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Homework Help: Calc workshop help

  1. Feb 8, 2004 #1
    Let f(x)=

    {3x - 2, if x < 0;
    {ax + b, if 0 <= x <= 1;
    {3x + 4, if x > 1.
    (its a piece wise function)

    It's telling me to Find a and b so that f(x) is continuous for all values of x. what's the easiest way to solve this? i was told that it's easier to graph it first. Help?
     
  2. jcsd
  3. Feb 8, 2004 #2

    jamesrc

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    Graphing is not necessary, but trying to visualize things is never a bad thing. Consider the points where the definition of the curve changes: at x = 0 and at x = 1. For a continuous function, the left and right hand limits of the function will be the same at these points (at all points to be more accurate, but we only need to check these two points since we know there are no potential continuity problems anywhere else).

    [tex] \lim_{x\rightarrow 0^+} f(x) = -2 [/tex]

    [tex] \lim_{x\rightarrow 0^-} f(x) = b [/tex]



    [tex] \lim_{x\rightarrow 1^+} f(x) = a + b [/tex]

    [tex] \lim_{x\rightarrow 1^-} f(x) = 7 [/tex]
     
  4. Feb 8, 2004 #3
    so am i just supposed to be able to solve for a and b now?
    a= 9
    b= -2
    is this right? or wrong?
    can you please explain how you got those limit equations?
     
    Last edited: Feb 8, 2004
  5. Feb 8, 2004 #4

    jamesrc

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    Yeah, that's right.

    [tex] \lim_{x\rightarrow 0^+} f(x) = \lim_{x\rightarrow 0^+} 3x-2 = 3*0-2 = 0 [/tex]

    and so on.
     
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