The Mean Value Theorem and Estimation The following problem appeared on the second exam: Given that F � (x) = 1+1x and F (0) = 1, the mean value theorem implies that A < F (4) < B for which A and B? y=1 y= 1 5 0 4 Figure 1: Graph of F � (x) = 1 1+x . To solve this, we ﬁrst apply the mean value theorem in such a way that the value F (4) appears, then use our knowledge of the formula for F � (c) to ﬁnd limits on that value. Remember that c is an unknown value between (in this case) 0 and 4. F (4) − F (0) F � (c)(4 − 0) 1 ·4 1+c = = We don’t know what 1 1+c (Use the MVT on F (4)) is, but we know that so: decreases from 0 to inﬁnity, 1 1 1 1 > > = . 1 1+c 1+4 5 1= Hence: 4> 1 4 ·4> . 1+c 5 We conclude that: 4 > F (4) − F (0) > and since F (0) = 1 we have: 5 > F (4) > Our ﬁnal answer is A = 1 x 9 5 and B = 5. 9 . 5 4 5 Now let’s compare this to what we can do with the fundamental theorem of calculus: � 4 dx F (4) − F (0) = 1 +x 0 Based on what we know about the graph of y = x1 and the area under it, we can deduce that: � 4 � 4 dx F (4) − F (0) = < 1dx = 4 0 1+x 0 and that � F (4) − F (0) = 0 4 dx > 1+x � 4 0 1 4 dx = . 5 5 So once again we have: 4 < F (4) − F (0) < 4. 5 � 4 dx Geometrically, we interpret 0 1+x as the area under a curve. We got an upper bound on the area by comparing it to the area of a rectangle whose height was the maximum value of 1+1x on the interval, and got a lower bound by comparing to a rectangle whose hight was the minimum of 1+1x on [0, 4]. � 4 dx We could think of this as estimating 0 1+x by comparing it to two diﬀerent Riemann sums, each with only one rectangle. � 4 dx lower Riemann sum < < upper Riemann sum 0 1+x

The Mean Value Theorem and Estimation

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