Business Calculus Chapter Zero

Transcription

1 Business Calculus Chapter Zero Are you a little rusty since coming back from your semi-long math break? Even worst have you forgotten all you learned from your previous Algebra course? If so, you are so fucked!!! Well, here s something to get you all caught up.

2 Table of Content Section 1: Algebra Factoring Simplifying Shorter way of FOIL with squares Section 2: Unlearning Bad Math Slope = Derivative 1/0=+-Infinity 0/0=Do More F%#King Algebra to cancel out the trouble maker causing that to occur Square root of 4 = 2 But x^2=4 Square root of x^2= Square root of 4 x=+-2 Section 3: Basic Math Memorize cube table Order of operation is the abbreviation "PEMDAS", which is turned into the phrase "Please Excuse My Dear Aunt Sally". It stands for "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction" Section 4: Number Lines Interval Notations Parentheses versus brackets (What s the meaning of this non-sense) Section 5: Log Properties (AKA Logarithmic Properties) Multiplication = Addition; Division = Subtraction; LNx^2=2LNx

3 Section 1: Algebra The hardest thing about calculus is the algebra. Anyone can easily learn calculus, even your 80 years grandma but the main hurdle for her is recalling her long forgotten algebra skills to finalize the questions. Factoring, now the good news for you is that for the purpose of Business Calculus you need to know only three factoring techniques: Basic pulling out a term factoring, the difference of perfect squares factoring, and trinomial factoring. We going to go over of each of these factoring techniques one by one starting from simplest to hardest. Basic pulling out a term Factoring For example: 2x 2 + 8x 2x(x + 4) Basically you first establish what is the smallest number that each term (2x 2 and 8x) has in common. In case the smallest but greatest number you could pull out is 2. Next you want to determine what is the least amount of x s from the two terms (2x 2 and 8x) which is quite easy to see is just one x.

4 The difference of perfect squares Factoring x 2 y 2 = (x y)(x + y) Or another it look at as is x 2 y 2 = ( x 2 y 2 ) ( x 2 + y 2 ) = (x y)(x + y) For example: x 2 4 = (x 2)(x + 2) Or another it look at as is x 2 4 = ( x 2 4) ( x 2 + 4) = (x 2)(x + 2) 49x x 2 100

5 Trinomial Factoring Easy Case x 2 + bx + c The key to factoring these trinomial is to read the numbers starting from the back working your way to the front. It should as follows: Factor of c that adds (if positive) or subtract (if its negative) to give you b. For example: x 2 5x + 6 You re looking for factors of 6 that adds up to give us negative 5. The general quick tip for finding the signs of these factors is that if the last number is positive then both of the factors will carry the same sign as the middle number. For example: x 2 5x + 6 a) 1, 6 b) 2,3 Which of these factors adds up to 5? Answer is 2, 3 results being 2+3=5 And since the last number of the trinomial is positive that means the two factors are negative due to the middle number being negative 5. (x 2)(x 3) When in doubt always check your work by FOILing out the equation to see if you derive the original equation you started off with.

6 x 2 7x + 12

7 In addition, if the last number in a trinomial is negative then that means that one of the factors will be negative while the other factor will be positive in order to such that subtraction of the two would result to the same number as the middle number. For example: x 2 5x 6 Which of these factors subtract to give us 5? a) 1, 6 b) 2,3 Answer is 1, 6 results being 6-1=5 The last number of the trinomial is negative 6, which means the two factors have will alternating signs (one negative / one positive). An easy trick to determine what number get the positive or negative sign, it to remember that the bigger number of the two factors will carry the same sign as to the middle number being. And finally since the middle number is negative 5, therefore leaving the bigger factor 6 also as a negative number. (x 6)(x + 1) When in doubt always check your work by FOILing out the equation to see if you derive the original equation you started off with.

8 x 2 11x 42

9 Harder Case ax 2 + bx + c There two different of factoring these harder cases. The most popular method involves brute force trial & error while the other least known approach is non-orthodox but is very systematic. Both ways have their various pros or cons base on the Individual trinomial problems. Step 1: Get rid of that annoying first number by multiplying it with the last number to make a much easier trinomial to factor. Example: 3x 2 + 4x 20 x 2 + 4x 60 Step 2: Now factor that trinomial normally Example: (x + 10)(x 6) Step 3: Put back the original first number that we remove by placing it front of both x s as co-efficient numbers. Example: (3x + 10)(3x 6) Step 4: Factor out any excess numbers and eliminated it. Example: (3x + 10)3(x 2) (3x + 10)(x 2) When in doubt always check your work by FOILing out the equation to see if you derive the original equation you started off with.

10 2x x + 12

11 FOIL = First Outer Inner Last

12 Longer way of FOIL with squares (x + y) 2 = (x + y)(x + y) = x 2 + xy + xy + y 2 = x 2 + 2xy + y 2 Shorter way of FOIL with squares (x + y) 2 = x 2 + 2xy + y 2 Longer Way (Very Common) (x 8) 2 Shorter Way (Much More Efficient) (x 8) 2

13 Section 2: Unlearning Some Bad Math Slope y 2 y 1 in most cases unless we are speaking of linear equations. In the real world nothing has x 2 x 1 the shape of a linear straight line unless we re speaking of wife s flat booty. Even current so called flat panel televisions are not really flat because of manufacturing imperfections it will have slight curve. The correct the way of finding the slope of curve from an equation or graph (also known as a tangent slope) is by using a calculus technique called the derivative. All your math life you been told to associate slope with that old school formula of y 2 y 1 x 2 x 1, but in this class you have to unlearn that and now on think of slopes as forms of derivatives. Slope derivatives (after plugging & chugging your target x number). OMG, what is this mysteries derivative thing you keep referring to. Well it s a pretty straight forward calculus method you soon learn about that will be first introduce using algebra but later you ll learned the much way of arriving to the same without algebra.

14 Another slight misconception is 1/0 is undefined. That was an okay statement is previous other math classes, but in calculus where we re dealing with limits when saying it s undefined would actually be incorrect. 1 0 = ± Or a better way of putting it is that Any Number 0 = ±

15 Section 3: Basic Math Computation The great new about this class is that there is no calculator, which can be quite an adjustment for those of you rely heavily on their calculator. But trust me, you would prefer math courses where calculators are not allow because if they were the questions would be harder and expect a lot more from you. Of course, you already know your times table and memorize them like the back of your hand. However, how many of you truly know your cube tables too? 2 3 = = = = = = = = = = = = = = = = = = 1000 Memorizing all your cubes will save you a lot of precious time during those final crucial plug and chug steps. Running out of time is one of common reason students report not doing so well on their recitation quizzes or exams. Bonus: Knowing most of the two s can also be very helpful. 2 4 = = = 64

16 Divisible by three rule A number is divisible by 3 if the sum of its digits is divisible by 3. See if the following number: is evenly divisible by three = 3 And 3 is divisible by 3 so the number 12 is also divisible by = 9 And 9 is divisible by 3 which means that 36 is also. Question: Is 792 divisible by three?

17 Fractions

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20 ( 6 11 ) 2 9 4

21 Order of operation is the abbreviation "PEMDAS", which is turned into the phrase "Please Excuse My Dear Aunt Sally". It stands for "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". This is what you do from left to right. 3[2(7 5) 4 ( 2) + ( 3) 3 ]

22 5(3 + 1) 2 2( )

23 Section 4: Number Lines Being comfortable with expressing your final answers you got using number lines into interval notations is extremely important for this class.

24 x < 1 x 7 x > 1 x 0

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27 Section 5: Log Properties

28 End each sections with practice problems. Finish this packet by giving real exam/quiz question they would encounter in this course. Making a few of those examples of business calculus cases where we apply a combination of the techniques from the various sections.