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Find the average speed of a car, starting at time 0, if it drives for 5 hours and its speed at time t (in hours) is given by s(t)=5t^{2}+7t+e^{t} in miles per hour.

This problem is done just like any other average value problem. Integrating the speed function over the interval gives the total distance traveled since s'(t)=x(t). Dividing this by the length of the interval gives you the distance traveled in 1 hour, which gives you the average speed per hour. Thus the average speed is

{\frac  {1}{5}}\int _{{0}}^{{5}}5t^{2}+7t+e^{t}\,dx={\frac  {1}{5}}\left[{\frac  {5}{3}}t^{3}+{\frac  {7}{2}}t^{2}+e^{t}\right]_{{0}}^{{5}}={\frac  {1}{5}}\left[{\frac  {5}{3}}5^{3}+{\frac  {7}{2}}5^{2}+e^{5}-1\right]={\frac  {1769+6e^{5}}{30}}\approx 88.65\, mph

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