Solution

Net Change: \[ \begin{{array}}{{rcl}} f(a+h)-f(a) &=& \displaystyle\frac{{7}}{(a+h)^2}-\displaystyle\frac{{7}}{a^2}\\ f(a+h)-f(a) &=& \displaystyle\frac{7\cdot {\color{{red}} a^2}}{(a+h)^2 {\color{{red}} a^2}} - \displaystyle\frac{7\cdot {\color{{red}} (a+h)^2}}{a^2{\color{{red}} (a+h)^2}}\\ f(a+h)-f(a) &=&\displaystyle\frac{7a^2-7(a+h)^2}{a^2(a+h)^2}\\ f(a+h)-f(a) &=&\displaystyle\frac{7a^2-7(a^2+2ah+h^2)}{a^2(a+h)^2}\\ f(a+h)-f(a) &=&\displaystyle\frac{7a^2-7a^2-14ah-7h^2)}{a^2(a+h)^2}\\ f(a+h)-f(a) &=&\displaystyle\frac{-14ah-7h^2)}{a^2(a+h)^2}\\ f(a+h)-f(a) &=&\displaystyle\frac{-7h(2a+h)}{a^2(a+h)^2} \end{{array}}\]

Horizontal Change: \(a+h-(a))=h\)

Average Rate of Change between \(s=a\) and \(s=a+h\): \[ \frac{-7{\color{{red}}h}(2a+h)}{a^2(a+h)^2}\div {\color{{red}}h} = \boxed{\frac{-7(2a+h)}{a^2(a+h)^2}} \]

This question is more tricky to check, but we can verify the formula by setting up some variables in Desmos. The slope will be the average rate shown above, and the point we pass the line through will be \((a,f(a))\) which is \(\left(a,\frac{{7}}{a^2}\right)\). We will also assume that \(h \gt 0\), although it doesn't strictly have to be.