Find the difference quotient and simplify your answer.
$$f ( s ) = - 2 s ^ { 2 } - 2 s , \quad \frac { f ( 4 + h ) - f ( 4 ) } { h } , h \neq 0$$
A) $6 + h$
B) $- 18 - 2 s - \frac { 16 } { s }$
C) $- 2 - 2 s - \frac { 16 } { s }$
D) $- 2 - 2 h$
E) $- 18 - 2 h$

Question

Quizplus · Accepted Answer

To find the difference quotient, we substitute $s = 4 + h$ and $s = 4$ into the function $f(s) = -2s^2 - 2s$ and simplify the expression $\frac{f(4 + h) - f(4)}{h}$.$$f(4 + h) = -2(4 + h)^2 - 2(4 + h) = -2(16 + 8h + h^2) - 8 - 2h = -32 - 16h - 2h^2 - 8 - 2h$$$$f(4) = -2(4)^2 - 2(4) = -2(16) - 8 = -32 - 8$$Substituting these into the difference quotient formula gives:$$\frac{(-32 - 16h - 2h^2 - 8 - 2h) - (-32 - 8)}{h} = \frac{-16h - 2h^2 - 2h}{h}$$$$= -16 - 2h - 2 = -18 - 2h$$Therefore, the correct answer is $-18 - 2h$, which is choice E.

Find the Difference Quotient and Simplify Your Answer f(s)=−2s2−2s,f(4+h)−f(4)h,h≠0f ( s ) = - 2 s ^ { 2 } - 2 s , \quad \frac { f ( 4 + h ) - f ( 4 ) } { h } , h \neq 0f(s)=−2s2−2s,hf(4+h)−f(4)​,h=0

Find the Difference Quotient and Simplify Your Answer $f ( s ) = - 2 s ^ { 2 } - 2 s , \quad \frac { f ( 4 + h ) - f ( 4 ) } { h } , h \neq 0$