The position of a moving car at time t is given by f(t)=a t2+b t+c, t0, 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 :

Question

The position of a moving car at time t is given by f(t)=a t2+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 : (A) a ( t 2- t 1)+ b (B) (t2-t1) / 2 (C) 2 a(t1+t2)+b (D) (t1+t2) / 2

Tardigrade Admin · Accepted Answer

(f(t2)-f(t1)/t2-t1)=2 a t+b (a(t22-t12)+b(t2-t1)/t2-t1)=2 a t+b ⇒ a(t2+t1)+b=2 a t+b ⇒ t=(t1+t2/2)

Q. 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 :

$a (t_{2} - t_{1}) + b$

$(t_{2} - t_{1}) /2$

$2 a (t_{1} + t_{2}) + b$

$(t_{1} + t_{2}) /2$

$\frac{f ( t _{2} ) - f ( t _{1} )}{t _{2} - t _{1}} = 2 a t + b$
$\frac{a ( t _{2}^{2} - t _{1}^{2} ) + b ( t _{2} - t _{1} )}{t _{2} - t _{1}} = 2 a t + b$
$\Rightarrow a (t_{2} + t_{1}) + b = 2 a t + b$
$\Rightarrow t = \frac{t _{1} + t _{2}}{2}$

Q. The position of a moving car at time t is given by f(t)=at2+bt+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 [t1​,t2​] is attained at the point :

a(t2​−t1​)+b

(t2​−t1​)/2

2a(t1​+t2​)+b

(t1​+t2​)/2