sections 1.3 - 1.4\nscore: 5/21 answered: 5/21\nquestion 6\ngiven ( f(x)=2 x^{2}+7 ), find the average rate…

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 = 9) and (b=9 + h), so we need to find (f(9 + h)) and (f(9)) first.

Step2: Calculate (f(9 + h))

Substitute (x=9 + h) into (f(x)=2x^{2}+7). [ \begin{align*} f(9 + h)&=2(9 + h)^{2}+7\ &=2(81+18h+h^{2})+7\ &=162 + 36h+2h^{2}+7\ &=2h^{2}+36h + 169 \end{align*} ]

Step3: Calculate (f(9))

Substitute (x = 9) into (f(x)=2x^{2}+7). [ \begin{align*} f(9)&=2\times9^{2}+7\ &=2\times81+7\ &=162 + 7\ &=169 \end{align*} ]

Step4: Compute the average rate of change

[ \begin{align*} \frac{f(9 + h)-f(9)}{(9 + h)-9}&=\frac{(2h^{2}+36h + 169)-169}{h}\ &=\frac{2h^{2}+36h}{h}\ &=\frac{h(2h + 36)}{h}\ &=2h+36\quad(h\neq0) \end{align*} ]

Answer:

(2h + 36)