A "Sandy" Free Response 2005FR 5

The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by

A pumping station adds sand to the beach at a rate modeled by the function S, given by
a) The integral of R(t) within the interval
will output the amount of sand that the tide removed.

PLUG IN f Int (2+5sin(4πt/25) , x, 0, 6)

Answer is approximately 31.815 cubic yards of sand.



b) The total number of cubic yards of sand on the beach would have to include the pumping AND removing of sand AND the initial 2500 cubic yards of sand at t=0, So...

y(t) = f Int( S(t)-R(t) dt ) +2500
= f Int( 15t/(1+3t) - 2+5sin(4πt/25) dt ) + 2500



c)TOTAL amt of sand at t=4.

f Int ( S(4)-R(4) dt), x, 0, 4) +2500
= f Int [ 15(4)/1+3(4) - 2+5sin(4(4)π/25 dt] +2500
= 4.6154 - 6.5241
= -1.908 cubic yards of sand

d) On my graphing calc, i input-ed Y1 as R(t) and Y2 as S(t). Then, i hit 2nd calc and pressed 5 to find the intersect of both graphs. The intersect will give me the minimum.
The two graphs intersect at 5.118, but this is not yet the answer. To find the exact value it must be plugged into our Y(t).

=f Int ( S(5.118)-R(5.118), x, 0, 5.118) + 2500
=20.93254581 - 28.56306324 + 2500
= -7.63051743 + 2500
= 2492.369 cubic yards of sand = MINIMUM!!!


Sorry if im not clear in some areas, its 1 AM!!

Mean Value Theorem

The Mean Value Theorem connects the average rate of change with an instantaneous rate of change. It states that if a function is continuous AND differentiable on an interval then there is at least one point in the function where an instantaneous rate of change is equal to the average rate of change.

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

1. Explain what this means graphically by showing an example.


The graph shown here is f(x)= x^2+1. First of all the graph is both continuous and differentiable so it is a possible candidate for the Mean Value Theorem. Now let's apply a secant line (or an average rate of change)

The green line [y=2x+1] represents f(b)-f(a)/(b-a). The two points where the green line intersects our parabola are the points "a" and "b".

The tangent line is parallel to the secant line, meaning that instantaneous rate of change is equal to the average rate of change.

2. Explain why this only works for continuous and differentiable functions.
The Mean Value Theorem only applies to continuous and differentiable functions because if a graph has a discontinuity of any kind, then it disrupts the normal flow of the graph and it would mean that the interval has stops. You cannot find instantaneous or average rates of change when there is a chunk of the graph missing. The function also has to be differentiable because there needs to be a slope in order for any graph to have a rate of change.

This graph, y= |x|, is not differentiable at x=o because it has a corner. A "corner" has an infinite amount of tangent lines.

The function f(x) from the graph f '(x)

1. Where is the function, f(x), increasing? Where is it decreasing? How can you tell from this graph? Explain.
   The function f(x) is increasing when the slope is greater than zero f '(x) > 0; therefore, at (-2, 0) U (0, 2). The function is decreasing when the slope is less than zero f '(x) <>; therefore, at (-∞, -2) U (2, ∞). The f '(x) graph, "outputs" the slope.
   
2. Where is there an extrema? Explain. (There are no endpoints.) There is a local minimum at (0,0), because at that point the slope changes from negative to positive.
   
3. Where is the function, f(x), concave up? Where is it concave down? How can you tell from this graph? The function has a positive concavity at (-∞, -1.25) U (0, 1.25) because that is where the slope of the graph of f '(x) is positive f ''(x) > 0. The function has a negative concavity at (-1.25, o) U (1.25, ∞) because it is where the slope of the graph is negative f ''(x) >0.
   
4. Sketch the graph f(x) on a sheet of paper. Which power function could it be? Explain your reasoning.

Mindset

1. Fixed Mindset and Growth Mindset. I believe that I am part of the Growth Mindset but I may sometimes express a Fixed Mindset in times of great stress. I think that I am of this mindset because I am able to learn well from criticism, I see effort as something positive, and I find inspiration in the success of others. I am not a person who takes things personally when they should not be taken personally. Whenever I receive critiques from classmates or teammates, I automatically gather their advice and I put it to good use. As for effort, I understand that one needs to put effort in order to achieve better things. Also, I do not overcome with jealousy if someone is smarter than I am. Instead, I look into more ways that I can improve my thinking. I have a strong belief that a person can do anything they set their mind to do. As long as I have goals, I know that I will get far in life.
2. The Growth Mindset has helped me in math because it prevented me from dropping the class. With this mindset I was able to take advice from classmates and make the decision to stay in the class.
   
3. It's a relief to hear that because it would be dreadful if intelligence could not be changed. I would probably cry if it were the other way around.
   
4. I think that as long as I remember the two mindsets and distinguish the good from the bad, good things will happen and I will be successful.

Algebra vs Calculus

1. 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?
- The difference is that when you find the limit of a function at x=c, you are aiming to find the y value as you approach c. However, the limit does not necessarily mean finding the limit of a point. A limit can still be found at a discontinuity such as a "hole". On the other hand, when PLUGGING in the number at x=c, there will be an exact point or output and there will be continuity at that point. They are the same when the function is continuous.


2.What are the SIMILARITIES between finding the derivative and finding the slope of a line? What are the DIFFERENCES between the two?
- The similarities are that a derivative and a slope are the change of y/change of x.

-The difference is that a derivative finds the slope of a curve's tangent line at one point and a slope finds the m of a simple line at two points. (when finding the derivative of a tangent line, you first have to find the slope of the secant line and then the take the limit of the secant line's slope to find the slope of the tangent)

I've reached my limit

I understand the general ideas and concepts from this chapter but there are a couple of problems that I did not find answers to.

1. For homework f2 (its not really about limits, i think, but its in chapter 2 ) I'm not sure how to do #30. I skipped a couple of questions on that homework.

2. On homework E3 I did not really know how to do part B for (27-30). I'm not really sure how I am supposed to explain the behavior of the function to the left and right of each vertical asymptote.

3. On homework f1 #18, I am not sure how the answer is infinity, I didn't get anything near that!

I think that's pretty much it..

Colleges!!! (Yay)

MAJORS AND COLLEGES

The top three majors I am interested in...

• NURSING (RN): I'm sure we have all been to a hospital and when we see people in scrubs or white coats we just think "oh they're all doctors". Well let me tell you now, that is not necessarily true. A registered nurse is not a doctor but instead are like doctor's assistants. RN's are basically the people who attend you in the emergency room or when you are not feeling well and go to the hospital. They have the ability to administer shots, check a patient's vital signs, and report everything to the head honcho (the doctor). It would be appropriate to say that nurses are like detectives; They have to figure out what is wrong with the patient and how to help get them better.
  

• Anesthesiologist: This major is very interesting but also requires several years of school. To become an anesthesiologist it takes about 12 years. Four years at a university, medical school, internships, and residency. Anesthesiologists are one of the many doctors working in the OPERATING ROOM. They have to administer anesthesia to the patient and are responsible for managing the medical care of patients before, during, and after surgery.
  

• BIOMEDICAL ENGINEERING: This major is for people who are interested in the medical field, biology, and engineering. Biomedical engineers do a lot of research but most importantly, create medical devices such as artificial hearts, pacemakers, and many machines that are seen in hospitals today. I thought this major was interesting because it involves helping people in a very cool way.

I think these majors fit me because I am a very "hands on" person and I enjoy learning about the human body. I also want to help people live longer and better lives.

I'm surprised I was able to narrow it down to only three colleges! My top choices are...

• UNIVERSITY OF WASHINGTON: The University of Washington is found in the state of Washington. It is known as one of the best schools for Nursing. Its mascot is the husky and the school colors are purple and gold. It is one of the oldest Universities in the west coast and by some is considered a "public ivy".

• UCLA: UCLA is known as one of the best UC's. It interested me because of its unique nursing program and also because it has a variety of majors. For the 2008 Freshman Admission Profile, UCLA's admit rate was only 22.7% for the 55,406 that applied. The average high school GPA is 4.15.

• UC BERKELEY: Another UC, which has a strong engineering program, is UC Berkeley. It is located in Northern California in a city called Berkeley. The admit rate for Berkeley in 2008 was 21.4% (less than UCLA) meaning that out of all 48,462 applicants, only 10,387 were admitted.

I really enjoyed writing this blog post and I'm looking forward to reading what everyone else wrote :)