is equal to zero, rendering the entire equation zero.

Trigonometry

Listen, I already wrote a lot of trig in the reference table so please go there if you can't find what you're looking for. There are a few things that were too basic, so I didn't bother. Here goes.

What is a sin? Take a right triangle, and pick an angle (not the 90 degree one). The sin is the length of the side opposite to the angle over the hypotenuse. The cos is the adjacent side over the hypotenuse. The tangent is the opposite over adjacent. A great way to remember this is the acronym sohcahtoa (pronounced Sokato'ah).

tan = sin/cos     cot = cos/sin    sec = 1/cos    csc = 1/sin

tan x is 0 at 0, ∏, 2∏, etc. It is infinity at ∏/2, 3∏/2, 5∏/2, etc. Thus you can see that it a repeating sequence every ∏.

When an equation is written such as sin2 x it is the same as saying (sin x)2. How can you square a sin? That doesn't even make sense. What you must conclude is that it means you are squaring the entire phrase, so any time you come across this in derivatives, which will be often, you will most likely want to reform the equation and standardize it to (sin x)2. The reason it is written the way it is is so that people should not confuse it with sin x2 in which the exponent 2 is on the interior of the sin function, on the x itself, and not on the outside, on the sin x.

 

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