2005 FR 5

Rate sand is removed: R(t) = 2+5sin(4πt/25)
Rate sand is added: S(t) = 15t/(1+3t)
Initial: 2500 cubic yards of sand

a.) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure.

₀∫⁶ 2+5sin(4πt/25)dt = (2t + (125/4π)cos(4πt/25))]⁶₀ ≈ 31.816 cubic yards of sand

b.) Write an expression for Y(t), the total number of cubic yards of sand on the beach at time t.

Y(t) = 2500 + 0 ∫ 6 (2+5sin(4πt/25)) - (15t/(1+3t))dt

c.) Find the rate at which the total amount of sand on the beach is changing at time t=4.

Rate sand is added - Rate sand is removed, when t=4
15t/(1+3t) - 2+5sin(4πt/25)
15(4)/(1+3(4)) - 2+5sin(4π(4)/25)
4.615 - 6.524
- 1.909 cubic yards of sand per hour

d.) For , at what time t is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.

Y'(t) = 0 and when t = 0 or 6 are critical points where as a minimum or a maximum may occur.
Y'(t) = S(t) - R(t) = 0
Y'(5.118) ≈ 0
Y(5.118) ≈ 2492.369 cubic yards of sand

Mean Value THeorem

1.) The purple line has a slope that is defined as [f(b) - f(a)]/(b - a) which is parallel to the yellow line which has a slope defined as f '(c).

A,B and C are outputs or points of y = f(x). The line that A and B produces is the secant line in which, according to the Mean Value Theorem, is parallel to a tangent line that is somewhere between point A and B on the graph of f(x).

2.) The figure below displays a removable discontinuity and an corner.
There is a jump between x = 0 and x = 3. The blue line illustrates the slope between points A and B which clearly shows that the slope of A and B cannot have a parallel tangent on the function. There is a corner between x = 3 and x = 7. The purple line illustrates the slope between points C and D which is 0. x = 5 is the center between these two points and has a slope that is undefined. THEREFORE, the Mean Value Theorem only applies to continuous and differentiable functions.

f(x) from f'(x)

1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Increasing - (-2, 0 )U( 0, 2 )
Since f '(x) displays the slope of f (x), whenever f '(x) is positive, is when f (x) is increasing.

Decreasing - ( -infinity, -2)U(2, infinity)
Whenever f '(x) is negative, f (x) is decreasing.
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2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A extrema occurs whenever f '(x) = 0 or undefined therefore in this derivative,when x = 0 and around -5/4 and 5/4 occurs a possibility of an extrema. Most likely when x = -5/4 and 5/4 are definite extremas because it is where f (x) changes its slope from negative to positive or positive to negative. When x = 0, there is probably no extrema because the slope does change after but rather says positive.
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3 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Concaves up - ( - infinity, -5/4)U( 0, 5/4)
Concaves down - ( -5/4, 0)U( 5/4, infinity)
f(x) concaves up whenever f "(x) is positive and concaves down whenever f "(x) is negative. Whenever the slope of f '(x) is positive is where f (x) concaves up and whenever the slope of f '(x) is negative is where f (x) concaves down.
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4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The power function is x^5. Since f '(x) is a x^4 function because the function changes its slope 4 times, the antiderivative of x^4 is x^5 thus the function has to at least reach a 5th power.
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My mindset

I think I am sort of in between of a fixed and growth mindset. I usually take on any challenges that come my way but I don't really try to find any. For obstacles, I tend to try it out for a while but then give up after because it's too time consuming. Sometimes I come back to it and try it again but would rather do something more "entertaining". I don't see effort as useless but rather see it as a opportunity to show my potential. Kind of hard to do since I'm a lazy guy. Criticism... Typically, I dislike negative criticism because it discourages me. After a while, I get over it and use that criticism to motivate myself to try better and sometimes don't take the criticism into consideration. For the success of others, I guess I feel sort of threatened. For a guy that lives on competition, the success of others tends to make me want to surpass them more.

Well, I can come up with one thing that had hurt me in calculus. In the homework, if there is a problem that I can't seem to figure out (meaning that it taking me about 30 minutes and still couldn't find a way to solve it), I just leave the work that I've done trying to figure it out and go on to the next problem.

The brain as a muscle? I guess it sort of works. A headache from doing word problems is kind of like soreness after running 4 - 6 miles except that the soreness lasts longer than a headache.

I honestly don't see this new info affecting my future. I just see it as extra information.

Concepts of Calculus

1.) "Finding the limit of a function" is basically saying, "What is the output approaching to as x gets closer and closer to a number?" while "plugging in c for x" is saying "what is the output when x = c". The limit may be different from when you plug in c for x.
When these two concepts are the same is when you are finding the limit of a continuous function.
For example,
f(x) = 4 - x^2 at x = 2
As a continuous function, the domain has no restrictions thus x can be any number therefore
the limit as x -> 2 of f(x) = f(2)

2.) Finding the derivative is basically finding the slope of the tangent line thus for both, you use your change of y over your change of x.
The difference is that finding the derivative is finding the slope of a curve at a point while finding the slope of a line, you are only finding the slope of that specified line.

Limits

3 ideas that elude me:

1.) Discontinuity when x approaches some number with a piece wise function
2.) Finding the extended function of the original equation
3.) The concept of integer x

Colleges?

Majors
1.) Game and Simulation Programming

You learn how to make video games! With this major you can also figure out how to create programs or technology that can provide reconstruction on crime scenes or provide simulations such as flying a jet for the Air Force. This requires a massive knowledge on mathematics. But more importantly, you learn how to make video games.

2.) Biomedical Engineering

With this major you learn how to craft artificial limbs or organs and learn how to create, fix or improve medical equipment to ensure stability of the success rate for a treatment. It requires a lot of mathematics on how to even create a device that would work with organic material.

3.) Electrical Engineering

With this major you learn how to work primarily with electricity. So like you learn how to fix the electrical cables in any house, build or fix a electronic device and more.

Colleges/Universities
1.) DeVry University

This university circles around with updated factors meaning that they use up to date methods on teaching such as using the internet for lecturing, sending assignments, or turning in assignments or the student works together with the professor. It provides flexibility on whether taking your classes online, in campus or both. If you thinking of taking a engineering major, you might need a laptop that has the minimum requirement for that major.

2.) UCLA

Nice school. It requires a minimum of 3.0 GPA. Provides free internet access to all students. You can house there. The university holds 419 acres of land...

3.) UC Berkeley

Known for being in the top engineering schools. 1232 acre campus. 32% tuition increase for all UCs...