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Thread: Sketch the graph of a continuous function

  1. #1
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    Sketch the graph of a continuous function

    Hi again I'm tackling a question that is very likely to be on a future exam, and I'm a bit stuck.

    The question:

    g(x){ 0 if x<2
    2 if 2<x<4
    -1 if x>4

    Sketch the graph of a continuous function f(x) knowing that f(0)=1, and f'(x) = g(x)

    Graphing g(x) is not a problem, but understanding how to formulate a seperate function with just the info given is a major roadblock.

    Any suggestions would be helpful.
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  2. #2
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    Quote Originally Posted by ughintegrals View Post
    Hi again I'm tackling a question that is very likely to be on a future exam, and I'm a bit stuck.

    The question:

    g(x){ 0 if x<2
    2 if 2<x<4
    -1 if x>4

    Sketch the graph of a continuous function f(x) knowing that f(0)=1, and f'(x) = g(x)

    Graphing g(x) is not a problem, but understanding how to formulate a seperate function with just the info given is a major roadblock.

    Any suggestions would be helpful.
    If $\displaystyle
    g(x) = \left\{
    \begin{array}{cl}
    0 & \text{if}\; x < 2\\
    2 & \text{if}\; 2 < x < 4\\
    -1 & \text{if}\; x > 4
    \end{array}\right.
    $

    then

    $\displaystyle
    f(x) = \left\{
    \begin{array}{cl}
    c_1 & \text{if}\; x \le 2\\
    2x + c_2 & \text{if}\; 2 < x \le 4\\
    -x + c_3 & \text{if}\; x > 4
    \end{array}\right.
    $.

    Then use the fact that $\displaystyle f(0) = 1$ (this gives you $\displaystyle c_1$) and that $\displaystyle f(x)$ is continuous at $\displaystyle x = 2$ and $\displaystyle x = 4$ (this gives the remaining constants.)
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  3. #3
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    ugh, im almost there.

    c1 was no sweat, but how am i supposed to use the x= 2 and x = 4 to solve the remaining constants? plug 2 into the 2x+c, or equate it to 2? or neither?

    im just a bit stuck there....sorry for being difficult.
    thanks for your help.
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