Free Res;ponmse

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


Both R(t) and S(t) have units of cubic yards per hour and t is measured in hours for . At time t=0, the beach contains 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.
*integral of R(t) from 0 to 6 of R(t) dx
-Since R(t) is the rate at which sand is REMOVED, the following is input into the calculator to figure out the amount of sand that is removed after 6 hours: fnInt(2+5sin(4πt/25),t,0,6) ≈ 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.

-R(t) is used to determine the amount of sand that is removed at t and S(t) is used to determine the amount of sand that is added at t. As a result, you subtract S(t) by R(t) to find the TOTAL number of cubic yards of sand on the beach at time t and 2,500 is added since that is the amount of sand that the beach originally contains: y(t) = fnInt(15t/1+3t)-fnInt(2+5sin(4πt/25))+2,500

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

-Since y(t) represents the total number of cubic yards of sand on the beach at time t, y'(t) represents the RATE at which the total number of sand on the beach is changing.
y'(t)=15/(1+3t)^2 - 5cos(4πt/25)(0.05024)
y'(4)≈2.1243 cubic yards/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.

-At t=5.1178, the amount of sand on the beach is at its minimum since both of the rates are equal which means that the same amount of sand is removed as the same amount of sand that is added.
-At t=5.1178, there is a minimum of 4.6943(+2,500) which adds up to 2504.6943 cubic yards of sand

Mean Value THeroijem

1)Mean Value Theorem states that when f(x) is continuous and differentiable at all points in a specific interval, there is a guaranteed point that will have the same slope as the average slope of that interval.
* To determine where the average slope is the same as point c, this equation can be used :f'(c)= f(b)-f(a)/b-a


An example in which this theorem works is found in f(x)=x^2, [0,2]
the average slope of the function could be found using f(b)-f(a)/b-a:
*4-0/2-0=2


to find c,plug in c into f'(x)and make it = to the average slope and
*2c=2 -> c=1
The slope at c=1 is equal to the average slope. As a result, the secant line(y=2x) is parallel to the tangent line(y=2x-1).


2)The mean value theorem is not guaranteed when the function isn't both continuous and differentiable, because the tangent line may not exist. If the tangent line doesn't exist.
Example:
Discontinuous:The Mean Value THeorem does not work on this graph because of the jump at x=0. It's discontinuity makes it impossible to find the tangent line at c.


Not DIfferentiable: y=x^(1/3)
For this graph, the mean value theorem works at all points except when x = 0 because there is a vertical slope which prevents it from being differentiable.

In order for the Mean Value Theorem to work, the slopes in the specific interval must actually exist. Also the graph must be "smooth" or without jumps, corners, or cusps in the specific interval.

F'(x) graph

1. Where is the function, f(x), increasing? Where is it decreasing? How can you tell from this graph? Explain.

-f(x) is increasing when f'(x)>0
-f(x) is decreasing when f'(x)<0>

Since this graph is f'(x), it graphs the slope of f(x). As a result, The function is increasing(positive slope) between (-2,0)U(0,2). The function is decreasing(negative slope) between (-∞,-2)U(2,∞).

2. Where is there an extrema? Explain. (There are no endpoints.)

-There's a possibility of having "critical points" when f'(x)=0

• There is a local minimum at x=-2, because the slope=0 and it changes from negative to positive.
  
• There is a local maximum at x=2, because the slope=0 and it changes from positive to negative.
• There isn't a critical point at zero, because there isn't a change of signs
  

3. Where is the function, f(x), concave up? Where is it concave down? How can you tell from this graph?

-f(x) is concave up when f''(x)>0
-f(x) is concave down when f''(x)<0

• Concave up:(-∞,-1)U(0,1)
• Concave down:(-1,0)U(1,∞)

4. Sketch the graph f(x) on a sheet of paper. Which power function could it be? Explain your reasoning.

• If you were to find the anti derivative of f'(x), f(x) would be an x^5 power function, because f'(x) has 4 slope changes.
  

Mindsets

1. I believe that I have both, a fixed and a growth mindset. For example, there are times when I tend to avoid challenges, because I just don't feel like trying although that is not always the case. The majority of the time I tend to test myself and take on the challenges no mater what the situation is. I also take criticism and use it in a way that benefits myself. There's always time to change yourself for the better.

2. I came into the class knowing it was going to be tough. My growth mindset helped me accept the challenge to prove others wrong. I was going to do more than just try and do the best that I could possibly do. My fixed mindset was also beneficial, because it let me know what I needed to change about myself.

3. I had already been told by past teachers that the brain is a muscle that can be trained. For example, reading is just like a workout that trains your English language, but it's not something that can happen instantaneously. Overtime, changes take place little by little.

4. This piece of information will help me out in the future, because from now on I will continuously remind myself that there is always time to change yourself. It would be a great idea to let others as well know that there is always room for change to build their confidence and hope, because those two things play a major role in a successful life.

¿¡Que Dijo!?

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?

• When you're looking for the limit of a function at x = c, you're looking for the closest output (y) as x gets closer and closer to the constant from both sides
  
• When you actually plug in the number x = c, you're looking for the exact output at the constant
  

Ex: f(x) = int x

The lim f(x) as x->2+ is 2, but the lim f(x) as x->2- is 1
The lim f(x) as x-> 3/2 is 1

• Both cases are the same when the lim f(x) as x->c = f(c). In other words, both cases are the same when the function is contiunous, because the limits from the positive and negative sides of the constant are the same as the limit when you plug in x=c .
  

Ex: f(x) = 1/xThe lim f(x) as x->0=0, lim f(x) as x->1=1, lim f(x) as x->-1=-1, e.t.c.

What are the SIMILARITIES between finding the derivative and finding the slope of a line? What are the DIFFERENCES between the two?

• The main similarity between finding the derivative and finding the slope of a line is that you basically use the same formula:
  

m = change of y/change of x which is also known as (y2-y1)/(x2-x1)

• When you're finding the derivative, you are looking for the slope of the tangent line for that specific point as h approaches 0, although there are plenty of other ones that could be found on the same curve of the graph..
  
• When you're finding the slope of a line, you are finding the slope one specific line unlike when you're finding the derivative...
  

Limits

What do you still not understand about limits? Choose 3 problems or types of problems anywhere from chapter 2 that were the most difficult and that you would like to get more help on. OR you can simply explain 3 ideas / concepts that still elude you.

Limits are pretty confusing, but after a while you start to understand them better.

1. One of the concepts of limits that confuses me at times is discontinuity. I understand that in order for a point to be continuous, it's limits from both sides must be the same. The answers to questions like, "what are the points of discontinuity," do not always come to me quickly. There are always problems that really make me think about the graph. Y=int x is one example. Although, the graph is pretty simple, I have to draw it myself so I could picture it well.

2. Finding end behaviors is one of the few concepts of limits that interest me. Maybe, it's because of the fact that I find them fun and fairly easy.

3. Every now and then I have problems finding the limits as x approaches positive and negative infinite. For example, f(x)=sin x/x is one of those problems. When I first looked at the graph I had no idea what the answer was. When I actually took the time to understand the graph, I noticed that the further out the graph went, it kept getting closer and closer to zero on both sides.

3. vert.horiz.

College

3 Majors

Architectural Drafting- Those who major in this, make drawings by hand and computer for the purpose of creating buildings. I find drawing very fun and doing so on a computer would be really interesting. Also, I would like help out in making sure people are safe by enhancing the structure of buildings.

Computer Engineer- Study computers for the purpose of enhancing their performance through the use of mathematics, computer science, and physics. Computers are very interesting and I would like to know more about how they function. Also mathematics and computers are two things that I like and combining them both would make me love this major even more.

Social Psychology- Study the mind and behavior of all kinds of people. Some people could be a lot different than everyone else and it would be very interesting to study why they act the way they do and finding the source.

3 Colleges

Northern Kentucky University

• 4 year public college in an suburban setting
• 78% acceptance of applicants
  
• 17% had a high school GPA of 3.75 or higher
• 68% are in-state students
• 58% in top half of graduating class
  

Arizona State University

• 4 year public university in an urban setting
  
• has a major in computer engineering
• 90% acceptance of applicants
  
• 85% in top half of graduating class
• 30% had a high school GPA of 3.75 and higher

St. Mary's College of California

• 4 year private university in a suburban setting
  
• has a major in social psychology
• 81% acceptance of applicants
• 24% had a high school GPA of 3.75 or higher