Net Change Inverse Property: If f(x) is integrated over [a,b] and f(x) = F' (x) , then Saf(x) ax = Saf(x)dx = [a F' ( x) dx = F (x ) = F(b ) - F(a ) Net Change Theorem Applications: Examples follow to help you capture the concept. V(t) = volume V '(t) = rate of liquid flow Se V'(t) dt = V(t2) - V(t1) change in amount of water from [t1, t2] N(t) population at time t N '(t) = rate of growth of population Se N'(t ) dt = N(t2) - N(t1) is net change in population or quantity from [t1, t2] S(t) = position function of an object that moves along a straight line V(t) = S'(t) = velocity of particle serv( t ) = Sex s' (t ) dt = s(t2) - s(t1) net change (displacement) in position of particle [t1,t2] Problem [3] (1) An object is moving with velocity v(t) = t2 + 4t-5. Find the displacement and the total distance travelled from t[0, 7] Displacement: s(t) Total Distance = Stil v(t) | dt = Serv(t)at + S? - v(t) at