Question

21. Construct and simplify the difference quotient for: f(x)=6–x^2

Answer 1

\[\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\] is the first step

Answer 2

since \(f(x)=6-x^2\) you have \(f(x+h)=6-(x+h)^2\)
using algebra you ge t
\[f(x+h)=x-(x^2+2xh+h^2)=6-x^2-2xh-h^2\]

Answer 3

then \[f(x+h)-f(x)=6-x^2-2xh+h^2-(6-x^2)\] and as usual, everything without an \(h\) in it add up to zero, giving
\[f(x+h)-f(x)=-2xh-h^2\]

Answer 4

divide that by \(h\) and you are left with
\[\frac{f(x+h)-f(x)}{h}=-2x-h\]