Free Responce

1. In order to find how much sand will be removed during a six hour period, one must find the integral between 0 to 6 hours of R(t)=2+5sin(4pi x/25)

Once you set that up, you solve and you end up getting about 31.815 cubic yds
2. We are given that when time is zero, there are 2500 cubic yards of sand. We are also given that the sand is added at a certain rate S(t)=15t/1+3t, and that it is removed at a rate of R(t)=s+5sin(4pi x/25). The formula would be the rate at which the sand is added minus the rate at which it is taken away plus the original amount. it would be fnInt(15t/1+3t)-(2+5sin(4pi x/25)+2500
3. in order to find the change at a certain time, all we have to do is replace the variable with the correct time, which in this casae would be 4 for t in fnInt(15t/1+3t)-(2sin(4pi x/25)+2500. It is S(4)-R(4) which ends up being -1.909 yards/hour
4. Besides the two end points, the only other critical point is at 5.118. When graphed, you can easily see that the end points are relative maximums rather than minimums. When inputs are plugged in before 5.118, the outputs of y' are negative, and the outputs of after 5.118 are positive, proving that this is certainly a minimum. now you plug in 5.118 instead of t. and you end up getting 2492.369 cubic yards

The Mean Value Theorem

1. The mean value theorem states that when given a function f(x), the slope of the secant line between two points f(b)-f(a) is equal to the slope of the tangent line at c

For example:
When f(x)=x² from [-4,4]
you find that the secant line is y=2x by using (f(b)-f(a))/(b-a) where b=2 and a=-2

in order to find the tangent line, you have to take the derivative of the original equation which would be f(x)=x²
which is equal to 2x this means that f(c)=2c
you set that equal to the slope which is 2 to get 2c=2
then you get c=1
now you imput that into the original equation
f(1)=1²=1
1 is the output or the y so the tangent is at point (1,1)

Now to find the equation for the tangent
y-y1=m(x-x1)
y-1=2(x-1)
y-1=2x-2
y=2x-1

and theres the equation

as you can see in the graph, both the tangent and the secant line are parallel

2. When a function is discontinuous or not differentiable, the mean value theorem will fail. there may be a secant line but there will not be a tangent parallel to it. For example if the graph is an absolute value graph, the secant line and the tangent lines will not be parallel. The slope of the secant would be 0. The only place where there could be a parallel tangent is at x=0 but there isn't because at x=0 it is non differentiable meaning there is no slope at that point. If the graph is discontinuous, the mean value theorem fails because there is no guaranteed point at x=c, meaning that not always will there be a tangent that is parallel to the secant line, because the point where the tangent line should be might not exist. There might be an equation like f(x)=1/x², in this function, there is no output at x=0. The slope of the secant line is 0 and there is no point in the function in which the slope of the tangent line can be 0 since there is no output at x=0

the Function f(x) from the graph f'(x)

1.The function is increasing from negative infinity to -1, and 0 to 1
the function is decreasing from 1 to 0, and 1 to infinity. The slope determines when the function is increasing or decreasing, and the given graph is the slope of the original function.

2. There is a local max at (1,4), because that is the point where the slope of the function changes from positive to negative.

3.The Function is concave up from x=0 to x=1, because that is where the slope of the given graph is positive, and the function is concave down at x=1 to infinity because that is where the slope of the given graph is negative.

4. The given graph is x^4 and that graph is the derivative of the original function which is not given. If you were to take the antiderivative, it could be x^5 or x^5+c

Test Review

In my calculus test I did a pretty horrible job with it. I made a silly mistake by confusing velocity and speed. I forgot that it was only speed that can't be negative. So in my free response, I only added the positive answer and ignored the negative, making me lose points. I learned that in a graph, the acceleration of a particle is zero when the graph goes completely horizontal and there in no tangent line in the velocity graph. I am still having a bit of trouble with graphing the acceleration in respect to time when given the graph of the velocity in respect to time.

Mindsets

1. I think i have a growth mindset. I am one of those people that learns from their mistakes. I don't keep repeating them
2. This mindset has helped me in math because whenever i get something wrong and i learn why i get it wrong i don't repeat the same mistake again. I may make other mistakes, but not the same one.
3. I already knew that the mind is a big muscle that can be trained since i took psychology so i wasn't all that surprised. The brain is something that is always growing with information. Nothing is ever really forgotten it is just stored away in the back of ones mind.
4. Well, with this i will attempt to find all my mistakes and even if i get criticized for them i will just take the criticism and use it to improve myself.