Suppose an object moves along the y axis so that its location is yequalsf(x)equalsxsquaredplusx at time x (y is in meters, x is in seconds). find (a) the average velocity (the average rate of change of y with respect to x) for x changing from 2 to 5. (b) the average velocity for x changing from 2 to 2plush. (c) the instantaneous velocity at xequals2 second. (a) the average velocity is 8 m/sec. (b) the average velocity is (nothing )m/sec. (c) the instantaneous velocity is 5 m/sec.

Explanation:

Given

y(x) = x² + x ( x represents time in seconds )

A )

value of y at x = 2

= 2²+2 = 6 m

value of y at x = 5

= 5²+5 = 30 m

displacement = 30 - 6 = 24 m

Time duration = 5 - 2 = 3 s

average velocity

= displacement /Time duration

= 24 / 3 = 8 m /s

B ) For x changing from x to x plus ie for small change in x  :

value of y at x = 2

= 5

For small change in x , the value of y will again be 5 so displacement during small change in x will be zero . Hence average velocity will be zero.

C )  

y(x) = x² + x

dy / dx = 2x + 1

dy / dx  represents instantaneous  velocity

so when x = 2

instantaneous  velocity = 2 x 2 +1

= 5 m /s