f(x+y) = f(x) + f(y) + 2xy

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Question from Marcia, a teacher:

For all real numbers x and y, let f be a function such that f(x+y) = f(x) + f(y) + 2xy and such that
the limit as h -> 0 of f(h) / h = 7, find: f(0), use the definition of the derivative to find f'(x), and find f(x).

Hi Marcia,

Write down the definition of the derivative f '(x) and simplify the expression using the fact that for any x and y

f(x + y) = f(x) + f(y) + 2xy (*)

Evaluate the limit to find f '(x). Use antidifferentiation to find f(x). Antidifferentiation introduces a constant but if you know the value of f(0) you can evaluate the constant. Can you use the expression (*) to find f(0)?

Penny

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