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Review Presentation: Equation of a Plane

In order to find the equation of a plane, you must use the formula:
a(x-x1) + b(y-y1) + c(z-z1) + 0
When asked to find the equation of a plane, you will be given three points.  The first step is to choose one point to be your initial, one point to be u, and one point to v.  But, since they are points, you must make them into vectors by using your initial point and doing: terminal - initial.
(1,2,3)v
(-1,5,4)u
(o,-3,7)i
u = <-1,8,-3>
v = <1,5,-4>

Now you must place your vectors into a cross product (u x v) so that you can find your a,b, and c.
     i     j     k     i     j
    -1   8    -3   -1    8
     1   5    -4    1    5
u x v = <-17,-7,-3>
Now you plug these into your original equation, with x1, y1, and z1, being your points from your initial.

Final answer = -17x - 7y -3z = 0

Limits -> Infinity

When presented with limits to infinity, you are usually given a fraction that is similar to p(x)/q(x). When this occurs, there are three rules you must follow that will make your answers so much easier!

If n > m your limit is 0
If n = m your limit is the numerator/denominator
If your n > m your limit does not exist

n = power of the numerator
m = power of the denominator

For example
lim       2x+1/x^2
x -> infinity

n < m = limit = 0

Limits

Limits of Polynomials are one of the easiest things of Math Analysis EVER! All you have to do is plug in the number!
For instance

lim 2x+1
x -> 4
 = 2(4) + 1
 = 9
WINNER WINNER CHICKEN DINNER

Limits of Rational Functions are a little big more difficult, because they usually equal 0/0 which means they are intermediate!
When this occurs, you must do cancellation. This is when you factor your given function, cancel out terms, and then plug in your x again.

lim        x-1/(x-1)(x+1)
x-> 1
 = cancel out x - 1
 plug 1 in for x
 = 1/2

Review Presentation: Equation of a Sphere

When finding the center of a sphere, you will be given an equation of a sphere. And all you have to do is complete the square essentially..but three times. For instance...

x^2 + y^2 - 2x + z^2 - 4z + 10y - 4 = 0
First you should combine the terms
x^2 - 2x + _ + y^2 + 10y + _ + z^2 - 4z + _ = 4
Now complete the square three times!
(x^2 - 2x + 1) + (y^2 + 10y + 25) + (z^2 - 4z + 4) = 4 + 1 + 25 + 4
Factor
(x - 1)^2 +  + 5)^2 +  - 2)^2 = 34
Pull out terms!
center: (1, -5, 2)
radius: √34

YAY

Review Presentations: Distance Between a Point and a Plane

When finding the distance between a point and a place, you will be given a point as well as an equation of a plane.  The formula for finding distance is: D = |PQ x n|/||n||
Q will be given to you as a point, and n will be the numbers you pull out from the equation.  For example Q (1,5,2) and 2x +3y -z + 5. n = <2,3,-1>
The next step deals with finding vector PQ. You will use Q as your terminal and you will create an initial by zeroing two variables from the plane equation. This will give you P = (0,0,-5) Then you must find the dot product of PQ and n which will equal -20, but the absolute value of that is 20. and you will divide that by the magnitude of n which is found by the square root of all the numbers squared of the vector which will equal root 14.
Your final answer will be 20/√14

The Limit Does Not Exist

In chapter 12 we learned about limits. Which is when a slope or curve has a boundary. But, there are times when limits don't exist.

1. f(x) approaches different numbers from left and right

2. f(x) increases or decreases without bound

3. f(x) oscillates between two fixed values

Review Presentation: Cross Product

Cross Products involve matrices with vectors. To find the cross product you must create a matrix using I, j, and k. You will be given two vectors: u and v, and you must find the cross product by using matrices and determinants, which will form an equation.

u: <2,1,5>

V: <-3,4,2>

these are your vectors and now you must place them into a matrix.

         i     j     k     i     j     

        2    1.   5.   2.   1

       -3.   4.   2.  -3.   4

u x v: 2i -15j + 8k + 3k -20i -4j

u x v: -18i -20j + 11k

u x v: <-18, -20, 11>

Review Presentation: Parametric Equations

Parametric equations consist of the variables, t, x, and y. When doing a parametric equation, we must remember that we must create a table, a graph, and then finally eliminate the parameter.

When making a table, we must use t, x, y. We can plug in any variable as long as we include negatives as well as positives. I usually use -2, -1, 0, 1, and 2. Once we place these numbers into our table, we simply plug them into the equation which is how we will get our x and y values. The next step is to plot the points onto the graph, and placing t = (number) near the point. This is when you will get a curve. 

The last thing you must do is eliminate the parameter. In the x equation, you must solve for t. And then plug t into the y equation, which is how you will eliminate the parameter. And that's how you do it!

Review Presentations: Mathematical Induction

Rebecca presented on mathematical induction, which is basically a proof. You must attempt to make e left hand side equal to the right hand side, this is how you begin.

First, when given an equation you must prove true for n=1. This is when you plug in 1 for each value of n. the left hand side and the right hand side must be equal.

Second, you must assume true for n = 1. Then prove true for n + 1. This is when you create a new equation. The right hand side of the equation + the left hand side (with n+1) = left hand side (with n+1). The end result is that each side must be equal to each other, and that is how you do mathematical induction.

The final thing that you have to do is state that the left hand side is equal to the right hand side through mathematical induction hence true for n + 1.