Given the position function s(t) = t cos t, find the velocity function.
Answer
v(t) = -sin t
v(t) = sin t
v(t) = cos t - t sin t
v(t) = cos t + t sin t
Ahh, calculus...
v= dx/dt= d/dt (tcost)=cost-tsint
To find the velocity function, we need to take the derivative of the position function, s(t) = t cos t, with respect to time, t.
Using the product rule, the derivative of t cos t is given by:
s'(t) = (d/dt)(t)cos t + t(d/dt)(cos t)
The derivative of t with respect to t is 1, and the derivative of cos t with respect to t is -sin t.
Therefore, the velocity function, v(t), is:
v(t) = 1 * cos t + t * (-sin t)
Simplifying,
v(t) = cos t - t sin t
So, the velocity function is v(t) = cos t - t sin t.
To find the velocity function, we need to take the derivative of the position function with respect to time.
Given the position function s(t) = t cos t, we can apply the product rule of differentiation to find the velocity function v(t):
v(t) = (d/dt)(t cos t)
= t(d/dt)(cos t) + cos t(d/dt)(t)
To differentiate cos t with respect to time, we can use the chain rule:
(d/dt)(cos t) = -sin t
To differentiate t with respect to time, we treat it as a constant:
(d/dt)(t) = 1
Now we can substitute these values into the equation for v(t) we found earlier:
v(t) = t(-sin t) + cos t(1)
= -t sin t + cos t
Therefore, the velocity function is v(t) = -t sin t + cos t.
So, the correct answer is v(t) = -t sin t + cos t.