You can
put this solution on YOUR website! What is the definition of Difference Quotient and
how do you arrive at an answer when you have
f(x) = x²+3x and
DQ = ((f(x+h)-f(x))/h
The difference quotient is defined as the slope of
the line through the two points (x, f(x)) and
(x+h, f(x+h))
y2 - y1
DQ = m = ---------
x2 - x1
f(x+h) - f(x)
DQ = ---------------
(x+h) - x
f(x+h) - f(x)
DQ = ---------------
x + h - x
f(x+h) - f(x)
DQ = ---------------
h
To find that for the given f(x):
f(x+h) - f(x)
DQ = ---------------
h
First we calculate the red part,
f(x) = x² + 3x
Replace all the x's by (x+h)
f(x+h) = (x+h)² + 3(x+h)
= (x+h)(x+h) + 3(x+h)
= x² + 2hx + h² + 3x + 3h
Now the blue part is given as
f(x) = x² + 3x
So in
f(x+h) - f(x)
DQ = ---------------
h
replace the red f(x+h) by (x²+2hx+h²+3x+3h) and
the blue f(x) by (x²+3x)
(x²+2hx+h²+3x+3h) - (x²+3x)
DQ = -----------------------------
h
(x²+2hx+h²+3x+3h) - (x²+3x)
DQ = -----------------------------
h
x² + 2hx + h² + 3x + 3h - x² - 3x
DQ = -----------------------------------
h
Combine like terms:
2hx + h² + 3h
DQ = ---------------
h
Factor h out of the top:
h(2x + h + 3)
DQ = ---------------
h
Cancel the h's
1
h(2x + h + 3)
DQ = ---------------
h
1
DQ = 2x + h + 3
Edwin
