Question

The position of a bug moving along a straight path is given by f(t) = 2 - (2/3)t. At what time is the instantaneous velocity equal to the average velocity of the bug on [0, 6]?

a) t = 2
b) t = 3
c) t = 4
d) t = 5

Answer 1

The instantaneous velocity of the bug is -(2/3) and is constant. The average velocity on the interval [0, 6] is -(1/3). Since they are not equal, none of the given options are correct.

Step-by-step explanation:

To find when the instantaneous velocity is equal to the average velocity for a bug moving along a path as defined by the position function f(t) = 2 - (2/3)t, we must first calculate the instantaneous velocity, which is the derivative of the position function. The derivative of f(t) with respect to t is f'(t) = -(2/3), which means the instantaneous velocity is constant at -(2/3) units per time interval.

We then calculate the average velocity over the interval [0, 6] by using the formula (f(b) - f(a))/(b - a), where a and b are the endpoints of the interval. So, the average velocity vavg on [0, 6] is (f(6) - f(0))/(6 - 0) = ((2 - (2/3)*6) - (2))/6 = (-2)/6 = -(1/3).

Since the instantaneous velocity is constant at -(2/3) and the average velocity is -(1/3), they are not equal at any time in the given interval. Therefore, the answer would be none of the above as no options provided match this scenario.