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Math Help - Using Properties to determine derivative and original equation

  1. #1
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    Using Properties to determine derivative and original equation

    A function f is defined for all real numbers and has the following properties.

    (i) f(1)=5
    (ii) f(3)=21
    (iii) for all real values of a and b, f(a+b) - f(a)= kab+2b^2 where k is a fixed real number independent of a and b

    a) Use a=1 and b=2 to find k.
    b) Find f'(3)
    c) Find f'(x) and f(x) for all real x

    So I found out that k=-4, but I'm stuck on how to find the original equation and thus the derivative by the given info.
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by ment2byours View Post
    A function f is defined for all real numbers and has the following properties.

    (i) f(1)=5
    (ii) f(3)=21
    (iii) for all real values of a and b, f(a+b) - f(a)= kab+2b^2 where k is a fixed real number independent of a and b

    a) Use a=1 and b=2 to find k.
    b) Find f'(3)
    c) Find f'(x) and f(x) for all real x

    So I found out that k=-4, but I'm stuck on how to find the original equation and thus the derivative by the given info.
    First of all, for (a), I think k=4, not -4. Here's why: 21-5=k*1*2+2*2^2. Thus 16=2k+8 and k=4.

    Use the definition of the derivative (using a and b for x and h): \lim_{b\to 0} \frac{f(a+b)-f(a)}{b}

    You know f(a+b)-f(a)=4ab+2b^2 so the above limit equals \lim_{b\to 0} \frac{4ab+2b^2}{b}=4a.

    Thus f'(a) = 4a. Thus f'(3)=12.
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  3. #3
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    Oh thanks!! I should have recognized that it was similar for the definition for a derivative. Sorry about getting k wrong, I tend to keep messing up my pos/neg signs. So the deriv of the original equation is f'(x)=4x?
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  4. #4
    Member arpitagarwal82's Avatar
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    Continuing with above solution

    Now we have f'(x) = 4x

    So f(x) = 2x^2 + c where c is constant of integration

    using f(1) = 5
    5 = 2 + c

    c = 3

    so function is f(x) = 2x^2 + 3
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  5. #5
    Super Member redsoxfan325's Avatar
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    No. f'(x)=4x. This makes f(x)=2x^2+C. Since f(1)=5, f(x)=2x^2+3.

    EDIT: What arpitagarwal82 said.
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  6. #6
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    YOU guys are awesome
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  7. #7
    Super Member redsoxfan325's Avatar
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    No problem.
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  8. #8
    Member arpitagarwal82's Avatar
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    Quote Originally Posted by ment2byours View Post
    YOU guys are awesome
    You are welcome.
    We are here to help.
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