This is from an early portion of Integral Calculus. So the Fundamental Theorem is about all that has been covered thus far.

If the value of a function p = 2 at time = 0 and the function is changing at a rate of -1 at t= 0, -1 at t=1, -1 at t=2 then at 0 at t=3 and 1 at t=4 and 1 at t=5 , Give the values of this function at t = 1 through t = 5

I assumed that I would just take the value of p at 0 and add to it or subtract from it by the rate of change at each t value.

In doing so, I would get p(1) = 2 + -1 = 1 However, a solution I saw showed the following:

p(0) + integral from 0 to 1 of p'(t) dt which the way I would use this method would yield an answer of 2 + (-1 - (-1)) which is equal to two. However, the solution shows:

2 + (-1 + -1) / (1/1) and therefore the answer is 0

I am trying to figure out my error in reasoning. Thanks for any help, Frostking