Find the difference quotient of f?
this might be simple but i don't know how to apply the h and the x+h:
f(x)= x^2 -9x +9
the equation is:
f(x + h) -f(x) / h
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this might be simple but i don't know how to apply the h and the x+h:
f(x)= x^2 -9x +9
the equation is:
f(x + h) -f(x) / h
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Answers & Comments
2x + 9 + h
2x + 9 + h
2x + 9 + h
[x + h]² - 9 [ x + h ] + 9 - [ x² - 9x + 9 ]
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h
x² + 2hx + h² - 9x - 9h + 9 - x² = 9x - 9
----------------------------------------------------
h
2hx + h² - 9h
--------------------
h
2x - 9 + h = 2x - 9 as h ->0
f(x) = x^2 -9x +9
f(x+h) = (x +h)^2 - 9(X + h) + 9
= x^2 + 2xh + h^2 - 9x - 9h + 9
= (x^2 - 9x + 9) + (2xh + h^2 + 9h)
f(x + h) - f(x) = (x^2 - 9x + 9) + (2xh + h^2 + 9h) - (x^2 -9x +9 )
= (2xh + h^2 + 9h)
[ f(x + h) - f(x)]/h = (2xh + h^2 + 9h)/h
= 2x + 9 + h
f(x + h) = (x + h)^2 - 9(x + h) + 9
= x^2 + 2hx + h^2 - 9x - 9h + 9
[f(x + h) -f(x)] / h
= [x^2 + 2hx + h^2 - 9x - 9h + 9 - (x^2 - 9x + 9)]/h
= (x^2 + 2hx + h^2 - 9x - 9h + 9 - x^2 + 9x - 9)/h
= (h^2 + 2hx - 9h)/h => factor h in the numerator:
= [h(h + 2x - 9)]/h => cancel h from top and bottom:
= h + 2x - 9
f(x+h) = (x+h)^2 -9(x+h)+9
f(x + h) -f(x) = (x+h)^2 -9(x+h)+9-(x^2 -9x +9 ) = ... = 2xh + h^2 - 9h
(f(x + h) -f(x)) / h = (2xh + h^2 - 9h)/h = ...simplify