exam #1\nscore: 24.33/40 answered: 15/21\nquestion 16\ngiven ( f(x)=x^{2}+2x - 7 ), find the average rate of…

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function (y = f(x)) on the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}). Here, (a = 6) and (b=6 + h), so the average rate of change is (\frac{f(6 + h)-f(6)}{(6 + h)-6}).

Step2: Find (f(6 + h))

Substitute (x = 6 + h) into (f(x)=x^{2}+2x - 7). [ \begin{align*} f(6 + h)&=(6 + h)^{2}+2(6 + h)-7\ &=36+12h+h^{2}+12 + 2h-7\ &=h^{2}+14h + 41 \end{align*} ]

Step3: Find (f(6))

Substitute (x = 6) into (f(x)=x^{2}+2x - 7). [ \begin{align*} f(6)&=6^{2}+2\times6-7\ &=36 + 12-7\ &=41 \end{align*} ]

Step4: Substitute (f(6 + h)) and (f(6)) into the average - rate - of - change formula

[ \begin{align*} \frac{f(6 + h)-f(6)}{(6 + h)-6}&=\frac{(h^{2}+14h + 41)-41}{h}\ &=\frac{h^{2}+14h}{h}\ &=h + 14\quad(h\neq0) \end{align*} ]

Answer:

(h + 14)