4.2 The Mean Value Theorem
Many of the results of this chapter depend on one central fact, which is called the Mean Value Theorem, but before we can get to that, we need the following result.
This seems plausible if we look at the graphs of some different possibilities (below), and the proof follows on p. 211.
1
Verify that the function satisfies the hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion.
f(x) = x3 ­ x2 ­ 6x + 2, [0, 3]
Let f(x) = x­2. Show that f(­1) = f(1) but there is no number c in (­1, 1) such that f '(c) = 0.
Why is this not a contradiction of Rolle's Theorem?
2
Rolle's theorem is used to prove the following theorem, which was first stated by Joseph­Louis Lagrange.
Again, this seems plausible if we look at the graphs of some different possibilities (below), and the proof follows on pp. 212­213.
3
Use the graph below to estimate the values of c that satisfy the conclusion of the Mean Value Theorem for the interval [0, 8].
If an object moves in a straight line with position function s = f(t), then the average velocity between t = a and t = b is and the instantaneous velocity at t = c is f '(c).
Thus, the MVT tells us that at some time t = c between a and b, the instantaneous velocity is equal to that average velocity. For instance, if a car traveled 140 miles in 2 hours, then the speedometer must have read 70 mph at least once!
4
Use the graph below to estimate the values of c that satisfy the conclusion of the Mean Value Theorem for the interval [­8, 10].
5
Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the MVT.
f(x) = x3 ­ 3x + 2, [­2, 2]
Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the MVT.
f(x) = sin x, [0, π/2]
6
Find a number c that satisfies the conclusion of the MVT on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at (c, f(c)). Are the secant line and the tangent line parallel?
f(x) = , [­3, 2]
7
Suppose that f(x) = (x ­ 4)­2. Show that there is no value of c in (2, 7) such that f(7) ­ f(2) = f '(c)(7 ­ 2). (i.e. f '(c) ≠ for any value of c.)
Why is this not a contradiction of the MVT?
8
The Mean Value Theorem can be used to establish some of the basic facts of differential calculus, including the following:
9