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Math Help - Help with average rate of change

  1. #1
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    Help with average rate of change

    Hi there.

    For g(x)=4x -3x +1 Find the rate of change for:

    A) x=1 and x=3
    B) (x, f(x)) and (x+h, f(x+h)

    I dont know how and would appreciate an explanation. Thanks so much.
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  2. #2
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    Perhaps you just need the definition for average rate of change?

    Between x=a and x=b, the average rate of change is (change in g(x) divided by the change in x):
    <br />
\displaystyle \frac{g(b) - g(a)}{b - a}<br />

    For example, average rate of change between x=0 and x=1 is
    <br />
\displaystyle \frac{g(1) - g(0)}{1 - 0} = \frac{(4(1)^2 -3(1) +1) - (4(0)^2 -3(0) +1)}{1 - 0} = 1<br />
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  3. #3
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    So, for the first part it would be: (4-3+1)-(36-9+1)/1-3
    Which becomes 12


    Also, I am confused about B).

    So for I got it to f(x+h)-f(x)/(x+h)-x

    Then I simplified it to 3(x+h)-3x/h

    Did I do it correctly?
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  4. #4
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    Quote Originally Posted by theridon View Post
    So, for the first part it would be: (4-3+1)-(36-9+1)/1-3
    Which becomes 12


    Also, I am confused about B).

    So for I got it to [f(x+h)-f(x)]/(x+h)-x brackets are important!

    Then I simplified it to 3(x+h)-3x/h - where did that 3 come from?

    Did I do it correctly?
    See red comments
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  5. #5
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    Sorry about the brackets, thats what I meant. I think you plug it into 4x -3x +1, correct?

    This is how I got there:
    1. [f(x+h)-f(x)]/(x+h)-x
    2. [[3(x+h)+1]-(3x+1)]/h
    2. 3(x+h)-3x/h
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  6. #6
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    Quote Originally Posted by theridon View Post
    Sorry about the brackets, thats what I meant. I think you plug it into 4x -3x +1, correct?

    This is how I got there:
    1. [f(x+h)-f(x)]/(x+h)-x
    2. [[3(x+h)+1]-(3x+1)]/h
    2. 3(x+h)-3x/h
    yes, but that's not what you did.

    first of all, is f(x) = 4x^2 - 3x + 1 ??? you called it g(x).

    if so,

    f(x+h) - f(x)

    the numerator should be ...

    4(x+h)^2 - 3(x+h) + 1 - (4x^2 - 3x + 1)
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  7. #7
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    Oh I missed the first part. So then it would become:
    [4x+8xh-3x-3h+1-4x+3x-1]/h


    (4h+8hx)/h

    4h+8x

    Which simplifies to 4(h+2x)

    I believe its correct
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  8. #8
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    Quote Originally Posted by theridon View Post
    Oh I missed the first part. So then it would become:
    [4x+8xh-3x-3h+1-4x+3x-1]/h


    (4h+8hx)/h

    4h+8x

    Which simplifies to 4(h+2x)

    I believe its correct
    numerator again ...

    4(x + h)^2 - 3(x + h) + 1 - (4x^2 - 3x + 1) =

    4(x^2 + 2xh + h^2) - 3(x + h) + 1 - (4x^2 - 3x + 1) =

    4x^2 + 8xh + 4h^2 - 3x - 3h + 1 - 4x^2 + 3x - 1 =

    8xh + 4h^2 - 3h  =

    h(8x + 4h - 3)
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