A particle moves on a line away from its initial position so that after t hours it is s=3t^2+t miles from its original position.
might speedy, I must say!
Oh -- was there a question in there somewhere?
To find the velocity of the particle, you need to differentiate the position function with respect to time. In this case, the position function is given by s(t) = 3t^2 + t.
To differentiate s(t), you can apply the power rule of differentiation. For any term of the form k*t^n, the derivative is obtained by multiplying the coefficient (k) by the power (n) and then decreasing the power by 1. In this case, we have:
s'(t) = (d/dt)(3t^2 + t)
= 3*(d/dt)(t^2) + (d/dt)(t)
Applying the power rule, we find:
s'(t) = 3*2t + 1
= 6t + 1
Hence, the velocity of the particle at any time t is given by v(t) = 6t + 1 miles per hour.