Graphs of sine and cosine

Here are the rules of graphing sine and cosine graphs:

1. Sine/ Cosine ​​​​​​​

2. Y = a sin (bx + c) + d

3. A = amplitude

4. B = period : 2∏/b

5. (bx+c) = Start: bx+c = 0; End: bx+c = 2∏

6. D = vertical translation; d units

Matrices, and elementary row operations

Elementary row operations:

1.Interchange 2 equations

2. Multiply an equation by a non zero constant

3. Add a multiple of an equation to another equation. 

Trig Identities

The Path to of verifying a trig identity:

You must only use one side, you can not work with bothe sides

Try to; factor, add fractions, square a binomial, create a monomial denominator. 

Look to use fundamental identities, they work best. 

Last resort, try to convert everything to sines and cosines

Trr anything, even mistakes help to solve the problem in the end. 

Converting Polar Coordinates

Converting between Rectangular Equations, (equations that contain x and y) and Polar Equations (equations that utilize r and theta (u) instead). When converting from x and y to r and u, use these equations:

 

x = rcos(u)

y = rsin(u)

 

When converting from r and u to x and y, use these equations:

 

tan(u) = y/x

r^2 = x^2 + y^2 (r squared = x squared + y squared)

 

Partial Fractions

When dealing with partial fractions, it is very important to follow the steps below! If you follow the steps, it will make the partial fraction problems very easy. 

The first step is to multiply by the LCD (least common denominator)

Second you need to distribute

Once you do those two steps, combine the terms

Then factor out the variable

When you do that you need to equate 

Next it is time to solve. 

Most people believe that they are done there, but dont forget to write it as a partial fraction.

Cross Product

The cross product of 2 vectors is not a hard topic. 

1. Make a 3x3 matrix, first column should be the variables i, j, k. Second column should be the values of vector U. Third column should be the values of vector V. 

2. Then take the first 2 rows of the matrix and rewrite them next to it. 

3. Then you multiply the diagonal values going forward (see example below, it is the green portion), and add the values you get from that. 

4. Then you multiply the diagonal values going backward (see example below, it is the purple portion), and subtract these values. 

Limits in Calculus

Limits are the core to calculus. The limit do a function is essential in many topics and situations in calculus. Limits are the building blocks of calculus and here is a link to read more. 

http://www.khanacademy.org/math/differential-calculus/limits_topic

Tangent Lines

Today in Mathland we learned about Tangent Lines and how to find their slopes! Finding the slope is simple, but not easy. There is a formula you must memorize. It's f(x+h)-f(x) all over h. First, you. It's plug the values into your formula. Then you foil. Then you combine like terms. After that, you plug in 0 for your h. Then you just solve. There is an example below. This lesson isn't to tough, just keep track of your terms. 

Evaluating Limits

Today in Mathland we learned how to evaluate limits! It's not that tough of a topic, but it takes sometime to finish. The first step is to use direct substitution and plug in the value of the variable. You should get 0/0. Then you take your original equation, and from there, 2 things can happen. If you have a square root in your numerator, then you multiply by the conjugate. If not, then you factor. And the final step is to substitute the value of the variable, into the factored equation. 

Limits can be tricky, but they are essential for the next levels of math. 

Polar Coordinates

Today in Mathland we learned about polar coordinates. It's a way of graphing similar to rectangular coordinates. First you fix a point O that is called the pole, and construct from it a ray called the polar axis. Then each point P in the plane can be assigned coordinates in (r, θ). 

You must also know how to convert between rectangular and polar coordinates. The formulas are down below. 

Hyperbolas

Today in Mathland we learned all about hyperbolas! They have a pretty interesting conic section. The formula for a horizontal parabola is (x-h)^2/a^2 - (y-k)^2/b^2=1 and for a vertical hyperbola the formula is (y-k)^2/a^2 - (x-h)^2/b^2. As long as you stick to the formulas you can't go wrong. 

Ellipses

Today in Mathland we learned about the conic known as the ellipse. An ellipse is the set of all points (x,y) the sum of whose distances from two focus points is constant. The standard equation of an ellipse is 

Here is a graph of an ellipse. 

Ellipse can be challenging at times, but if you stick to the formula you can't go wrong. 

Parabolas

Today in Math Land we learned about Parabolas. Parabolas have a lot of rules and a lot going on. 

A parabola is a set of all points (x,y) that are equidistant from a fixed line, and a fixed point on the line.

The formula for a vertical parabola is: (x-h)^2=4p(y-k) and the directrix: y=k-p

The formula for a horizontal parabola is: (y-k)^2=4p(x-h) and the directrix:'x=h-p

Make sure you keep your eye on all the variables and make sure you don't make any silly mistakes. 

Lightning!

The chances of being struck by lighting are 1 in 3000. Yet, Lightning is one of the leading weather related causes of death. And don't think rubber shoes will help. Rubber shoes actually do nothing to protect from lightning. Here is a link to an article explaining the probability of getting struck by lightning: 

http://news.nationalgeographic.com/news/2004/06/0623_040623_lightningfacts.html

Measures of Central Tendency

Today we learned the !easures of Central Tendency. They are mean, median, and mode

Mean: is the average

Median: is the middle number, the set must be in order of least to greatest. 

Mode: the number that occurs most often. 

They are pretty simple to learn, but make sure you don't make any silly mistakes like forgetting to order your sets from least to greatest before searching for the median. 

Standard Deviation

Today in Mathland we learned about the measures of dispersion. Measures of dispersion will give us an idea of how much the numbers in the set differ from the mean of the set. The two measures of dispersion are called the variance of the set and the standard deviation of the set. Finding the standard deviation of a set and the variance of a set take a little effort. But as long as you are careful you can't go wrong. 

Well Ordering Principle

The well ordering principle states that every non empty subset of positive numbers contains a least element. So a subset of all positive natural numbers has a least element. Here is a link explaining more:

http://www.math.wustl.edu/~chi/310notesIV.pdf

Big Data

Big data sets are becoming increasingly more prevelant in today's world. We are constantly looking for new and better ways to analyze these data sets. New tactics and technology is being made to help improve this. By identifying patterns you can speed the process up. Here is a link to an article relating to this topic: http://www.sciencedaily.com/releases/2014/02/140218185128.htm

Sequences

Sequences are great but they can be very challenging at times. There are several formulas to learn and different types of sequences. There are geometric and arithmetic. There can also be infinite solutions. But, if you stick to the formulas and plug in the right numbers you can't go wrong. For arithmetic sequences. Here are the formulas:

Factorials

Today in Mathland we learned about sequences and factorial a. My favorite was the factorials! They are so neat. You take a number and multiply it with all the numbers that come before it. So if you have 4! you multiply 4x3x2x1. And you get 24. You can also divide and multiply factorials together. That's when it gets really fun. I can't wait for a quiz on factorials! 

Here is an example of dividing and multiplying factorials.