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Question 896991: 1) Find the difference quotient and simplify your answer.
f(x) = x2 − x + 8, f(4 + h) − f(4)/h, h ≠ 0
2) Find the difference quotient and simplify your answer.
f(x) = 7x2 − 8x, f(x + h) − f(x)/h, h ≠ 0
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! -----
problem number 1:
f(x) = x2 − x + 8, f(4 + h) − f(4)/h, h ≠ 0
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f(x) = x^2 - x + 8
f(4) = 4^2 - 4 + 8 = 16 - 4 + 8 which becomes:
f(4) = 16 + 4 which becomes:
f(4) = 20
f(4 + h) = (4+h)^2 - (4+h) + 8 which becomes:
f(4 + h) = 16 + 8h + h^2 - 4 - h + 8 which becomes:
f(4 + h) = h^2 + 7h + 20
f(4+h) - f(4) = h^2 + 7h + 20 - 20 which becomes:
f(4+h) - f(4) = h^2 + 7h
(f(4+h) - f(4)) / h = (h^2 + 7h) / h
when h is not equal to 0, this can be simplified to:
(f(4+h) - f(4)) / h = h + 7
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problem number 2:
f(x) = 7x2 − 8x, f(x + h) − f(x)/h, h ≠ 0
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f(x) = 7x^2 - 8x
f(x+h) = 7(x+h)^2 - 8(x+h) which becomes:
f(x+h) = 7(x^2 + 2hx + h^2) - 8x - 8h which becomes:
f(x+h) = 7x^2 + 14hx + 7h^2 - 8x - 8h.
f(x+h) - f(h) = 7x^2 + 14hx + 7h^2 - 8x - 8h - (7x^2 - 8x) which becomes:
f(x+h) - f(h) = 7x^2 + 14hx + 7h^2 - 8x - 8h - 7x^2 + 8x which becomes:
f(x+h) - f(h) = 7h^2 + 14hx - 8h
the 7x^2 and minus 7x^2 cancel out and the 8x and minus 8x also cancel out.
(f(x+h) - f(x)) / h = (7h^2 + 14hx - 8h) / h
when h is not equal to 0, you can simplify this to get:
(f(x+h) - f(x)) / h = 7h + 14x - 8
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