The displacement (in feet) of a particle moving in a straight line is given by
s = 1/2t^2 − 5t + 15, where t is measured in seconds. Find the instantaneous velocity (in ft/s) when t = 8.
s = 1/2t^2 - 5t + 15
ds/dt = t - 5
ds/dt = 8 - 5
ds/dt = 3
Therefore, the instantaneous velocity of the particle after 8 seconds is 3 ft/s.
differentiating ... ds/dt = 2 t - 5
plug in 8 for t
ds/dt should have been
t - 5
yup ... go with the update ... blizzard blindness ...
To find the instantaneous velocity when t = 8, we need to find the derivative of the displacement function with respect to t and evaluate it at t = 8.
The displacement function is given by s = 1/2t^2 − 5t + 15.
To find the derivative of the displacement function, we can apply the power rule of differentiation. The power rule states that if we have a function of the form f(t) = t^n, then the derivative is given by f'(t) = n*t^(n-1).
Applying the power rule of differentiation to our displacement function:
s'(t) = [d/dt](1/2t^2) - [d/dt](5t) + [d/dt](15)
The derivative of 1/2t^2 with respect to t is:
[d/dt](1/2t^2) = (1/2) * [d/dt](t^2) = (1/2) * 2t = t
The derivative of 5t with respect to t is simply 5.
The derivative of 15 with respect to t is 0, since it's a constant.
Putting it all together, we get:
s'(t) = t - 5
Now, we can evaluate the derivative at t = 8 to find the instantaneous velocity:
s'(8) = 8 - 5 = 3 ft/s
Therefore, the instantaneous velocity when t = 8 is 3 ft/s.