A particle is moving in a straight line such that the distance covered by it in t seconds from a point is `((t^(3))/(3)-t)` cm. find its speed at t=3 seconds.

Explanation

Here `f(t)=(t^(3))/(3)-t`
The speed at t=3 sec will be obtained from f'(3).
`f'(3)=underset(hrarr0)"lim"(f(3+h)-f(3))/(h)`
`=underset(hrarr0)"lim"({(1)/(3)(3+h)^(3)-(3+h)}-{(1)/(3)(3)^(3)-3})/(h)`
`=underset(hrarr0)"lim"((1)/(3)(27+27h+9h^(2)+h^(3) -27)-h)/(h)`
`=underset(hrarr0)"lim"(8h +3h^(2)+(1)/(3)h^(3))/(h)`
`=underset(hrarr0)"lim"(8+3h+(1)/(3)h^(2))=8`
`therefore` speed of the particle at t=3 sec is 8cm/sec.