If you were my teacher, how would you have graded this problem?

Question

I'm going to use 'S' as the integral sign. The interval is [x,0] so x=b and 0=a

***WORTH 10 POINTS**
Question: Identify all local extrema of f(x)=S[x,0] (t^2 -3t +2) dt.

It's kind of funny because I handed in my quiz and this problem was worth 10 points. I knew for sure I was going to get a zero on this problem, even though we just went over it like 2 minutes ago. The thing is though, I was trying to evaluate another problem that I thought for sure was going to be on the quiz. How wrong I was! Anyways, I asked the TA after class if i could re-do it and he gave me the heads up. So this is what I wrote:

(x-1)(x-2)
local max @ x=1
local min @ x=2

In addition, I had some scribbled out (still recognizable) work I did previously which involved the integration of the original equation. All in all, that's what I had as my final answer. I knew I blew it and if I'm ever on the borderline of committing suicide, I'll just think back to this instance and my mind will be made up.

Answer 1

Your notation is a little confusing. Generally, when writing the interval of integration, the lower endpoint is written first, so the integral from a to b is written as the integral on [a, b], not on [b, a]. However, your text (and your answer) seems to suggest that you mean for x to be the upper endpoint. I'm going to assume that the text is correct, and that you meant to write [0, x]∫t² - 3t + 2 dt. In that case, you actually go the problem right! Here's why:

The critical points of the function [0, x]∫t² - 3t + 2 dt are precisely the points where the derivative of that function equals 0. However, one of the fundamental theorems of calculus (different textbooks disagree on whether it is the first or the second) states that whenever f is continuous, d/dx [0, x]∫f(t) dt = f(x). So indeed, the critical points of the function are where f(x) = 0, i.e. the roots of x² - 3x + 2. The only thing you left out was how you determined that x=1 was specifically a maximum and that x=2 was a minimum -- this follows from the fact that x² - 3x + 2 is decreasing at 1 and increasing at 2, and that is easy to see visually (so you don't really need to compute the derivative of x² - 3x + 2 explicitly), but I would still have required you to note that x² - 3x + 2 is decreasing at 1 and increasing at 2, or otherwise provide some justification that x=1 was a maximum and x=2 was a minimum. I'd probably give you an 8 on this problem.

Of course, this is assuming that you really did mean for x to be the upper endpoint. If the problem was what you wrote, with x on the bottom, then your answer is wrong. d/dx [x, 0]∫f(t) dt = -f(x), not f(x), so the derivative of [x, 0]∫t² - 3t + 2 dt would be -t² + 3t - 2. The critical points work out to be the same, but in this case x=1 is a minimum and x=2 is a maximum. If that was the problem, I would give you only 5 points.

P.S. -- even though your answer was imperfect, I would hardly say that you "blew it", and in any case, even if you had failed the entire class (instead of just one quiz), that's hardly a reason to give up all hope. Many people have failed one or more classes in high school and went on go to college and become highly successful. What matters most is not whether you do well on one quiz or one test or even one class, but whether you're willing to keep trying. So don't try to become "an hero" over this, okay?

Answer 2

hmm, actually get your mother and father in contact. colleges incredibly much constantly cave whilst mother and father start up calling. Plus this instructor is likewise screwing including your GPA. maximum faculties right this moment have an information superhighway gradebook that the two the student, determine, or the two have get right of entry to to. undecided in the adventure that your mother and father have get right of entry to to a minimum of something like that yet whilst they do examine in on that and discover out why the grade is so low. Get your mother and father to attempt to get a grade checklist sheet from the instructor. in fact have your mother and father call the college and get a grade checklist from all your instructions.