Adding or Subtracting Radical Expressions

We start by factoring 32 and 72 the rewriting in a way that helps us determine if the radicals are like terms. After we know they are like terms and we've simplified each term we can combine them (add their rational parts and keep sqrt(2) as it is)

Add or subtract Polynimials

Solution:
We can simplify by combining the like terms. Like terms are the ones that are alike in every way except for maybe their coefficients. To simplify subtraction we can convert to addition (and eventually reverse the signs on every term of the subtrahend)

Write 0.00000000012 in Scientific Notation

Suppose the distance from Earth to Planet-X is 4,502,233,434,224,445,657,213,455,467,634,345,654,634,234,333,345,547,665,754,234,647,673,535,345 light years away and you needed to write that distance 20 times in various places in a book report. Wouldn't it be nice if there was a way to "abbreviate" it? Scientific notation is how we can write numbers in a "shorter" way. In this case we could round the number to say 4,500,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, but we'd still have deal with all of those zeros. Using scientific notation we could write the distance as "approximately 4.5 x 10^72" and capture the size of the number without needing to write all of its digits. Here is how it works:

So... 0.00000000012 = 1.2x10^-10

Limit Definition of Derivatives

Use the definition of derivatives to find the derivative of sqrt(x)

Solution:
Multiplying by a conjugate can be a useful strategy and it helps in this problem. We also notice that sqrt(2) is only defined for real numbers greater than or equal to 0 (for calculus). This means the absolute value of an expression is also greater than or equal to 0. In the denominator we must be careful not to multiply because factoring out the delta-x is easier.

Factoring a degree 4 polynomial

Factoring polynomials is as much an art as it is a skill and we can get better with practice. It helps to know some important facts about them to make our work easier. For instance, if all the coefficients are real and we are factoring over the real numbers, we'll always be able to factor down to degree 1 or degree 2 polynomials. We can "cheat" by graphing the polynomial and checking when it crosses the x-axis. And we can "cheat" by using a bunch of tools from Pre-Calculus topics: Descartes' Rule of signs, synthetic division, etc. Finally there is the option of using software on a calculator, computer, or the internet. Then going back to another strategy with the foreknowledge of the actual solution (shameful).

Here we employ a couple of handy strategies.  Grouping is our most important tool and we exploit it ruthlessly. Next we notice that if we replace 7x^2 with 5x^2 + 2x^2  (on line 2 of the solution) we  get two polynomials that can be factored - each with a common factor of x+1.  On line six we use the same strategy by writing 5x^2 as 2x^2+3x^2

Polar Equations Example

a.) Find the eccentricity, and directrix of the polar equation then classify the conic section
b.) Convert this Polar Equation to Rectangular Form
c.) Sketch the plot

Actual plot (from Wolfram Alpha):

Limit of an exponential function

Limits of exponential functions sometimes call for L'Hopital's Rule. In this example the limit is much easier to compute after we apply the rule.

Limit of a sequence

Limit of a sequence. We us the handy strategy of re-writing the limit in a more convenient way that does not change it's value. And by using the conjugate of the expression, we are able so simplify. Alternately we can apply the direct comparison test. 1/n is a sequence that we know converges, so we compare our sequence to it. After some algebra, we can see that the inequality will hold.

Summations

Notice how each summation appears very different, but they all have the same sum and in fact the same terms (The last summation should start at 0)

Bearings

Most of us don't use bearings very much - if at all. Here is the convention:

The direction to an object is described as the number of degrees east or west of north or south.

A is read "30 degrees East of North"
B is "60 degrees West of North"
C is "70 degrees East of South"
D is "80 degrees West of South"

Derive Area Formula for a Trapezoid

Use Triangles to derive the formula for a trapezoid

So we denote the trapezoid area as A-sub-T and subtract the area of the small triangle on top from the entire triangle. We can also use the idea of similar triangles to replace parameters from the top triangle.

Finally we rename h-sub-2 as h so that it looks more familiar.

first post...

This is my first post of my "first blog". The actual first blog was somewhat of an experiment. It's called "What's This Blogging all about?" and I used it to learn how to set up a blog. I think I'll keep it and use it for something...

So this blog will be where I post examples of math problems that I've worked out. I'm sure it wont be great at first, but I hope to get good, fast.