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.