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), which is integral (2+5sin(4 pie t/ 25)from 0,6, and the amount of sand removed comes out to be 31.816. the reason for finding the integral of R(t) is because it outputs how much sand was removed from 0,6, which is 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.
since R(t) is the rate of how much sand is being removed and S(t) is how much sand is being added, and 2500 is how much sand there is at the start, you would have to subtract how much is removed from how much is added, then add it to 2500 since that's how much sand there is, Y(t)= 2500 + (S(t)- R(t)) or Y(t)= 2500 + ((15t/1+3t) - (2+5sin((4 pie t)/25)
(c) Find the rate at which the total amount of sand on the beach is changing at time t=4.
since we want to find the total rate of how much the sand is changing at t=4, then we would plug in 4 into Y(4)=((15(4))/1+3(4)) - (2+5sin((4 pie (4))/25) without 2500, and we would get the total rate of how much sand is changing because we subtract how much sand is being removed which is 6.524 from how much sand is being added which is 4.615, and get -1.909, but since we want the total, doesn't matter if its negative or positive, its 1.909 cubic yards of sand
(d) For 0,6, at what time t is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.
At t=5.1, because that's where it intercepts and it looks that both equations are Even, the rate is nearly the same. the minimum value would be Y(5.1)= 2500 +((15(5.1))/(1+3(5.1))- (2 + 5sin((4 pie (5.1)/25) which equals 2499.961 cubic yards of sand which is the minimum value.
Sunday, April 4, 2010
Saturday, March 6, 2010
mean value theorum( revised)
The mean value theorem is f'(c)= [f(b)-f(a)]/(b-a) .
What this means is that from [a,b] , it must be continuous and differentiable, the secant lines between them is Parallel to the tangent line of c, the midpoint or average of [a,b], which is c.
For example in the equation y=sin(2x)+3, there is a secant line on the interval [0,1.56], that secant lines slope,green, will be the same as c tangent lines slope,blue, the midpoint of that interval, which makes them parallel.
2. it doesn't work for non differentiable and/or continuous functions because for non differentiable and non continuous functions is the equation y=2/x^2
since the secant line would be at 4, the midpoint of that interval,c,would be at 0,due to the discontinuity, there is not midpoint and is not differentiable, meaning c has no tangent line because of the discontinuity, and since we cant find the slope of tangent of c, then the mean value theorem is not applicable to functions that are not differentiable and not continuous.
a function that is continuous but not differentiable would be abs(-x/2)+1
It is continuous , but at x=0, there is a corner. As x approaches 0 from -1 from the negative side, its slope is not the same as x approaches 0 from the positive side from 1, they do not have the same tangent line. Since they do not have the same tangent line, it is not differentiable at point c=0, because the slopes are different and the tangent line from the negative and the positive side are not the same.
What this means is that from [a,b] , it must be continuous and differentiable, the secant lines between them is Parallel to the tangent line of c, the midpoint or average of [a,b], which is c.
For example in the equation y=sin(2x)+3, there is a secant line on the interval [0,1.56], that secant lines slope,green, will be the same as c tangent lines slope,blue, the midpoint of that interval, which makes them parallel.
2. it doesn't work for non differentiable and/or continuous functions because for non differentiable and non continuous functions is the equation y=2/x^2
since the secant line would be at 4, the midpoint of that interval,c,would be at 0,due to the discontinuity, there is not midpoint and is not differentiable, meaning c has no tangent line because of the discontinuity, and since we cant find the slope of tangent of c, then the mean value theorem is not applicable to functions that are not differentiable and not continuous.
a function that is continuous but not differentiable would be abs(-x/2)+1
It is continuous , but at x=0, there is a corner. As x approaches 0 from -1 from the negative side, its slope is not the same as x approaches 0 from the positive side from 1, they do not have the same tangent line. Since they do not have the same tangent line, it is not differentiable at point c=0, because the slopes are different and the tangent line from the negative and the positive side are not the same.
Saturday, February 13, 2010
f(x) and f'(x)
1.the function is increasing from (-2,0)U(0,2) because the output is positive and is decreasing because the output is negative. i can tell because if the slope outputs a positive number, even if the slope is negative but the output is still positive, then the function is increasing, if th output is negative, then it is decreasing.\
2.The local minimum at -2 because its before the output changes from negative to positive.
The local minimum would be at 2 because that's before the output turns from positive to negative.
3. Its concave up when its at (-infinity, -1) U (0,1)
concave down (-1,0)U(1, infinity)
concave up when f''(x) > 0
concave down when f''(x)<0
4. its x^5 because slope changes 4 times, since its the derivative of f(x) which would be x^4 which is -1 less than x^5 when u take the derivative of it, then it mus be x^5.
2.The local minimum at -2 because its before the output changes from negative to positive.
The local minimum would be at 2 because that's before the output turns from positive to negative.
3. Its concave up when its at (-infinity, -1) U (0,1)
concave down (-1,0)U(1, infinity)
concave up when f''(x) > 0
concave down when f''(x)<0
4. its x^5 because slope changes 4 times, since its the derivative of f(x) which would be x^4 which is -1 less than x^5 when u take the derivative of it, then it mus be x^5.
Thursday, January 14, 2010
fiiixxxed mindset!
1. 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?
2. How has this mindset helped or hurt you in math?
3. What is your reaction to finding out that the brain is just a big muscle that can be trained?
4. How do you see this new piece of information affecting your future?
1. my mindset about intelligence is a fixed mindset. What tells me im this mindset is the effort part. After working on something for a long time i end up in the same place i began, then that's a back breaker and i just stop and move on to doing something else.
2. this mindset hasn't really helped me much, since i try to get everything right at the first try, and if i try something a few times i just get frustrated and move one to the next thing. usually i try to find a way so i can remember how to do something easier , so i wont confuse myself, but its hard since sometimes i cant solve some problems and get frustrated.
3. I already knew that is was a big muscle that can be trained, and i also know that you have to train it. By training it is practicing something the right way, or understanding something and practicing it so it stays in the brain and i can easily remember it. Its just the effort that i fall sometimes short off, i get lazy or thing that i will do good enough with the basics and don't really push myself to train my brain.
4. It will affect it i think a lot. It sort of give me motivation to try harder in things like my homework and not just try it and stop, but actually try it and try to figure out why i don't understand the homework. I think i will start to have better habits, like study habits, since studying is a big part of passing or understanding anything, or it makes it a lot easier!
2. How has this mindset helped or hurt you in math?
3. What is your reaction to finding out that the brain is just a big muscle that can be trained?
4. How do you see this new piece of information affecting your future?
1. my mindset about intelligence is a fixed mindset. What tells me im this mindset is the effort part. After working on something for a long time i end up in the same place i began, then that's a back breaker and i just stop and move on to doing something else.
2. this mindset hasn't really helped me much, since i try to get everything right at the first try, and if i try something a few times i just get frustrated and move one to the next thing. usually i try to find a way so i can remember how to do something easier , so i wont confuse myself, but its hard since sometimes i cant solve some problems and get frustrated.
3. I already knew that is was a big muscle that can be trained, and i also know that you have to train it. By training it is practicing something the right way, or understanding something and practicing it so it stays in the brain and i can easily remember it. Its just the effort that i fall sometimes short off, i get lazy or thing that i will do good enough with the basics and don't really push myself to train my brain.
4. It will affect it i think a lot. It sort of give me motivation to try harder in things like my homework and not just try it and stop, but actually try it and try to figure out why i don't understand the homework. I think i will start to have better habits, like study habits, since studying is a big part of passing or understanding anything, or it makes it a lot easier!
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