Read this section to learn about the Mean Value Theorem and its consequences. Work through practice problems 1-3.
Geometrically, the Mean Value Theorem is a "tilted" version of Rolle's Theorem (Fig. 5). In each theorem we conclude that there is a number so that the slope of the tangent line to at is the same as the slope of the line connecting the two ends of the graph of on the interval . In Rolle's Theorem, the two ends of the graph of are at the same height, , so the slope of the line connecting the ends is zero. In the Mean Value Theorem, the two ends of the graph of do not have to be at the same height so the line through the two ends does not have to have a slope of zero.
Fig. 5
Mean Value Theorem: If is continuous for and differentiable for ,
then there is at least one number , between and , so the tangent line at is parallel to the secant line through the points and .
Proof: The proof of the Mean Value Theorem uses a tactic common in mathematics: introduce a new function which satisfies the hypotheses of some theorem we already know and then use the conclusion of that previously proven theorem. For the Mean Value Theorem
we introduce a new function, , which satisfies the hypotheses of Rolle's Theorem. Then we can be certain that the conclusion of Rolle's Theorem is true for , and the Mean Value Theorem for
follows from the conclusion of Rolle's Theorem for .
First, let be the straight line through the ends and of the graph of . The function goes through the point so . Similarly, . The slope of the linear function is so
for all between and , and is continuous and differentiable. (The formula for is with .)
Define for (Fig. 6). The function satisfies the hypotheses of Rolle's theorem:
and
is continuous for since both and are continuous there, and is differentiable for since both
and are differentiable there, so the conclusion of Rolle's Theorem applies to : there is a , between and , so that .
Fig. 6
The derivative of is so we know that there is a number , between and , with . But so .
Graphically, the Mean Value Theorem says that there is at least one point where the slope of the tangent line, , equals the slope of the line through the end points of the graph segment,
and . Fig. 7 shows the locations of the parallel tangent lines for several functions and intervals.
Fig. 7
The Mean Value Theorem also has a very natural interpretation if represents the position of an object at time represents the velocity of the object at the instant , and represents the average (mean) velocity of the object during the time interval from time to time . The Mean Value Theorem says that there is a time , between
and , when the instantaneous velocity, , is equal to the average velocity for the entire trip, . If your average velocity during a trip is
miles per hour, then at some instant during the trip you were traveling exactly miles per hour.
Practice 2: For on the interval , calculate and find the value of so that .