Mathematics

Milan Patel, Michael Nguyen, Ramses Carranza, Mark Do

General Mathematics (sec 1)

• 3 topics in it

• Simple and Compound interest

• Counting with permutations and Combinations

• Probability of equally likely events

1.1 Simple and Compound InterestSimple Interest• A situation is described as using simple interest if the

amount of interest charged doesn’t change from time period to time period.

• Example: your friend has to borrow $50 to pay his cell phone bill. You loan him the money provided that every month it takes to pay you back is $5. If your friend doesn’t pay for six months and then pays it back, he would owe $80

• Formula: Interest= P times r times t. P=initial amount of money P. r=interest rate. t=time periods. Furthermore, to find the total amount of money owed at the end of the loan: P+P times r times t, or P(1+r times t).

Compound Interest

• A situation is described as using compound interest if the interest earned during each time period is then added to the principal amount when determining the amount of interest earned during the next time period.

• Example: When banks advertise a particular interest rate, they generally report the annual interest rate (APR). The interest is usually compounded (at each compounding, accounts earn the appropriate fraction of interest for the year).

• In a situation using compound interest, an initial amount of money P invested at an interest rate r compounded n times per year for a total of t years will be valued P x (1+ r/m)^nt. Note that r is an interest rate and so generally is less than 1. Also note that the exponent is equal to the total number of compounding.

1.2.1 The Multiplication Principle

-The Multiplication Rule used to determine the number of ways to select objects from different groups

-Multiplication Rule Formula: total number of possibilities is n1 * n2 * n3 …n (k-z) where n(k) is the number of ways to select the k.th object.

Permutation

• Is a selection of objects where no object can be selected more than once and the order of selection matters

• Factorial notation: n! read aloud “n factorial” and means to multiply all of the counting numbers 1 to n together

Permutations formula

• n / (n-k)!Where k out of n objects in the group are being selected

Combination

• Is a selection of objects where no objects can be selected more than once, and the order of selection doesn’t matter

• Combination formula: n! / (n-k)*k!

1.3 Probability of Equally Likely Event And Binomial Distribution

• If all the outcomes have the same likelihood of occurring, then the probability of an event, E, occurring is given by the formula: P(E)= Number of ways E can occur / total number of outcomes

Binomial distribution

• Used to solve specialized probability problems where an event with the same probability of occurring each time happens multiple times in succession

Binomial distribution formula

• An event with n outcomes has p outcomes that are considered a “success” and q outcomes that are considered a “failure.” If this event is repeated r times, then the probability of getting s successes and f failures is given by the formula: (r s) * P^s * q^f / n^r

Geometry

• All diagonals are of a square is related to the area of a square. For every area of one there is a diagonal measure of the square root of 2. so if the area is 2 the diagonal is equal to 2 * the square root of 2.

Phythagorean Theorem

• The square of one side plus another square of the other side is = to the square of the hypotenuse

• A^2+B^2=C^2• The Phythagorean theorem of a 3-D Figure is the square

root of l^2+w^2+h^2

Special right triangles

• In a 30-60-90 triangle all sides are related• By x, x times the square root of 3, 2x

• In a 45-45-90 it is• X, x, x times the square root of 2

Formulas

• Midpoint formula: (first x+ second x)/2 , (first y+ second y)/2

• Slope: (Second y – First y)/ (Second x – first x)• Point slope formula: y– y (coordinate) = m(slope) * (x-

x(coordinate))• Distance formula: Square root of (the difference between

the x)^2+ (The difference of the y)^2• If needed in a 3-D shape add the difference of z ^2• To find the sum of the interior angles you use (n-2) * 180

where n= the number of sides.• To find about of diagonals: n* (n-3)/2 where again n= the

amount of sides.

Volume & Surface Area Formulas

• Cylinder: 2*pi*r^2+2pi*r*h• Pyramid: Base area * .5* the perimeter* Slant height.• Cone: pi*r^2+ pi*r* (slant height)• Sphere: 4* pi* r^2• Cube: 6s^2• Volume: • Cylinder: pi* r^2* h• Pyramid: (1/3) Area of Base* h• Cone is the same as pyramid• Sphere (4/3)pi*r^3• Cube s^3

Introduction to Calculus

• Slope of a function is defined as m (y=mx+b); also written as either (change in y/change in x) or (y2-y1/x2x1)

• Slope is the rate of change

• Example: y=3x-5 f(x)is the function y, while 3 is the slope and -5 is the y-intercept

Derivatives

• The derivative of a function requires taking every part of the equation one at a time; 1) multiply the exponent of the variable by the multiplicative factor of the corresponding variable, 2) reduce the exponent by 1, 3) repeat for each part of the equation.

• When you take the derivative of a constant by itself, it is 0.• The derivative of a function is usually written in the form

f’(x)=such and such• Example: f(x)=3x^3 +2x^2. Find f’(x).

You would first take the coefficient, in this case 3, and then multiply it by the multiplicative factor. So 3*3=9, then the x^3 reduces by 1 to x^2. Same for the 2x^2, 2*2=4, and then x^2 becomes x. So, f’(x)=9x^2 + 4x.

Antiderivatives

• Taking the antiderivative of a function is basically doing the opposite of what you would do if you were taking the derivative of a function.

• Take f(x)=3x^2 for example:

You would first add 1 to the exponent. So 3x^2 becomes 3x^3.

Then, you would divide the entire designated part of the equation by the new exponent. So, (3x^3)/3 equals x^3. Therefore, the antiderivative of f(x)=3x^2 is x^3.

Calculating Limits• The first thing to do when you see a limit equation (lim x2

3x-4/x) is to use direct substitution, a method where you directly plug in the value x is approaching into the function. So, (3(2)-4)/2 equals 1. Therefore, the lim x2 3x-4/x is equal to 1.

• If you use direct substitution and you get a number over 0, the answer is undefined because you cannot divide by zero.

• If you use direct substitution and get an answer of 0/0, it means the limit statement has a removable discontinuity. This means that you can factor a quantity from both the numerator and denominator first, then directly substitute the designated value x is approaching.

• Example: lim x3 (x^2-9)/(x-3). If you plug in 3, the answer is 0/0, so you must factor. There is a (x-3) in common in both the numerator and denominator, so factor that quantity out. So, you are left with lim x 3 (x+3). Now that you do not have a discontinuity to deal with, directly substitute x for 3. So, 3+3=6. Therefore, the limit as x3 of the function is equal to 6.

Determining Asymptotes

• There are two types of asymptotes: horizontal and vertical. The vertical asymptotes are the asymptotes that cross only the x-axis once. The horizontal asymptotes cross the y-axis once. In an equation, the horizontal asymptotes are found in the denominator, vertical asymptotes in the numerator. Sometimes when you graph of function, the graphs are piece-wise functions where the function itself is approaching two different domains and/or ranges. Asymptotes determine this.