Problem: The position of a body is given by

Find the:

a) velocity as a function of time
b) velocity at t = 2.5 seconds
c) acceleration as a function of time
d) acceleration at t = 3.2 seconds
e) position at 4.1 seconds

Taking a derivative twice, I get

This takes care of sections a and c. It is important NOT to substitute numbers if the general relationships - v as a f(t) and a as g(t) - are desired. (Functional notation: sometimes we use f₁(t) and f₂(t); sometimes we use f and g, or other letters, to indicate two different functional relationships.)

For sections b, d, and e, all that's needed is to substitute the times:

Notice the functional notation? a(3.2) does not mean "a times 3.2"! It means "a evaluated at 3.2 sec)".