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PHY 241
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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Q(1) for Startup Part 4 B4
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I just have a couple of questions about Startup Part 4 B4 - Calculus
Problem 009: The problem concerns the product of two linear functions, f(x) and g(x) for which only two points are given for each function. A portion of the question asks whether the product function, h(x) ever exceeds 4. I said No - but that was wrong. Sure enough, once I find the equations of the lines (which we were discouraged from doing in answering the problem) and determine h(x) I see that we have a maximum of 4.1667 at x = 5.667.
While I understand how I can determine without finding the equations of f(x) and g(x) that h(x) is quadratic, I don't see how I can determine that the apex of the quadratic is between x = 2 and x = 6 given just the four points. My memory isn't all that it used to be ... so could you refresh it?
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To give a good answer to this question I would need a copy of the problem, in order to give a very specific answer. This is why a copy of the problem is requested by the form. (Judging by the dates you gave on your enrollment request form I"m probably a year ot two older than you are, and I can confirm the phenomenon of memories not being what they once were)
However I can give you a general answer.
To do this numerically you could test the behavior of the product function on a small increment to the left of x = 6. Let's pick a number, say .01. More generally we would want to denote this number by `dx (standing for delta-x) or maybe epsilon.
We know the values of the two functions at x = 4.
We also know the slopes of the two functions, which are constant.
So we can determine how much each function changes between x - .01 and x.
Adding these changes to the known values and multiplying the results, we get a number greater than the product at x = 4.
If we use epsilon instead of .01, we again confirm that for sufficiently small epsilon the product is greater than 4.
Knowing that the function is quadratic, and that its value at x = 2 is less than at x = 4, we conclude that its vertex lies between x = 2 and x = 6.
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Problem 011: The problem concerns three scenarios that describe a car rolling down a hill. You comment that in my response to the problem I don't specify to which graph I refer, and say that each situation calls for two graphs. I don't understand why there are two graphs for each situation. The instructions say Describe the graph of the rate of change of the position of a car vs. clock time, given each of the above conditions. This seems to say that there is only one graph for each of the conditions. What is the other graph in each of the conditions?
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I believe the question also asks for a description of the graph of velocity vs. clock time.
If I'm wrong, or even if I'm right, it's no problem. I'll double check this question when I encounter it again, as I'm likely to do when reviewing today's submitted work.
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P.S. Thanks for giving me the comments in these problem sets. The comments and the commentary on previous student questions helps a lot.
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Check my inserted responses and let me know if anything remains unclear.
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