Free Response =D

a) To find out how much sand will be removed, you must find the interval of the function from 0 to 6; which is about, 31.816 cubic yards.

b) Y(t)= S(t)-R(t)+2500
S(t) is the sand being added subtracted by R(t) which is the sand being taken away + the initial sand the beach originally had, (2500).

c) To find out the rate of sand being taken out at time 4, we have to plug it into Y(t), which is -1.909 cubic yards/an hour.

d) Graph both S(t) and R(t) into the graphing calculator and find the point where they intersect. The point would be, (5.1178, 4.6943), so 5.1178 is the minimum, and the output at that time would be 4.6943, since we are trying to find out how much sand we have at that time, we add 2500 (initial amount of sand)+4.6943=2504.694.

revision!

1. Mean Value Theroem
f'(c)=f(b)-f(a)/b-a
What this means is that the slope of the tangent line is parallel to the slope of the secant line. Both of the lines will have the same slope and be equal to one another.

An example would be where:
f(x)=cos(x)+1the blue line would be the tangent line (y=2) and the green line would be the secant line (y=1) the interval would be (-2,2). they are parallel to each other and their slopes equal on another as well.

2. Why does the Mean Value Theorem only work for continuous and differentiable functions?
- A continuous function has the same limit from both the negative and positive sides.
- A diferentiable function has a slope at a certain point, If f is differentiable at a point c, then f is continuous at that point c.

Here are some examples:

Differentiable but not continuous:

In the function: x^(-1)+1, there is a hole at x=0, making it differentiable but not continuous because there is a hole discontinuity in the graph.

Continuous but not differentiable:The function:x^(1/2)+1 on the interval [-1,1] is continuous, but not differentiable because there is a cusp at x=0.

A very MEAN value theorem,

The mean value theorem shows how the slope of the tangent line and the slope of the secant line are parallel to each other. The equation that we would use would be x^2.
The blue line would be the tangent line (derivative) and the green line would be the secant line.
by using the mean value theorem, we can prove that the tangent and secant line are parallel to one another.

If we use the interval [0,2], then the equation of the tangent line is 2x-1.
The derivative would be 2x.
They would be parallel at x=1.


The mean value theorem fails when:

There is a point when the function is not differentiable.
such as: f(x)=abs(x)And where a function would be discontinuous
an example would be at f(x)=1/x

The function would be discontinuous at x=0.

f(x) on f'(x)

1. The function increases at [-2,0]u[0,2] since f'(x) is a velocity graph and that is when it is above the x axis. It is decreasing at (-∞,2]u[2,∞) since that is when it is below the x axis.

2. Local minimum at x=0 and Local maximum at x=+-1.25

3. You need the graph of f(x).

4. The graph changes its slope 4 times, so by getting the derivative of the function, it makes the power function 5.

Mindsets

1. I honestly think that I am more on the growth mindset side because I am always trying to be better at something. I am usually pretty competitive when it comes to anything. I do not let failure discourage me, on the other hand, I usually try harder to make sure that I can do things right the next time around.

2. I think that this has helped me in math because I didn't drop calculus even though i finished the last mester with a D. If i had a fixed mindset, I would have most likely dropped out of it. So this mindset has really helped me.

3. It makes me feel like nothing is impossible. If I keep working at it, I can achieve anything i want and i would have control over it. The only thing that could stop me is time.

4. Well it will affect me because now i know that I can do anything. Now i know that I can learn, and that nothing physical is going to stop me. Like now I know that if i try hard enough, I can learn anything I set my mind to, and I'll have one buff brain, lol =].

Differences

The difference is that when you are trying to find the limit as x approaches c, you are trying to find where the function is discontinuous coming from either the positive or negative or both sides. It can give you two different outputs. When you are plugging in c for a function, you get an exact point. The two cases are the same when the discontinuity is removable, you can get an exact point like when you plug in the constant.

Some similarities are that in the end, you are still looking for a slope. You use a formula in order to find both of the slopes. A difference would be that when you find the slope of a derivative, you are looking for the slope of the tangent line, and when you are looking for the slope of a line, you are just looking for the slope of a specific line.

LIMITS

Well, limits have been the hardest thing and most confusing thing for me to learn. I hope I end up understanding them.

Top three problems:
1) How do you find the limit of infinity if you don't know how to graph it?

2) How to solve a limit with sin or cosine or anything like that.

3) Graphing trigonometric functions when there is more than just a variable.

Well I hope I understand it better soon.