2005 FR 5

(a) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure.
How much sand is begin removed: use the formula R(t)=2+5sin(4piet/25)
nInt(2+5sin(4piet/25)),t,0,6) = 31.815931 cubic yards of sand [exactly how you input it into a TI-89 calc.]

(b) Write an expression for Y(t), the total number of cubic yards of sand on the beach at time t.
At time t=0, the beach contains 2500 cubic yards of sand.
Create a new formula with the formulas given.
Y(t)=(15t/(1+3t))-2+5sin(4piet/25) dt +2500 (add the 2500)

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

find the derv. of y(t)
Y'(t)=(15t/(1+3t))-2+5sin(4piet/25)
plug in 4 for t
Y'(4)=(15(4)/(1+3(4)))-2+5sin(4pie(4)/25)
4.6153846- 6.5241353
= -1.908yard^3 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.
need to find where t is a minimum...
equal Y'(t)= 0 to find critical points.
nDeriv or Y(t)
in the interval of 0 to 6...t= 5.117
min value at 2492

im not sure if that's right though...

Mean Value Theorem

f'(c) = [f(b) - f(a)]/(b - a)

1. Explain what this means GRAPHICALLY by showing a good example!


The average rate of change of a function is the slope of the secant line. the secant line connects the function values at the endpoint of the interval. endpoints of the function on [a,b]are (a,f(a)) and (b,f(b)).
m=y2-y1/x2-x1 = f(b)-f(a)/b-a

the instantaneous rate of change at x=c is the slope of the tangent line to the function.


The function must be continuous on [a,b] and differentiable on (a,b) then the mean value theorem must have at least one value x=c between a and b == f'(c)=f(b)-f(a)/b-a. soooo at least one x value on the interval where the tangent line is parallel to the secant line that connects the endpoints of the interval.






the function= x^3
secant line= 3x ===> f'(x)=3x [0,2]
tangent line= 3x-2

tangent line is parallel to the secant line



Explain why this only works for continuous and differentiable functions.




sqr root of (x^2)-2
this is an example of a function that is not differentiable because it has a corner at x=0 within the interval of (-1,1) it is not differentiable but this function is continuous.

this function does not satisfy the theorem because it does no have a tangent that can be parallel to secant.




this function has a discontinuity but is differentiable.
piecewise function of
f(x)= {x^3+3 for x<1
{x^2+1 for x>(or equal to) 1
close hole at one and at the same time of open hole at one.

the function is not continuous within the interval of [-1,1]

Mindset: Growth

1. Which mindset do you think you are a part of when it comes to "intelligence"?
I believe that I am more of a growth mindset when it comes to intelligence because i tend to look for the challenges and rather than quitting I fight through the end to become stronger in that sense.

According to the reading, what tells you that you are of this mindset?
2. How has this mindset helped or hurt you in math?
3. What is your reaction to finding out that the brain is just a big muscle that can be trained?

2) This has helped me in math because although it is not my strongest subject I have gone on through it and tried out those new challenges. At first it may seem hard but I don't give up that easy and so practice until i get it right. Instead of giving up i keep my feet on the ground and tell myself it will all pay out in the end.

3) I was surprised that their was another being who thought the same as me. I thought it was as every other muscle in the human body which can grow and become "stronger" The statement was taken for granted by others but im glad there are others who see it as a muscle that can be trained. [:

4. How do you see this new piece of information affecting your future?
4)It just gave me much more hope in succeeding in math one way or another! Like it says, it has inspired me!

Calculus vs. Algebra; Algebra vs. Calculus

i thought i forgot my password but it seems that my email was .com rather than .net -_____- att.com? -___-

# What is the DIFFERENCE between finding the limit of a function at x = c and actually plugging in the number x = c? When are the two cases the SAME?

finding the limit of a function at x=c is when f(x), y, gets closer as it gets closer to c. So your finding the value of y as it approaches. Never will there be an exact point because sometimes there can be a hole at that point.
Plugging in relates to the exact point. The exact point for the value of y even if it does not really approach.
With a continuity, both are the same. with the limit of f(x) as x> c =(c) then these cases are the same. Y values are the same with the same points

# What are the SIMILARITIES between finding the derivative and finding the slope of a line? What are the DIFFERENCES between the two?
the change of y over the change of x. The formulas to finding them differ but are the same concepts.
The difference for finding a derivative is that we use limits. We use the limit as H approaches zero. We use h as the difference point, as h+ or h- The derivative is used to find the slope of a tangent line.
Finding the slope of a line is only finding the slope of that line. Finding the slope of a line at a curve is finding the tangent line..

Reaching the Limit

I asked my friend to help me understand limits yesterday (Tuesday). He did and believe I have a different perspective on it and understand it a little bit more using the table on the calculator. But, maybe calculators can't be used on the AP test so i need to learn it more and be proficient. I think if I study more and do more problems then ill get better at it. As they say, to understand calculus is to get yourself working on more and more problems.

I think I need more understanding when the limit approaches infinite as well as discontinuity and how to remove a discontinuity, removable.
p.84 #20 With piecewise functions. Need help determining if its removable or not.

Colleges O_O.....^-^

Majors:
Preveterinary Studies:
There are only 28 veterinary schools in the United States resulting in high competition. In most of the schools people cannot major in preveterinary but must major in other courses such as Biology to be exposed to similar courses. The programs given in preveterinary studies help prepare for admission to vet colleges by giving a strong understanding in science and other studies. For preveterinary studies, the Graduate Record Examination (GRE) or another standardized entrance exam will be needed to be taken by students.

Vetenarinary Technology: The major only takes two years to learn how to do things a vet does. In training a VT will learn how to take x-rays, anesthetize an animal for spaying or neutering and analyze blood for samples. VT majors will learn how to help with medical care to animals and assist vets in laboratories and practices. VT's can do anything as vets can except prescribe medicine, perform surgery on animals and make diagnoses. VT's run the risk of injury: getting bite, scratched, or kicked! ^____^ VT's will care for campus animals! Man:"Please help! my fish isn't breathing!" me: "well maybe if you placed it in water then..." Man:"oh...THANKS!" me: -__- i thought this was a good college filled with....

Biology and Biotechnology Laboratory Technology:
Biologist studying living creatures: look at organisms, their biochemistry, and their genes. This major will train people to assist biologist and help them conduct researches, practices and etc. They help with experiments at the universities for industry and the government. These technicians help scientist in medical research, for example, the cure for AIDS. This filed requires students to write lab reports!

Colleges:
Following two are for Biology and biotechnology Laboratory Technology
State University of New York at Brockport:
A liberal arts college that have 48 undergraduate majors. Founded in 1835, the school provides all types of sports like indoor track and field and x-country. This schools provides students with a variety in studies to transform them into well-rounded and multidimensional students who will be successful on their own. Undergraduates are provided with many undergraduates programs such as the Honors program, Delta college program, and tradition general education program. Students are provided with many services such as student advisement, health services and student accounts.

California Lutheran University:
Founded in 1959 this private university with 67 majors and minors, the academic programs help assist in helping choose a career. Professors, beside teaching scholars, are advisers that will help in committing to help with your career. Freshman year, for all students, will be provided with professors rather than assistants. There are 12 residence halls that ensure students a home because 85% of time is spent outside. The average class size is 22 students which gives the opportunity for professors to get to know their students resulting in establishing a real relationship.

California State Polytechnic University Pomona:
This school provides 48 academic departments; Animal and Veterinary Sciences Department
This program provides students to study cellular DNA and up. They study genetics, breeding principles, reproductive physiology and produce embryos and clones. The students would perform artificial insemination, monitor the pregnancy, and witness birth! hand out the pink or blue flag!

Approximately 3200 students live in traditional dorms, residential suites, and university apartments. Current annual fees are around $4,551 for undergraduates.

Tips and Hints

1. Share how you remember Transformations. Do you have any tips or hints that help you remember?
   

Flips, Slides, Stretches and Squishes!!

SLIDES:

When starting with the function of f(x) focus on where the points will go then focus on the the graph is going to look like. When something is added or subtracted from a function then the function shifts the graph vertically. Lets say you add 3 to the function of "P" then this indicates that P will move 3 times up and if you subtract 4 then the graph will move vertically 4 units down. Therefore, the point on the graph of f(x)-3 should be plotted 3 units down the original graph of f(x)
Adding parentheses to a function determines the operation shifting horizontally on the graph. Subtracting moves the graph to the right and adding will shift the graph to the left. The graph of f(x-3) is the same as f(x) but the only difference is that it will move 3 units to the right. The parentheses with f(x-3) moves the function RIGHT not left but will adding then it moves LEFT not right. Merely think of this as the opposite...the opposite of right is left and left is right.
FLIPS:
Basically the graph of -f(x) is the reflection of f(x) across the x-axis. The trick here is to keep in mind a mirror image of the graph. How would it look like if you folded it, turned it, had it overlap and etc. Multiplying x by negative one has the graph reflect across the y-axis; if f(x) has the point (a,b) then f(-x) has the point (-a,b)
SQUISHES:
Multiplying the y-values by 1/2...affects the y values. If the graph of f(x) has the point (x,y) then the graph of f(x)timesE has the point (x,Etimesy) In this case each point on 1/2 f(x) is half of the distance from the x-axis as the point of the graph of f(x). When you have a number outside the parentheses such as 2f(x) then you are simplying making it smaller.
STRETCHES:
Multiplying f(x) by a number affects the distance of the points from the x-axis which leads to stretching it away from the x-axis. Multiplying x by a number inversely affects the distance of a function's coordinates from the y-axis. When you have a number inside with the x then you are stretching that out.

2. Share how you remember or understand trigonometry. Do you have any tips or hints that you remember/ memorize all those facts?

Unit Circle

Okay some of the tricks I learned by myself were the values of the unit circle. Think of 2pie as your starting point. since its on the right side of the unit circle and on the horizontal line then visualize it like a graph. On the right of the horizontal line you your positive numbers...well 2pie is point (1,0) because its on the right side of the graph on the horizontal line...Now the opposite of 2pie is pie which is on the left side of the graph, so e know that pie's point is (-1,0) because all numbers to the left of the horizontal line on a graph are negative. Now the vertical line as we know is y...and for the unit circle you can use it as so to assist you. The top of the unit circle is pie/2...think of this as whats half of pie...so the top has the points of (0,1) because its on y. The opposite of pie/2 is 3pie/2 which has the point (0,-1) Okay going back to 2pie, from that point going counter-clock wise we will hit pie/6, then pie/4 then pie/3 from this point on its an order...after pie/3 we come across pie/2 which is half of pie. okay since our last number was pie/3 then we will come across 2pie/3. Okay now the sequence i am trying to show you is: 6, 4, 3...3, 4, 6...6, 4, 3...3, 4, 6. This indicates that the unit circle is divided into four sections. think of the unit circle as a division of 4 quadrants. Okay know that you have those denominators its time to add the numerators. the right upper quadrant of the unit circle all consists of just pie on top of the denominator such as pie/6 and pie/4 and pie/3. The second quadrant which is the left upper one will be filled up as the following: the number over 3 will be 2. The next number over 4 will be 3pie/4 now the trick for these is that in Q1 you have pie/4. okay count this one as one and next count pie/2 as number two then in Q2 3pie/4 will be three hence "3"pie/4..then pie will be number 4...the in Q3 5pie/4 will be number five hence "5"pie/4 then 3pie/2 will be counted as number six and 7pie/4 will be number 7 because "7"pie/4....Now i rather explain the rest visually...MOVING ON
the values for the domains are quiet simply....anything with a denominator of four corresponds to the square root of 2/2
Anything with a denominator of 6 will be square root of 3/2, 1/2
Anything with the denominator of 3 will be 1/2 first THEN square root of 3/2

3. What still confuses you or worries you about trigonometry?
Well so far i understand everything...i know my unit circle even with the pressure of writing it in 2.5 time...I believe i know how to graph them...possibly the inverses of the functions along with the domain and the range...but i know i will learn that during the weekend because i have a a book that will help me.