The position, p(t), of an object at time t is given. Find the instantaneous velocity at the indicated time c. Also determine whether it is moving forward, backward, or neither at time c.
p(t)=2-1/t at c=3
To find the instantaneous velocity, we need to find the derivative of the position function with respect to time.
Given: p(t) = 2 - 1/t
To find the derivative, we can use the power rule for differentiation. The power rule states that if we have a function of the form f(t) = x^n, then the derivative with respect to t is given by f'(t) = nx^(n-1).
Let's apply the power rule to find the derivative of p(t):
p'(t) = d/dt (2 - 1/t)
= 0 - d/dt(1/t)
= 0 - (-1/t^2) (using the power rule with n = -1)
= 1/t^2
Now we have the derivative of p(t) with respect to t. To find the instantaneous velocity at time t = c, we substitute c into the derivative expression:
p'(c) = 1/c^2
So the instantaneous velocity at time t = c is 1/c^2.
To determine whether the object is moving forward, backward, or neither at time c, we can examine the sign of the velocity. If the velocity is positive, the object is moving forward; if it is negative, the object is moving backward; if it is zero, the object is not moving.
In this case, since the velocity is given by 1/c^2, we need to substitute the value of c = 3:
p'(3) = 1/3^2
= 1/9
Since 1/9 is positive, it means the object is moving forward at time c = 3.