Question

The position of an object at time t is given by s(t) = -1 - 13t. Find the instantaneous velocity at t = 8 by finding the derivative (you must use the limit of the difference quotient). Show all work.

Answer 1

To find the instantaneous velocity of an object given its position function s(t) = -1 - 13t, we calculate the derivative using the limit of the difference quotient. The derivative of s(t) with respect to t is -13, which means the instantaneous velocity at any time t is -13 m/s. For t = 8, the velocity v(8) is -13 m/s.

Step-by-step explanation:

The student is asking to find the instantaneous velocity of an object at time t = 8 when the position is given by s(t) = -1 - 13t. To do this, we need to find the derivative of the position function using the limit of the difference quotient. The derivative, s'(t), represents the instantaneous velocity.

To find s'(t), we'll start with the difference quotient:
s'(t) = limit as h approaches 0 of [s(t+h) - s(t)]/h. Plugging the position function into this formula gives us:
s'(t) = limit as h approaches 0 of [-1 - 13(t+h) - (-1 - 13t)]/h.
Simplify within the brackets to get lim as h approaches 0 of [-13h]/h. As h cancels out, we have lim as h approaches 0 of -13, which is simply -13. Therefore, the instantaneous velocity at any time t is -13 m/s. Specifically, at t=8, v(8) = -13 m/s.