Question

Consider the function below.
f(x) = 3 - 3x - x^2
Evaluate the difference quotient for the given function. Simplify your answer. f(2+h)-f(2)/h

Answer 1

\[\large f(x) = 3 - 3x - x^2\]


\[\large f(2) = 3 - 3(2) - (2)^2\]


\[\large f(2) = 3 - 3(2) - 4\]


\[\large f(2) = -7\]


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\[\large f(x) = 3 - 3x - x^2\]


\[\large f(2+h) = 3 - 3(2+h) - (2+h)^2\]


\[\large f(2+h) = 3 - 3(2+h) - (4+4h+h^2)\]


\[\large f(2+h) = 3 - 6-3h - 4-4h-h^2\]


\[\large f(2+h) = -7-7h-h^2\]


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The key things to draw out from the work above is that \[\large f(2) = -7\] and \[\large f(2+h) = -7-7h-h^2\]




\[\large \frac{f(2+h)-f(2)}{h}\]


\[\large \frac{-7-7h-h^2-(-7)}{h}\]


\[\large \frac{-7-7h-h^2+7}{h}\]


\[\large \frac{-7h-h^2}{h}\]


\[\large \frac{h(-7-h)}{h}\]


\[\large \frac{\cancel{h}(-7-h)}{\cancel{h}}\]


\[\large -7-h\]


\[\large -h-7\]


So \[\large \frac{f(2+h)-f(2)}{h}=-h-7\]