The position of a moving car at time t is given by f ( t ) = a t 2 + b t + c , t 0 , where a , b and c are real numbers greater than 1 . Then the average speed of the car over the time interval t 1

Question

The position of a moving car at time t is given by f ( t ) = a t 2 + b t + c , t > 0 , where a , b and c are real numbers greater than 1 . Then the average speed of the car over the time interval t 1 , t 2 is attained at the point:

TestHub · Accepted Answer

Given: position of the moving car at time $t is = f (t) = a t^{2} + b t + c$

So,

$v_{avg} = \frac{f (t_{2}) - f (t_{1})}{t_{2} - t_{1}}$

$\Rightarrow v_{avg} = \frac{a (t_{2}^{2} - t_{1}^{2}) + b (t_{2} - t_{1})}{t_{2} - t_{1}}$

$\Rightarrow v_{avg} = a (t_{1} + t_{2}) + b$

The instantaneous speed is given by:

$f^{'} (t) = 2 a t + b$

So, to get the point where average speed is equal to car's actual speed,

$f^{'} (t) = v_{avg}$

$\Rightarrow a (t_{1} + t_{2}) + b = a t + b$

$\Rightarrow t = \frac{t_{1} + t_{2}}{2}$

Mathematics - Application of Derivative Question with Solution | TestHub

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