Question

1) The average value of the function g(x) = 3^cos x on the closed interval [ − pi , 0 ] is:






2)The change in the momentum of an object (Δ p) is given by the force, F, acting on the object multiplied by the time interval that the force was acting: Δ p = F Δt . If the force (in newtons) acting on a particular object is given by F(t) = cost, what's the total change in momentum of the object from time t = 5 until t = 7 seconds?




3) An ant's position during an 8 second time interval is shown by the graph below. What is the total distance the ant traveled over the time interval 2<=t<=8?

What is the total distance traveled by the ant over the time interval 0 <= t <= 8?

Answer 1

#1. Recall that the average value is

1/π ∫[-π,0] 3^cos(x) dx
use your calculator to find that the integral is 4.16348

Divide that by π and you get 1.32528

#2 is just another integral. Just as distance is ∫ v(t) dt, here the total change in momentum is

∫ cost dt

since Δ p = F Δt, in the continuous case, dp = F dt

#3 hard to say from the graph, but I expect it is just the arc length of the curve illustrated.

Or, if you can figure the velocity function, just integrate its absolute value.

Or, you can do it as in this video

https://www.khanacademy.org/math/ap-calculus-ab/derivative-applications-ab/rectilinear-motion-diff-calc-ab/v/total-distance-traveled-by-a-particle