12.4 Limits at Infinity and Limits of Sequences

A limit at infinity can be used to solve the area problem!

What a limit at infinity looks like: 

The limits as f(x) approaches the vertical asymptote x=2 is less important to what we are learning.

The limits as x approaches positive and negative infinity are what concerns us. 

As you can see, the limit of f(x) as x approaches negative infinity is 3. The limit of f(x) as x approaches infinity is 3. These limits mean that the value of f(x) gets arbitrarily close to 3 as x decreases or increases without bound. The limits give us the horizontal asymptotes to the left and to the right. 

Side-note: The graph of a rational function need not have a horizontal asymptote. If it does, however, its left and and right asymptotes must be the same. 

Something to keep in mind when evaluating limits at infinity:

When 

• If  r  is a positive real number, then the limit as x approaches infinity is 0. There is a limit toward the right.

• If   x^r  is defined when  x < 0,  then the limit as x approaches negative infinity is 0. There is a limit toward the left. 

Evaluating a limit at infinity: 

Examples:

1. Find the limit:

                                    

                                   

                                   

2. When finding the limit as x approaches infinity for a function such as

You can begin by dividing both the numerator and denominator by the highest-powered term in the denominator, in this case x^2.

In this case the limit is 0.

General Rules:

• When the degree of the numerator is less than the degree of the denominator, the limit is 0.
• When the degree of the numerator and denominator are equal, the limit is the ratio of the coefficients of the highest-powered terms. 
• When the degree of the numerator is  greater than the degree of the denominator, the limit does not exist.

If you are more of a visual and audio learner this video explains what I have just stated and shows some very helpful examples. 

Limits of sequences:

Limits of sequences have many of the same properties as limits of functions.

The limit of a sequence as it approaches infinity asks what the terms of the sequence get closer and closer to, or converge towards, as n increases without bound. 

A sequence that does not converse, for instance a sequence that oscillates, is said to diverge.

Let's say that:

and we are evaluating:

If   is a sequence such that  for every positive integer n

Then

Examples:

Here is a helpful video that shows how to evaluate the limit of a sequence:

I hoped this helped you with evaluating limits at infinity and evaluating the limit of a sequence!

GOOD LUCK TO ALL MY HONORS PRECALC BROTHERS AND SISTERS ON THIS UPCOMING FINAL.

STAY POSITIVE.

WORK HARD.

FOR WILHELM!  (AND BECAUSE WE ALL LOVE MATHS SO MUCH)

SHOUT OUT TO MR. WILHELM FOR BEING THE BEST MATH TEACHER OF ALL TIME. 

#legendary 

Thank you, 

Leah 

12.3 The Tangent Line Problem

The slope of a curve at a point is defined to be the slope of the tangent line. Thus the slope of a curve is found using the derivative.

Everyone knows slope is the change of y over the change of x

You use that to find the AVERAGE slope between two points. But some may ask how to find the slope at a GIVEN POINT.

They may think it is impossible by combining what they have learned and common knowledge. But they are wrong.

With derivatives you shrink the point (two points) to the smallest possible distance... but then you eventually shrink it down to zero.


How to Find a Derivative!

You start off with the original slope formula like the one above, but with a little twist.

f(x+h)-f(x)= y
              h=x

Once you simply that formula, that gives you the average slope of the graph. Then from that you plug in a point and voila! the slope at that given point.

Like I said earlier, eventually you turn "h" into 0
So the final average slope of the formula above is "2x"

For more help visit: http://www.coolmath.com/precalculus-review-calculus-intro/precalculus-algebra/24-the-difference-quotient-01.htm

Thank You,

Peter Kessel

12.2- Techniques for Evaluating Limits

Limits of Polynomial and Rational Functions

--1. If p is a polynomial function and c is a real number, then  



--2. If r  is a rational function given by r(x)= p(x)/q(x), and c is a real number such that q(c) does not equal 0, then:
     as x approaches c,   lim r(x) = r(c) = p(c)/q(c)

Evaluating Limits by Direct Substitution

      - when evaluating a limit by direct substitution, you insert the c value in for the x value of the function and the resulting number is the limit as x approaches c.

     - This method only works if the function is continuous, ehich means there are no holes in the graph

Evaluating a limit by Dividing Out

  

     - To evaluate a limit by dividing out, you can factor the numerator and divide out the common factors.

     - This particular example has a limit of 7, and has a hole at the point (1,7) because the factor of (x-1) is divided out, and at that point, y=7.

Evaluating a Limit by Rationalizing

    - To evaluate a limit by rationalizing, you have to first rationalize the numerator of the function.

   - In this example, once you get the numerator rationalized, you can plug in the c value for x and that gives the limit as x approaches c, where at c there will be a hole in the graph.

Evaluating One-Sided Limits

To evaluate a one sided limit, you find what y value is approached from either side as x approaches c.  In this graph, as x approaches c (2) from the left, y also approaches 2.  As x approaches c from the right, however, the y value approaches 1.  These can be solved by simply looking at the graph, or applying direct substitution to the 2 functions for their correct sides of the graph.  

Some helpful videos:

http://www.youtube.com/watch?v=0u7NtyGGlv4

http://www.youtube.com/watch?v=mGPyMDi1Mdc

Thanks,

Michael 

Chapter 12.1: Introduction to Limits

This next chapter is all about limits. So the first question was, what is a limit?
The book defines a limit as:

If f(x) becomes arbitrarily close to a unique number L, as x approaches c from either side, the limit of f(x) as x approaches c is L.

More simply, a limit is the value y approaches as x approaches a specified value.

 

 

Today we learned two different ways of finding the limit:

1. Tables

2. Graphs



 

First, insert the equation into your calculator.

Then insert values into the table that are close to 2.

From this table, you can tell that as x approaches 2, y approaches 5. So the limit is 5.

From the graph, we can see that as x approaches 0, y approaches 1. All these types of problems are solved by just looking at the graph.

Limits That Do Not Exist:
There are also limits that don't exist. There are 3 scenarios where this happens.

1. When y approaches two different numbers
-In the example above, if you try to solve for the limit when x approaches 1, you get two values that y approaches. Therefore, there is not a limit at all.

2. When y approaches infinity or negative infinity
-In the graph below, as x approaches 1, y approaches both infinity and negative infinity. In this example, the limit DNE.
 

 3. In oscillating graphs
-In the graph of sin(1/x), the wave only goes between 1 and -1 so there is no limit.

That's mostly everything we learned today. Hope it helped!
-Natalie

 

10.6 Polar Coordinates

Hey everyone, so today we learned a new way to graph with the polar coordinate system.


We are all very familiar with the Cartesian Plane or the rectangular coordinate system.

Points are labeled (x,y)
Each set of points represent the directed distances from the coordinate axes to the points.  Each set of points is unique.














Now we are learning about a new coordinate system.
The polar coordinate system consists of concentric circles.  The polar axis is similar to the x axis.

Each point P can be assigned polar coordinates (r,θ)
r represents the distance from the origin (it can be negative)
θ represents the rotation from the polar axis (also can be negative)

Unlike the rectangular coordinate system, points on the polar plane are not unique. The same point can be represented by infinite sets of polar coordinates.  









Now that we know about the polar coordinate system, we can now relate both of the graphs.  Points can be converted from (x,y) to (r,θ) and vice versa.  

Coordinate conversions 

x=rcosθ

y=rsinθ

tanθ=(y/x)

r^2=x^2+y^2

Thats about it!!!

 Helpful video

-Jen Kendall

10.5 Parametric Equations

This section was all about parametric equations

Most equations just use x and y variables, but parametric equations use a third variable, which is often t for time and is called a parameter.  You can then write x and y as functions of the parameter to get two parametric equations.

Graphing Parametric Equations:

Here's an example of a parametric equation and its graph:

x = t - 2

y = t + 1

To graph this, you can pick values of t and plug them into the equations to calculate the coordinate points of the graph.  For example, when t = -2, the x value is -4 and the y value is -1 so the coordinate is (-4, -1).  This is what the graph should look like:

Eliminating the Parameter:

Once you've graphed the parametric equations, you can rewrite it as an equation using only x and y.  The equation for the line above is y = x + 3.

You can also eliminate the parameter algebraically without graphing to get, as the book says, a rectangular equation, which is just a normal equation with two variables.  You do this by solving for the parameter in one of the equations, then plugging that into the other equation.

x = t -2
t = x +2

y = t + 1
y = x + 2 + 1
y = x + 3


Finding Parametric Equations:

 

You can  find parametric equations from a rectangular equation.

ex.  x² + y² = 1

There are many different ways to rewrite this equation parametrically.  One of them is...

x = sinθ

y = cosθ

Calculator:

You can graph parametric equations on your calculator by going to MODE and selecting PAR, then you just type the equations in and graph.

That's basically it.
-Olivia R

9.7 Blog Post

9.7: Probability, Dude

Merriam-Webster (rough riders don’t use dictionary.com) defines probability as the ratio
of the number of outcomes in an exhaustive set of equally likely outcomes that produce
a given event to the total number of possible outcomes or the chance that a given event
will occur.

Definin’ Some Termz:

Experiment-activity under consideration
Outcome-Each possible observation
Sample space/Outcome set-set of possible outcomes
Event-subset of a sample space

No Mo Termz

How ‘Bout We Do Some Scenarios???

Lets suppose you’re holding a coin. The type of coin isn’t vital, but lets call it a quarter,
cause quarters are big and yeah. So you’re ready to flip that coin and you ask yourself,
what are the chances that this quarter will land with Mr. Washington looking up at me? If
you are seriously thinking about the answer to this question, you probably need to re-
evaluate your life, but this is just an example so stop nit picking my flawless and
awesome blog post. The point is that the CHANCE you’re trying to calculate is the
PROBABILITY that the quarter will land heads up. You do so by creating a ratio of the
number of favorable outcomes (heads), divided by the number of possible outcomes
(heads and tails). This gives you ½. That’s easy, right? YEAH! But, I doubt Thad will put
it on a test because he likes to use his green pen.

Lets look at a trickier situation, shall we? Say you got Drew’s left shoe, and a six sided
die. The shoe can land either laces up or sole up. If you were to flip the shoe and then
role the die, what is the probability that you get laces and a 2? Lets break it down,
dawg. You gotta ½ chance of flippin’ a lace, right? Of course I’m right. Then, you would
multiply that by the ratio of favorable die outcomes (1) over total possible die outcomes
(6). Thus giving you a 1/12 shot of gettin’ laces and a 2. Done. Next.

Just throwing this out there… P(A)= the probability of A

Almost forgot, mutually exclusive means that two sample spaces do not share a
common event (usin’ dem termz).

Bring in the formulas!!!!!

P(A U B)= P(A) + P(B)*
P(A U B)= P(A) + P(B) - P(A “upside down U” B)
P(A “upside down U” B)= P(A) x P(B)**

Also, independent means that the events don’t influence each other, like how flipping
Barton’s shoe doesn’t effect the roll that the die will land on.

Now we get off track…

http://www.youtube.com/watch?v=eVtDSFgeqPY

http://www.youtube.com/watch?v=jsKpazuC0RY

This is too help Barton and Pete learn how to count:

http://www.youtube.com/watch?feature=endscreen&NR=1&v=85M1yxIcHpw

Back on task! YAY!

Complements are cool too. The complement of event A are all of the possible outcomes
in the sample space that are not in A. I’m tired so I’m not going to explain it in more
detail. The book does an okay job.
A’= Complement A
P(A’)= 1- P(A)

*Only if A&B are mutually exclusive
**Only if A&B are independent

My Closing Statement.

By now you have realized that this is the greatest blog post of all time, followed closely
by all of my other blog post, and then Oran’s famous “I’m going to be as creepy as
possible” post of 2010. I’m going to be completely honest, and I think I speak for
everyone when I say, I am pretty damn impressive.

Pete *****

Barton *****

Seaglass can’t do maths

Austin’s cool

Geller’s too smart

Thad’s a beast

Barton *****

And insert Noah’s blog post here:

I was just too tired to write this blog post. I am sorry that I could not supplement
you with more maths knowledge. Maths is forever.

NNNNNNNNNNNNOOOOOOOOOOOOOAAAAAAAAAAAAAHHHHHH!!!!!!!!!!!

K bye.

9.6 Counting Principles

Fundamental Counting Principle- E₁ and E₂ are two different events. E₁ can occur in m₁ different ways and E₂ can occur in m₂ different ways. The number of different ways that the two events can occur together is m₁ x m₂.

The fundamental counting principle is true for any number of events that occur together.

When counting the number of possible outcomes, it is important to know if you are counting combinations or permutations.

When counting permutations, order matters. This means that choosing option 1, 2, and then 3 is different than choosing option 1, 3, and then 2. Lets look at an example…

If you randomly choose 3 digits (between 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9), where each digit can only be used once, how many numbers can you make?

This is asking for the number of outcomes. Because order matters, we have to count the number of permutations.

According to the fundamental counting principle, we have to find the product of the number of possible outcomes for each event (m₁ x m₂ x m₃).

1. The first digit can be any of the numbers listed above so there are 10 different possible outcomes.
2. The second digit can be any number listed above other than the first digit so there are 9 possible outcomes.
3. There are 8 possible outcomes for the third digit.

So to find the total number of permutations when you choose three out of ten options...

Our equation for counting the number of permutations is...

Where 

Now let’s look at how to count combinations. When counting combinations, order does not matter. This means that choosing option 1, 2, and then 3 is the same as choosing 1, 3, and then 2. Here’s an example of counting the number of possible combinations…

When ordering a pizza with toppings, it does not matter what order you choose the toppings in. A cheese, ham, pineapple pizza is the same thing as a pineapple, ham, cheese pizza. Let’s say there are ten toppings to choose from. How many ways can you choose 3?

If order mattered we would find the answer like this...

However, because order doesn't matter, this number is too big because it counts

cheese, ham, pineapple

cheese, pineapple, ham

ham, cheese, pineapple

ham, pineapple, cheese

pineapple, cheese, ham

and

pineapple, ham, cheese

as separate outcomes, when in reality, they are all the same thing.

To get the answer we want we just divide by 3! or 6. 

We divide by 6 because (as we saw above) any combination of three outcomes has 6 different permutations (because 3!=6)

We can find the number of combinations using...

Remember 

Hope this is helpful!!!

Binomials

Lets expand binomials!!!

WOAH!
      Can you imagine what a pain it would be to
      foil all of those?!

But wait, THERE ARE PATTERNS!

First, notice that as the power of x goes down; the power of y goes up.
Each term follows the pattern (coefficient) x ^n-r y ^r
where n is the power or the whole binomial
    and r is the number of the term minus 1 (for the fourth term, r = 3)

.

Here Be-ith Pascal's Triangle.
Behold the magnificence.
Each row corresponds with the power of the binomial  -- (x + y)ⁿ --  where the top is n = 0, second n = 1, third n = 2, etc. Each number in the row corresponds with the coefficient of each of the terms of the expanded binomial.
(Compare the binomials' coefficients to the numbers in the triangle at the right. It will make sense.)
One can find the coefficient by using the number in the triangle from the n^th row and the r^th diagonal.
.
One can find the numbers in the next row by adding together the two numbers directly above it in the previous row.  Unfortunately, Pascal's Triangle, although an incredible feat of wizardry, is not the most effective method. Especially when dealing with the eighty somethin-ith term.

You can figure out what the coefficient of any term is by using
The Binomial Theorem. It's pretty great.

If the terms of the compounded binomial already HAVE coefficients, you put THAT to the powers of n-r and r also and then multiply it by the coefficient you get from compounding it.

That's it, bloggy friends.
Olivia Miller

Chapter 9.4 Mathematical Induction

                                                          Chapter 9.4 Mathematical Induction

What is Mathematical induction???????

Mathematical induction is a method of proof in which a statement is proved for one step in a process, and it is shown that if the statement holds for that step, it holds for the next.

The reasoning behind mathematical induction can be linked to a infinite chain of dominos.  It is impossible to go and push the dominos down one by one.  However, if it is given that by pushing down one domino it will push down that domino next to it then it can be inferred that simply by pushing down one domino you are pushing them all down.

It is very important to note that this process must be a legitimate proof, and is not simply jumping to conclusions.

Mathematical induction is a three step process.

1. Show that the claim is true for n=1

2. Assume that the claim is true for n.

3. Show that it is true for n+1

Example

1 + 3 + 5 + ... + (2n-1) = n²

1. Show it is true for n=1

1 = 1² is True

2. Assume it is true for n=k

1 + 3 + 5 + ... + (2k-1) = k² is True

3. Now, prove it is true for k+1

1 + 3 + 5 + ... + (2k-1) + (2(k+1)-1) = (k+1)² 

 1 + 3 + 5 + ... + (2k-1) = k² (the assumption above), so we can do a replacement for all but the last term:

k² + (2(k+1)-1) = (k+1)²

Now expand all terms:

k² + 2k + 2 - 1 = k² + 2k+1

And simplify:

k² + 2k + 1 = k² + 2k + 1

It's good!

So:

1 + 3 + 5 + ... + (2(k+1)-1) = (k+1)² is True

And that's mathematical induction In a nut shell.

www.youtube.com/watch?v=lOGqZAMHS-M

Matthew Silbergleit

Partial Faction Decomposition (Included in 7.3)

Partial fraction decomposition is essential to success in Calculus next year.  PFD involves breaking down a rational faction with a polynomial denominator into the sums of smaller fractions.

Before "decomposing" a fraction, you first have to take the degrees of the numerator and denominator of the rational function into consideration.

• If the degree of the numerator is greater than that of the denominator (improper fraction), divide the numerator into the denominator by long division.  This will result in a polynomial followed by the remainder of the long division divided by the original denominator.  Then proceed to decompose the latter using the steps that will be explained later.
• If the degree of the numerator and the denominator equal, follow the steps above and use long division to break the function down further.
• If the degree of the numerator is less than that of the denominator (proper fraction), proceed using the steps below.

Here is an example:

   First, factor the denominator. 

Next, make this fraction equal to a sum of two fractions with the factored denominators.

Working with the right side, proceed as you would to find a common denominator to add the fractions together.  Add the like terms and begin to simplify the equation.














As you can see the  term has a coefficient of 1 and therefore  equals 1.  Consequently,  must then equal 5.  Create a system of equations from here and solve.







Your final answer:




Side note- In cases where the denominator has repeated factors such as , be sure to decompose as followed:





Thanks for reading!

Julia Wilkins