17. -/1 points let f(x)=-x^{2}+4x - 7. find and simplify the difference quotient, \\(\\frac{f(x +…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: [ \begin{align*} f(x + h)&=-(x + h)^2+4(x + h)-7\ &=-(x^{2}+2xh+h^{2})+4x + 4h-7\ &=-x^{2}-2xh - h^{2}+4x + 4h-7 \end{align*} ]

Step2: Calculate $f(x + h)-f(x)$

[ \begin{align*} f(x + h)-f(x)&=(-x^{2}-2xh - h^{2}+4x + 4h-7)-(-x^{2}+4x - 7)\ &=-x^{2}-2xh - h^{2}+4x + 4h-7 + x^{2}-4x + 7\ &=-2xh - h^{2}+4h \end{align*} ]

Step3: Calculate $\frac{f(x + h)-f(x)}{h}$

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{-2xh - h^{2}+4h}{h}\ &=\frac{h(-2x - h + 4)}{h}\ &=-2x - h+4 \end{align*} ]

Answer:

$-2x - h + 4$