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Math Help - Calc FTC with piecewise?

  1. #1
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    Calc FTC with piecewise?

    I'm super confused about what is going on here. I got an email from my prof with an explanation that left me even more lost. Here's the question:

    Let f(x) =
    0 if x<-4
    4 if -4<= x < -1
    -5 if -1 <= x < 4
    0 if x >= 4

    and

    g(x) = int_(x at top, -4 at bottom) f(t) dt

    Determine the following:

    g(0) = ?
    g(5) = ?

    The absolute maximum of g(x) occurs when x = -1 at what value?

    The explanation from my prof used things like "m1(a-b) and m2(b-x)" but I have no idea what this means. Could someone please explain to me the process of solving this? I tried just figuring out that when x = 0 the answer (given above) is -5, but this isn't right.

    Thanks!
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  2. #2
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    anybody? please?
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  3. #3
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    Bumping is not polite.
    \begin{gathered}<br />
g(0) = \int\limits_{ - 4}^0 {f(t)dt} = \int\limits_{ - 4}^{ - 1} {(4)dt} + \int\limits_{ - 1}^0 {( - 5)dt} \hfill \\<br />
g(5) = \int\limits_{ - 4}^5 {f(t)dt} = g(0) + \int\limits_0^4 {( - 5)dt} \hfill \\ <br />
\end{gathered}
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  4. #4
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    I have very little idea what that means. What I was hoping for was an explanation of what was going on and how to find an answer. I do not know what to do with what you have given me, although I am grateful for the help. Perhaps an English explanation?

    Thanks!!!
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  5. #5
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    Plato just used a basic rule regarding definite integrals, it's just \int_a^bf+\int_b^cf=\int_a^cf, the additivity with respect to the interval. Hence, consider your piecewise function and that'd yield what Plato did.
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  6. #6
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    Quote Originally Posted by littlejodo View Post
    I have very little idea what that means. What I was hoping for was an explanation of what was going on and how to find an answer. I do not know what to do with what you have given me, although I am grateful for the help. Perhaps an English explanation?
    Maybe a sit down with your instructor is in order!
    We are not a tutorial service.
    Please arrange to see your instructor.
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  7. #7
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    Plato - I didn't think you were a tutoring service. I simply meant that I was not understanding the notation. I am more interested in learning how things work than simply getting the answer.
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  8. #8
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    Recall that an integral represents the net area under a curve.

    So when we look at g(0) = \int_{-4}^0 f(t) dt, we are finding the net area under the curve from x = -4 to x = 0. This should be fairly easy by considering as we are essentially working with straight line segments. No complicated formulas here besides that of finding the area of a rectangle.

    Note that it was split into 2 integrals in order to work with both line segments individually.
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  9. #9
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    THANK YOU!

    I just couldn't see what was going on (that's a common problem for me). Thank you for the explanation, it really helped.
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