The Fundamental Theorem and the Mean Value Theorem Our goal is to use information about F � to derive information about F . Our ﬁrst example of this process will be to compare the ﬁrst fundamental theorem to the Mean Value Theorem. We’ll use the notation ΔF = F (b) − F (a) and Δx = b − a. The ﬁrst fundamental theorem then tells us that: � b ΔF = f (x) dx. a If we divide both sides by Δx we get: � b ΔF 1 = f (x) dx Δx b−a a � �� Average(f ) the expression on the right is the average value of the function f (x) on the interval [a, b]. Why is this the average of f and not of F ? Consider the following Riemann sum: � n f (x) dx ≈ f (1) + f (2) + · · · + f (n). 0 This is a cumulative sum of values of f (x). The quantity: �n f (1) + f (2) + · · · + f (n) 0 f (x) dx ≈ n n is an average of values of f (x); in the limit, the average value of f (x) on the �b interval [a, b] is given by b−1 a a f (x) dx. We’ll rewrite the ﬁrst fundamental theorem one more time as: ΔF = Average(F � )Δx. In other words, the change in F is the average of the inﬁnitesimal change times the amount of time elapsed. We can now use inequalities to compare this to the (a) mean value theorem, which says that F (b)−F = F � (c) for some c between a b−a and b. We can rewrite this as: ΔF = F � (c)Δx. The value of Average(F � ) in the ﬁrst fundamental theorem is very speciﬁc, but the F � (c) from the mean value theorem is not; all we know about c is that it’s somewhere between a and b. Even if we don’t know exactly what c is, we know for sure that it’s less than the maximum value of F � on the interval from a to b, and that it’s greater than the minimum value of F � on that interval: � � � � � � � min F (x) Δx ≤ ΔF = F (c)Δx ≤ max F (x) Δx. a