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Math Help - Find Average Rate of Change

  1. #1
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    Find Average Rate of Change

    Find the average rate of change for f(x) = x^2
    from x = delta(x) to x = x + delta(x).

    USE delta(y)/delta(x) = [f(b) - f(a)]/(b - a)
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  2. #2
    Senior Member mollymcf2009's Avatar
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    Quote Originally Posted by magentarita View Post
    Find the average rate of change for f(x) = x^2
    from x = delta(x) to x = x + delta(x).

    USE delta(y)/delta(x) = [f(b) - f(a)]/(b - a)

    You are just being asked to find the slope of some secant line on the function. It looks like you can probably just pick whatever points on the graph that you want. So, you know this parabola goes through (1,1) AKA (a, f(a)) Find another point on the graph to use for your "b" and then work it out. Not sure what your question is. Hope that helps!
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  3. #3
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    ok but......

    Quote Originally Posted by mollymcf2009 View Post
    You are just being asked to find the slope of some secant line on the function. It looks like you can probably just pick whatever points on the graph that you want. So, you know this parabola goes through (1,1) AKA (a, f(a)) Find another point on the graph to use for your "b" and then work it out. Not sure what your question is. Hope that helps!
    Ok but can you do this one for me step by step?
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  4. #4
    Senior Member mollymcf2009's Avatar
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    Quote Originally Posted by magentarita View Post
    Ok but can you do this one for me step by step?
    Draw the graph of your function y = x^2  Then draw a straight line through the function. Do this on graph paper if you need to so that you can draw a line through points that you can see the value of. I have attached a graph that I made to show you what I mean by this.

    SLOPE-AVERAGE RATE OF CHANGE.doc

    It is a parabola with its vertex at the origin. "a" & "b" are x-values somewhere on the graph. f(a) and f(b) are the y-values associated with those x-values.

    The average rate of change of a function is the slope of a secant line of the function. A secant line is a straight line that goes through the graph of the function intersecting at two separate points.

    The slope of the secant line is:

    slope = \frac{change.in.y.values)}{change.in.x.values}

    OR

    m = \frac{f(b) - f(a)}{b-a}

    So, choose any value of x, let's say x=2. Now, to find the y-value associated with that x-value, plug the x-value into your function.

    y = x^2

    which can also be written as

    f(x) = x^2

    So,
    y= (2)^2<br />
    or

    f(2) = (2)^2 = 4

    So when x=2 on the graph of this function, y=4

    Does that make sense?

    So,

    a = 2
    f(2) = 4

    Now pick another value on your function, let's say x = -1

    Plug -1 into y = x^2, you get y = (-1)^2 = 1

    So,
    b = -1
    f(b) = 1

    So from your problem:

    \frac{\delta y}{\delta x} = \frac{f(b)-f(a)}{b-a} = \frac{-1-2}{1-4}

    = \frac{-3}{-3} = 1

    So, the average slope of your function is m = 1

    Does that help you understand now?
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  5. #5
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    ok.....

    Quote Originally Posted by mollymcf2009 View Post
    Draw the graph of your function y = x^2 Then draw a straight line through the function. Do this on graph paper if you need to so that you can draw a line through points that you can see the value of. I have attached a graph that I made to show you what I mean by this.

    SLOPE-AVERAGE RATE OF CHANGE.doc

    It is a parabola with its vertex at the origin. "a" & "b" are x-values somewhere on the graph. f(a) and f(b) are the y-values associated with those x-values.

    The average rate of change of a function is the slope of a secant line of the function. A secant line is a straight line that goes through the graph of the function intersecting at two separate points.

    The slope of the secant line is:

    slope = \frac{change.in.y.values)}{change.in.x.values}

    OR

    m = \frac{f(b) - f(a)}{b-a}

    So, choose any value of x, let's say x=2. Now, to find the y-value associated with that x-value, plug the x-value into your function.

    y = x^2

    which can also be written as

    f(x) = x^2

    So,
    y= (2)^2<br />
    or

    f(2) = (2)^2 = 4

    So when x=2 on the graph of this function, y=4

    Does that make sense?

    So,

    a = 2
    f(2) = 4

    Now pick another value on your function, let's say x = -1

    Plug -1 into y = x^2, you get y = (-1)^2 = 1

    So,
    b = -1
    f(b) = 1

    So from your problem:

    \frac{\delta y}{\delta x} = \frac{f(b)-f(a)}{b-a} = \frac{-1-2}{1-4}

    = \frac{-3}{-3} = 1

    So, the average slope of your function is m = 1

    Does that help you understand now?

    Wonderfully done!
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