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Math Help - Theorems about Definite Integrals!!

  1. #1
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    Theorems about Definite Integrals!!

    The graph of a derivative f '(x) is shown in the figure below. This is not the graph of f(x).


    Fill in the table of values for f(x) given that f(0) = 8.

    i need some help on doing this, its still a little fuzzy haha, thanks

    mathlete
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  2. #2
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    Quote Originally Posted by mathlete View Post
    The graph of a derivative f '(x) is shown in the figure below. This is not the graph of f(x).


    Fill in the table of values for f(x) given that f(0) = 8.

    ...
    I assume that the graph of f is continuous and once (but only once!) differentiable - even though that isn't stated in your problem, but otherwise you can't calculate the missing values.

    Divide the graph of f' into 4 parts concerning the intervals [0,1), [1, 3), [3, 5) and [5, 6]:

    to [0, 1): The graph of f is a parabola whose quadratic term is -\frac12 x^2. Therefore  f(1)= \frac{15}2:.... f_1(x)=-\frac12 x^2+8

    to [1, 3): The graph of f is a straight line whose linear term is - x. Therefore f(3)= \frac{11}2:.... f_2(x) =-(x-1)+7.5

    to [3, 5): The graph of f is a parabola whose quadratic term is \frac12 x^2. Therefore f(5)= \frac{11}2:.... f_3(x)=\frac12 (x-4)^2+5

    to [5, 6]: The graph of f is a straight line whose linear term is  x. Therefore f(6)= \frac{13}2:.... f_4(x) = (x-5)+5.5
    Attached Thumbnails Attached Thumbnails Theorems about Definite Integrals!!-stckwse_4parts.jpg  
    Last edited by earboth; April 18th 2008 at 08:30 PM.
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