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Math Help - help with derivatives...

  1. #1
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    help with derivatives...

    The graph of a derivative f '(x) is shown in Figure 5.61.




    ok, im given this graph of the derivative of f(x)...
    can some help me find the coordinates to f(x)...i was given that f(0)=5
    thanks...

    mer1988
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  2. #2
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    Hello, mer1988!

    This takes quite a bit of work (but simple stuff).
    I will also assume that f(x) is continuous.


    The graph of a derivative f'(x) is given.


    i was given that: f(0)\,=\,5
    We'll examine the function over the different intervals.



    On [0,\,1], we see that: . f'(x)\:=\:-x
    Integrate: . f(x) \:=\:-\frac{1}{2}x^2 + C
    Since (0,\,5) is on the graph: . 5 \:=\:-\frac{1}{2}\!\cdot\!0^2 + C\quad\Rightarrow\quad C\,=\,5
    Hence: on {\color{blue}[0,\,1],\;f(x) \:=\:-\frac{1}{2}x^2 + 5}
    . . Note that: . f(1) \:=\:\frac{9}{2}



    On [1,3],\;f'(x) \:=\:-1

    Integrate: . f(x)\:=\:-x + C

    Since \left(1,\,\frac{9}{2}\right) is on the graph: . \frac{9}{2}\:=\:-1 + C\quad\Rightarrow\quad C \:=\:\frac{11}{2}
    Hence: [color=blue}on [/color] {\color{blue}[1,\,3],\;f(x) \:=\: -x + \frac{11}{2}}
    . . Note that: . f(3) \,=\,\frac{5}{2}



    On [3,\,5],\;f'(x) \:=\:x - 4

    Integrate: . f(x)\;=\;\frac{1}{2}x^2 - 4x  + C
    Since \left(3,\,\frac{5}{2}\right) is on the curve: . \frac{5}{2}\:=\:\frac{1}{2}\!\cdot\!3^1 - 4(3) + C\quad\Rightarrow\quad C \:=\:10
    Hence: on {\color{blue}[3,\,5],\;f(x) \:=\:\frac{1}{2}x^2 - 4x + 10}
    . . Note that: . f(5) = \frac{5}{2}



    On [5,\,6],\;f'(x) \:=\:1

    Integrate: . f(x) \:=\:x + C

    Since \left(5,\,\frac{5}{2}\right) in on the curve: . \frac{5}{2}\:=\:5 + C\quad\Rightarrow\quad C = -\frac{5}{2}
    Hence: on {\color{blue}[5,\,6],\;f(x) \:=\:x - \frac{5}{2}}



    We have the piecewise function: . f(x) ;=\;\left\{\begin{array}{ccc}-\frac{1}{2}x^2+5 & \;\; & 0 \leq x \leq 1 \\-x + \frac{11}{2} & \;\; & 1 \leq x \leq 3 \\\frac{1}{2}x^2-4x+10 & \;\; & 3 \leq x \leq 5 \\x - \frac{5}{2} & \;\; &5 \leq x \leq 6\end{array}\right\}

    And you can graph f(x) now . . .

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