Description
Simplify the difference quotient f(x+h) - f(x) all over h.
(1) y = 4x - x^2
(2) f(x)= 2x + 1 over x + 3
Explanation & Answer
y = 4x - x2
f(x+h) = 4(x+h)-(x+h)2
f(x+h)-f(x) = 4(x+h)-(x2+2hx+h2)- 4x +x2 = 4h -2hx-h2
[f(x+h)-f(x)]/h = 4-2x-h
(2) f(x)= 2x + 1 over x + 3
f(x) = (2x+1)/(x+3)
f(x+h) = (2x+2h+1)/(x+h+3)
f(x+h)-f(x) = (2x+2h+1)/(x+h+3) –(2x+1)/(x+3)
=[ (2x+2h+1)(x+3) –(2x+1)(x+h+3)]/[(x+h+3)(x+3)]
=[ (2x+1)(x+3) +2h(x+3) –(2x+1)(x+3) -h(2x+1)]/[(x+h+3)(x+3)]
= h(2x+6-2x-1)/ [(x+h+3)(x+3)]
= 5h/[(x+h+3)(x+3)]
[f(x+h)-f(x)]/h = 5/[(x+h+3)(x+3)]
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