Fundamental theorem help

Question

let ft(t)=F'(t), with F(t)=7t^2 + t, and let a=1 and b=3. write the integral and evaluate it using the Fundamental theorem of Calculus so far I was able to write Integral=(14t +1) but need help

dt=_________ -___________=_________

Answer 1

if you are evaluating the integral at a and b, 1 to 3

then 7(3)^2 + 3 - (7(1)^2 + 1) = 63+3 - 7-1 = 66- 8 = 58

= the area under the parabolic curve from t=1 to t=3

F(t) is the integral of f(t)

f(t) = the derivative of F(t) = F'(t) = 14t+1 = the slope of the curve

Not clear why you want to evaluate the deriviative

but slope of the curve at t=3 = 14(3)+1 = 43

slope of the curve at t=1 = 14(1) + 1 = 15

change in slope from t = 1 to t=3 = 43-15 = 28