2005 FR 5 ( i edited my mistakes)

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.

fnInt(2+5sin(4π t/25),t,0,6)= 31.816 cubic yards

(b) Write an expression for Y(t), the total number of cubic yards of sand on the beach at time t.

Y(t) = fnInt(s(t))- fnInt(R(t)) + 2500

Y(t) = fnInt((15t/(1+3t)) - (2+5sin(4π t/25)) + 2500

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

(Y(t))=nDeriv (2500+fnInt (S(t)-R(t), t, 0, x) )=S(t)-R(t).

S(4)-R(4)=-1.909 cubic yards/hours.

(d) For , at what time t is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.

S(t)-R(t) = 0 to find critical pts.

critical pt at t=5.118,and 0 and 6

y(5.118)= 2492.369 cubic yards

t=5.118

minimum value at t=5.118 is 2 492.369 cubic yards

Thank you!!!

for correcting my mistakes!!!

very MEAN value Theorem (edited version)

Hmmm soo. from the comment i got from miss hwang... I am going to add to the second question...

The very MEAN value theorem states, that given f(x) of a smooth continuous and differentiable curve, there is at least one point on f(x) at which the derivative (slope) of the curve is parallel to the "average" derivative of the arc.

on the interval [a,b] when a≤c≤b... it is guaranteed that at one point,

Example

for the graph y=x^2 on [0,2]

the goal is we must find where point c lies on the interval...

by using this formula

where a=2

and b= 0

First finding the secant line

slope of the secant line is 2

y-y1= m (x-x1)

so from f(2)=(2)^2 = (2,4)

y-4=2(x-2)

y=2x >>>> is our secant line equation

Now we can find where point c lies

the derivative of f(x) = x^2 is 2x , therefore f'(c) = 2c

2c= m

2c= 2

c=1

Now that we have found where c lies (which is at x=1)

Now we can find the equation of the tangent line

f(c) = c^2

we have found that c=1

therefore, f(1)= 1^2 =1

the tangent line equation has to pass through (1,1)

We know that the tangent line passes through point c,

and it is paralell to the secant line (which is y=2x)

the slope of y=2x is 2

y-1=2(x-1)

y-1= 2x-2

y= 2x - 1

YAY!!!!

This only works for continuosus and differentiable functions...

for example: (square root (x^2)) +1

from [-2, 2]

this function is not differentiable at x=0

the mean value theorem FAILS here!!!

there is no tangent an x=0 because corners and cusps do not have a tangent line

so from the interval [a,b] in this case is [-2,2]

m= (f(2)-f(-2)) /2--2

= [(|2|+1)-(|-2|+1)]/4

=0/4

m=0

f(2)=3

y-3=0(x-2)

y=3 >>> this is suppose to be your secant line equation

but... sadly :( there's no secant line here.

because of the corner at x=0

but if.. you use the same function f(x) but on the interval [0,2] then the secant line is y=x+1

f'(c)= 1

the tangent line is also y=x+1

YAY!!!!!

i hope that this time my explanation is clearer

:)

now i gotta go comment more blogs

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?

f(x) is increasing from (-2,0)U(0,2). because f(x) is increasing when f '(x) > 0.

f(x) is decreasing from (-∞,-2)U(2,∞). because f(x) is decreasing when f '(x)<0.

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

extrema is at (0,0). because an extrema can only be found at a critical point, and a critical point is where f '(x)= undefined or 0. as we can see... f '(0)=0

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

it concaves up where the slope of f '(x) is positive and concaves down where the slope of f '(x) is negative.

concaves up: (-∞,-1.25)U(1.25,∞)

concaves down: (-1.25,0)U(1.25,2)

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

looks like an x^5. because it changed directions 4 times. also, f '(x) looks like an x^4 function. so briefly, the antiderivative of x^4 is somewhere around x^5 something... so i am assuming that f(x) is 5th power function.

Mindset!!!

FIXED OR GROWTH MINDSET?

Which mindset do you think you are a part of when it comes to "intelligence"? According to the reading, what tells you that you are of this mindset?

Can I say that..I am a mixture of both?

But rather towards the Fixed Mindset. However, I believe that people are born with different sets of genes. and its true. Genes express themselves by affecting an individual’s urge to learn, build and develop mental abilities throughout their life. Basically, I believe that people are born with a certain amount of intelligence, but it can be changed/developed as they grow. Because geniuses aren't only born, but can also be made.

Something that might affect a person's adult intelligence could be how they were raised as a kid, how their parents may have pushed them extra hard to learn new things vs. parents who do not encourage their kids to do anything.

I am a fixed mindset..
I believe because I like doing things that i am good at and avoid the things that I do not like.

Real life example..
I dropped out of Miss An's AP class because I didnt like History, however, my scores weren't that bad, I got an A in the first mester. But i was afraid that if i stayed in that class for longer, I might be too stressed/ overwhelmed and not be able to push myself all the way. I was scared that I might score low in the AP exam. So I just didnt bother trying for another 2 mesters for that class.
I also dropped out of AP Bio, AP English.. for the same reason. I dont like challenges in the fields that I am not good at. However, I like math, although I am not good at math.. but I am willing to accept the challenge.. because I like it, and I want to see how I will do.

How has this mindset helped or hurt you in math?

Well.. having a fixed mindset will hurt me. because I will just stop trying like i did before. But Now its too late to drop. And why drop. perhaps I like Math anyway! Math is exciting!!! I can never run out of problems to solve!!!!!

What is your reaction to finding out that the brain is just a big muscle that can be trained?

I can feel myself wanting to change already!!!

How do you see this new piece of information affecting your future?

I see myself in the future, with a whole new perspective. I will now open up my mind to challenges, accept criticism(this is hard- I admit), stop being discouraged by obstacles, put in 200% effort!!!! For a better future!!!!

We'll never get anywhere if we never start!!