12.2: Evaluating Limits

There are three main ways to evaluate limits.

1) direct substitution (plug-in) 

2) cancellation technique (factor and cancel) 

3) rationalizing technique (multiply by the conjugate-only used with problems with radicals) 

You should use the direct substitution first. If using direct substitution gives you 0/0, then it is in intermediate form. If it is in intermediate form, then you have to either use the cancellation technique or the rationalizing technique to get a new equation. Then plug the number from the original equation into the new equation. 

Example of the cancellation technique

 

Example of the rationalizing technique

One-sided limits

Sometimes there are problems like this:

In these types of problems, you have to plug it in to both eqautions. If it the result is the same for each equation then there is a limit. In this case, the limit is 3. If the result for each eqaution is different, then it is a one sided equation and you can solve for each side separately. 

In other cases, the problem has a f(x) in it and you have to first solve the function before solving the main equation to find the limit. 

Math joke of the day:

Interesting Math Stuff #13: John Nash

John Forbes Nash is a modern mathematician who made considerable advances in many contributions to the fields of game theory, differential geometry, no partial differential eqautions. 

One of his most interesting contributions was involved encryption which we studied earlier this year. In 2011, the NSA revealed some of Nash's letters where he discussed a new encryption/decryption machine. 

Nash was also diagnosed with paranoia and schizophrenia which caused his hospitalization. His mathematical genius and his mental disorders inspired a 2001 Hollywood film named A Beautiful Mind. 

Math joke of the day:

11.2: 3-D Vectors

Component form: v=<v1, v2, v3>

Unit vector form: v=v1i+v2j+v3k

Length formula: 

Length can also be called tension or magnitude. 

Unit Vector:

Vector addition and dot product:

Angle between two vectors formula:

Orthogonal if...

-The dot product between the two vectors is 0

-The angle between the vectors is 90 degrees

Parallel if...

u=cv

Vector 1=(constant)(vector 2) 

-the points are collinear 

Math joke of the day:

11.1: 3-D graphs introductions

Today we learned about 3-D graphs. 

With the points (x1, y1, z1) and (x2, y2, z2) 

The distance and midpoint formula:

The eqaution of a sphere:

r=radius

center=(h, k, j) 

Math joke of the day:

3-D Graphs

Check out these cool 3-D graphs I made on the apps quick graph and good graph! 

Interesting Math Stuff #12: Pythagoras

Pythagoras was a Greek mathematician, philosopher, and the founder of the religion Pythagoreanism. He is of course, best known for the pythagorean theorem: a2=b2=c2. 


The only writings about him were written many years after his death, so we are not sure of the validity of the facts about his life. It is known in Samos, Greece. It is said that a prophet told his pregnant mother that her child would be supremely wise, beautiful, and beneficial to mankind although this is debated. Pythagoras was not only a brilliant mathematician, but also made discoveries in the fields of mathematics, astronomy, and music. He was also said to have practiced divination and prophecy, often being associated with Apollo. According to some writings, he married a woman named Theano who bore him children including a son Telauges and three daughters Damo, Arignote, and Myia.
Source: http://en.wikipedia.org/wiki/Pythagoras

Math joke of the day: 
I strongly dislike the subject of math, however  I am partial to fractions. 

10.7: Graphs of Polar Equations

Today we learned how to make graphs of polar equations.

Symmetry tests for polar coordinates 

1. Replace θ with -θ. If an equivalent equation results, the graph is symmetric with respect
to the polar axis.
2. Replace θ with -θ and r with -r. If an equivalent equation results, the graph is symmetric
with respect to θ = π .
3. Replace r with -r. If an equivalent equation results, the graph is symmetric with respect
to the pole.

Note: It is possible for a polar equation to fail a test and still exhibit that type of symmetry.

There are four main types of graphs of polar equations:

Circles
1. r = a cos θ is a circle where “a” is the diameter of the circle that has its left-most edge at
the pole.

 2. r = a sin θ is a circle where “a” is the diameter of the circle that has its bottom-most edge
at the pole.

Limaçons (Snails) 
 1. r = a ± b sin θ, where a > 0 and b > 0
2. r = a ± b cos θ, where a > 0 and b > 0

-The limaçons containing sine will be above the horizontal axis if the sign between a and b is plus or below the horizontal axis if the sign if minus.
-If the limaçon contains the function cosine then the graph will be either to the right of the vertical axis if the sign is plus or to the  left if the sign is minus.
-The ratio of a to b will determine the exact shape of the limaçon.

Rose Curves 
A rose curve is a graph that is produced from a polar equation in the form of:

 r = a sin nθ or r = a cos nθ, where a ≠ 0 and n is an integer > 1

The number of petals that are present will depend on the value of n. The value of a will determine the length of the petals.

Lemniscates 
Lemniscates have the general polar equation of:
 r2 = a2 sin 2θ or r2 = a2cos 2θ, where a ≠ 0

A lemniscate containing the sine function will be symmetric to the pole while the lemniscate containing the cosine function will be symmetric to the polar axis, to θ = 2π , and the pole.

Math joke of the day:
A team of engineers were required to measure the height of a flag pole. They only had a measuring tape, and were getting quite frustrated trying to keep the tape along the pole. It kept falling down, etc. A mathematician comes along, finds out their problem, and proceeds to remove the pole from the ground and measure it easily. When he leaves, one engineer says to the other: "Just like a mathematician! We need to know the height, and he gives us the length!"