$5.(10$ points$)(*)$ In an experiment the total number of cars that pass a position x$0$ after t $=0$ is

recorded and the points have been smoothed over into a continuous function that approximates

this data, M $($x$0,$ t$)$ for t $0.$ Using M $,$ find an expression for q$($x$0,$ t$)$ in two ways:

$($a$)$ Derivative: From the definition of flow in the notes consider the number of cars that pass

x$0$ from t to t $+$t$.$ Write this out using M defined above and take the limit as $$t $0.$

$($b$)$ Integral: Write M $($the number of cars$)$ using the integral of flow then use the Fundamental

Theorem of Calculus.