The average rate of change from x 1 {\displaystyle x_{1}} to x 2 {\displaystyle x_{2}} , where x 1 < x 2 {\displaystyle x_{1}<x_{2}} , is
E ( x 2 ) − E ( x 1 ) x 2 − x 1 {\displaystyle {\frac {E(x_{2})-E(x_{1})}{x_{2}-x_{1}}}}
Setting x 1 = 0 {\displaystyle x_{1}=0} and x 2 = t {\displaystyle x_{2}=t} , we obtain
E ( t ) − E ( 0 ) t − 0 {\displaystyle {\frac {E(t)-E(0)}{t-0}}}
Recall from part b) that
E ( t ) = 4 t − 1 2 t 2 + 1 48 t 3 {\displaystyle E(t)=4t-{\frac {1}{2}}t^{2}+{\frac {1}{48}}t^{3}}
and that E ( 0 ) = 0 {\displaystyle E(0)=0}
Using this formula, we find that the average rate of change is
E ( t ) − E ( 0 ) t − 0 = 4 t − 1 2 t 2 + 1 48 t 3 t = 4 − 1 2 t + 1 48 t 2 {\displaystyle {\frac {E(t)-E(0)}{t-0}}={\frac {4t-{\frac {1}{2}}t^{2}+{\frac {1}{48}}t^{3}}{t}}=\color {blue}4-{\frac {1}{2}}t+{\frac {1}{48}}t^{2}}
to 
, where 
, is
and 
, we obtain
