course Mth-173 Sorry that I just submitted this form twice, I have a nasty habit of hitting enter instead of tab to change lines. Qvjbٷfassignment #003
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11:02:25 Query class notes #04 explain how we can prove that the rate-of-depth-change function for depth function y = a t^2 + b t + c is y' = 2 a t + b
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RESPONSE --> we can prove this by averaging to points plugged into the depth function, and when simplifying the problem we get y=2at+b+c(delta)t as (delta) T nears 0 then so does c(delta)t and so that part of the problem is no longer needed, therefore we end up with y'=2at+b
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11:03:57 ** You have to find the average rate of change between clock times t and t + `dt: ave rate of change = [ a (t+`dt)^2 + b (t+`dt) + c - ( a t^2 + b t + c ) ] / `dt = [ a t^2 + 2 a t `dt + a `dt^2 + b t + b `dt + c - ( a t^2 + b t + c ) ] / `dt = [ 2 a t `dt + a `dt^2 + b `dt ] / `dt = 2 a t + b t + a `dt. Now if `dt shrinks to a very small value the ave rate of change approaches y ' = 2 a t + b. **
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RESPONSE --> That is essentially what I had except I didn't write out the whole equation I just explained it. I will try to write out the whole thing next time.
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11:08:38 explain how we know that the depth function for rate-of-depth-change function y' = m t + b must be y = 1/2 m t^2 + b t + c, for some constant c, and explain the significance of the constant c.
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RESPONSE --> y' = 2at+b is the universal rate function, so we divide M by two which gives us (1/2)m t^2 + bt + c = y. C is not given to us in the equation so we must substute in a number for c as long as it is the same number through out the work it will correctly work the problems. But if you are given information that says that at time=0 it is known that the depth is 130 then c= 130, etc.
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11:12:44 ** Student Solution: If y = a t^2 + b t + c we have y ' (t) = 2 a t + b, which is equivalent to the given function y ' (t)=mt+b . Since 2at+b=mt+b for all possible values of t the parameter b is the same in both equations, which means that the coefficients 2a and m must be equal also. So if 2a=m then a=m/2. The depth function must therefore be y(t)=(1/2)mt^2+bt+c. c is not specified by this analysis, so at this point c is regarded as an arbitrary constant. c depends only on when we start our clock and the position from which the depth is being measured. **
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RESPONSE --> What is the best way to assign a value to C if it is not given to you? Is it best to just choose a random number or to stick to a small number that won't throw off the answer from the equation much?
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11:15:07 Explain why, given only the rate-of-depth-change function y' and a time interval, we can determine only the change in depth and not the actual depth at any time, whereas if we know the depth function y we can determine the rate-of-depth-change function y' for any time.
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RESPONSE --> Because with only the rate of depth change function we can't make a complete depth model you can only find A and B you can't find C therfore anything that you sub for C will throw the equation off +or- the difference of the actual C variable, where as with a depth function we can make a complete rate function.
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13:40:07 ** Given the rate function y' we can find an approximate average rate over a given time interval by averaging initial and final rates. Unless the rate function is linear this estimate will not be equal to the average rate (there are rare exceptions for some functions over specific intervals, but even for these functions the statement holds over almost all intervals). Multiplying the average rate by the time interval we obtain the change in depth, but unless we know the original depth we have nothing to which to add the change in depth. So if all we know is the rate function, have no way to find the actual depth at any clock time. ANOTHER EXPLANATION: The average rate of change over a time interval is rAve = `dy / `dt. If we know rAve and `dt, then, we can easily find `dy, which is the change in depth. None of this tells us anything about the actual depth, only about the change in depth. If we don't know rAve but know the function r(t) we can't use the process above to get the exact change in depth over a given interval, though we can often make a pretty good guess at what the average rate is (for a quadratic depth function, as the quiz showed, you can actually be exact the average rate is just the rate at the midpoint of the interval; it's also the average of the initial and final rates; and all this is because for a quadratic the rate function is linear--if you think about those statements you see that they characterize a linear function, whose average on an interval occurs at a midpoint etc.). For anything but a linear rate function we can't so easily tell what the average is. However we do know that the rate function is the derivative of the depth function. So if we can find an antiderivative of the rate function, all we have to do to find the change in depth is find the difference in its values from the beginning to the end of the interval. This difference will be the same whichever antiderivative we find, because the only difference that can exist between two antiderivatives of a given rate function is a constant (whose derivative is zero). We have to develop some machinery to prove this rigorously but this is the essence of the Fundamental Theorem of Calculus. You might not understand it completely at this point, but keep coming back to this explanation every week or so and you will soon enough.**
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RESPONSE --> A linear depth function will make a quadratic rate function and a quadratic rate function will make a linear depth function, only with a linear function is the average equal to the actual rate.
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13:49:22 In terms of the depth model explain the processes of differentiation and integration.
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RESPONSE --> Differentiation If given a depth model you can plug into the universal rate equation or you can do it the hard way. Easy way: y = ax^2 + bx + c plug into rate equation 2ax + b Hard way: y = ax^2 + bx + c enter and average 2 points ((y=ax^2 + bx + c)-(y=ax(sub2)^2 + bx(sub2) + c))/ ((x - x(sub2)) ultimatly yielding the slope or the average rates of the point in the middle of the points x and x(sub2). Integration: knowing that 2ax+b is the equation format for the rate equation, we can plug that into the quadratic depth formula of y = ax^2 + bx + c. But we don't have c so we assign a value to c and keep it at that, unless there is something otherwise pointing out the value of C.
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13:50:24 ** Rate of depth change can be found from depth data. This is equivalent to differentiation. Given rate-of-change information it is possible to find depth changes but not actual depth. This is equivalent to integration. To find actual depths from rate of depth change would require knowledge of at least one actual depth at a known clock time. **
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RESPONSE --> I left out that you can't find the actual depth without c but you can find the depth changes.
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