We motivate this section with the following question: Suppose you leave your house and drive to your friend’s house in a city \(100\) miles away, completing the trip in two hours. At any point during the trip do you necessarily have to be going \(50\) miles per hour?
In answering this question, it is clear that the average speed for the entire trip is 50 mph (i.e. \(100\) miles in \(2\) hours), but the question is whether or not your instantaneous speed is ever exactly 50 mph. More simply, does your speedometer ever read exactly 50 mph? The answer, under some very reasonable assumptions, is “yes.”
Let’s now see why this situation is in a calculus text by translating it into mathematical symbols.
First assume that the function \(y = f(t)\) gives the distance (in miles) traveled from your home at time \(t\) (in hours) where \(0\le t\le 2\text{.}\) In particular, this gives \(f(0)=0\) and \(f(2)=100\text{.}\) The slope of the secant line connecting the starting and ending points \((0,f(0))\) and \((2,f(2))\) is therefore
The slope at any point on the graph itself is given by the derivative \(\fp(t)\text{.}\) So, since the answer to the question above is “yes,” this means that at some time during the trip, the derivative takes on the value of 50 mph. Symbolically,
How about more generally? Given any function \(y=f(x)\) and a range \(a\le x\le b\) does the value of the derivative at some point between \(a\) and \(b\) have to match the slope of the secant line connecting the points \((a,f(a))\) and \((b,f(b))\text{?}\) Or equivalently, does the equation \(\fp(c) = \frac{f(b)-f(a)}{b-a}\) have to hold for some \(a \lt c \lt b\text{?}\)
Let’s look at two functions in an example.
Example3.2.2.Comparing average and instantaneous rates of change.
with \(a=-1\) and \(b=1\) as shown in Figure 3.2.3. Both functions have a value of \(1\) at \(a\) and \(b\text{.}\) Therefore the slope of the secant line connecting the end points is \(0\) in each case. But if you look at the plots of each, you can see that there are no points on either graph where the tangent lines have slope zero. Therefore we have found that there is no \(c\) in \([-1,1]\) such that
The graph illustrates the function \(f_1(x) = \frac{1}{x^2}\text{,}\) showcasing a scenario where the Mean Value Theorem is not applicable. As \(x\) approaches zero, the function demonstrates a vertical asymptote, highlighting its discontinuity and the tendency of the values to become infinite.
Within the interval \([-1, 1]\text{,}\) although the function’s value is the same at both endpoints, there is no point where the function’s derivative, which represents the slope of the tangent, is zero. This absence violates the Mean Value Theorem’s requirement for the function to be both continuous on the closed interval and differentiable on the open interval, specifically at \(x = 0\text{.}\)
(a)A graph of \(f_1(x) = 1/x^2\)
Displayed is the function \(f_2(x) = \abs{x}\text{,}\) chosen to highlight a case where the Mean Value Theorem cannot be applied. The graph forms a V shape, characteristic of the absolute value function, and is continuous over the interval \([-1, 1]\text{.}\)
At the endpoints of the interval, the function attains the same value, yielding a secant line with a slope of zero. However, due to the sharp corner at the origin \(x = 0\text{,}\) the function is not differentiable at this point. This lack of differentiability means that there does not exist a point in the interval where the slope of the tangent line is equal to the slope of the secant line, as required by the Mean Value Theorem. Thus, the function \(f_2(x)\) serves as an example of when the Mean Value Theorem’s conditions are not fulfilled.
(b)A graph of \(f_2(x) = \abs{x}\)
Figure3.2.3.Graphs of two “misbehaving” functions
So what went “wrong”? It may not be surprising to find that the discontinuity of \(f_1\) and the corner of \(f_2\) play a role. If our functions had been continuous and differentiable, would we have been able to find that special value \(c\text{?}\) This is our motivation for the following theorem.
Theorem3.2.4.The Mean Value Theorem of Differentiation.
Let \(y=f(x)\) be a continuous function on the closed interval \([a,b]\) and differentiable on the open interval \((a,b)\text{.}\) There exists a value \(c\text{,}\)\(a \lt c \lt b\text{,}\) such that
That is, there is a value \(c\) in \((a,b)\) where the instantaneous rate of change of \(f\) at \(c\) is equal to the average rate of change of \(f\) on \([a,b]\text{.}\)
Note that the reasons that the functions in Example 3.2.2 fail are indeed that \(f_1\) has a discontinuity on the interval \([-1,1]\) and \(f_2\) is not differentiable at the origin.
We will give a proof of the Mean Value Theorem below. To do so, we use a fact, called Rolle’s Theorem, stated here.
Theorem3.2.5.Rolle’s Theorem.
Let \(f\) be continuous on \([a,b]\) and differentiable on \((a,b)\text{,}\) where \(f(a) = f(b)\text{.}\) There is some \(c\) in \((a,b)\) such that \(\fp(c) = 0\text{.}\)
Consider Figure 3.2.7 where the graph of a function \(f\) is given, where \(f(a) = f(b)\text{.}\) It should make intuitive sense that if \(f\) is differentiable (and hence, continuous) that there would be a value \(c\) in \((a,b)\) where \(\fp(c)=0\text{;}\) that is, there would be a relative maximum or minimum of \(f\) in \((a,b)\text{.}\) Rolle’s Theorem guarantees at least one; there may be more.
The image showcases a graph of the polynomial function \(f(x) = x^3 - 5x^2 + 3x + 5\text{,}\) which is used to illustrate Rolle’s Theorem. The function is plotted to show the existence of at least one point \(c\) in the open interval \((a, b)\) where the derivative \(\fp(c)\) is zero, indicated by the horizontal tangent at the peak of the curve. The points labeled \(a\) and \(b\) on the x-axis denote the interval on which the function’s endpoints have equal values, satisfying the precondition for Rolle’s Theorem.
Rolle’s Theorem is stated, asserting that for a function continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\text{,}\) where the function values at the endpoints are equal (\(f(a) = f(b)\)), there exists at least one point \(c\) within \((a, b)\) where the derivative \(f'(c)\) is zero. This theorem is a specific case of the Mean Value Theorem.
Figure3.2.7.A graph of \(f(x) = x^3-5x^2+3x+5\text{,}\) where \(f(a) = f(b)\text{.}\) Note the existence of \(c\text{,}\) where \(a\lt c\lt b\text{,}\) where \(\fp(c)=0\text{.}\)
Rolle’s Theorem is presented here as a stepping stone toward the Mean Value Theorem, but it’s a useful result in its own right. It often turns up as a tool in mathematical problem solving. The video in Figure 3.2.8 illustrates one such use of Rolle’s Theorem.
Figure3.2.8.Using Rolle’s Theorem to show a polynomial has at most one real root
Rolle’s Theorem is really just a special case of the Mean Value Theorem. If \(f(a) = f(b)\text{,}\) then the average rate of change on \((a,b)\) is \(0\text{,}\) and the theorem guarantees some \(c\) where \(\fp(c)=0\text{.}\) We will prove Rolle’s Theorem, then use it to prove the Mean Value Theorem.
Proof of Rolle’s Theorem.
Let \(f\) be differentiable on \((a,b)\) where \(f(a)=f(b)\text{.}\) We consider two cases.
Case.
Consider the case when \(f\) is constant on \([a,b]\text{;}\) that is, \(f(x) = f(a) = f(b)\) for all \(x\) in \([a,b]\text{.}\) Then \(\fp(x) = 0\) for all \(x\) in \([a,b]\text{,}\) showing there is at least one value \(c\) in \((a,b)\) where \(\fp(c)=0\text{.}\)
Case.
Now assume that \(f\) is not constant on \([a,b]\text{.}\) The Extreme Value Theorem guarantees that \(f\) has a maximal and minimal value on \([a,b]\text{,}\) found either at the endpoints or at a critical value in \((a,b)\text{.}\) Since \(f(a)=f(b)\) and \(f\) is not constant, it is clear that the maximum and minimum cannot both be found at the endpoints. Assume, without loss of generality, that the maximum of \(f\) is not found at the endpoints. Therefore there is a \(c\) in \((a,b)\) such that \(f(c)\) is the maximum value of \(f\text{.}\) By Theorem 3.1.15, \(c\) must be a critical number of \(f\text{;}\) since \(f\) is differentiable, we have that \(\fp(c) = 0\text{,}\) completing the proof of the theorem.
We know \(g\) is differentiable on \((a,b)\) and continuous on \([a,b]\) since \(f\) is. We can show \(g(a)=g(b)\) (it is actually easier to show \(g(b)-g(a)=0\text{,}\) which suffices). We can then apply Rolle’s theorem to guarantee the existence of \(c\) in \((a,b)\) such that \(g'(c) = 0\text{.}\) But note that
Going back to the very beginning of the section, we see that the only assumption we would need about our distance function \(f(t)\) is that it be continuous and differentiable for \(t\) from \(0\) to \(2\) hours (both reasonable assumptions). By the Theorem 3.2.4, we are guaranteed a time during the trip where our instantaneous speed is 50 mph. This fact is used in practice. Some law enforcement agencies monitor traffic speeds while in aircraft. They do not measure speed with radar, but rather by timing individual cars as they pass over lines painted on the highway whose distances apart are known. The officer is able to measure the average speed of a car between the painted lines; if that average speed is greater than the posted speed limit, the officer is assured that the driver exceeded the speed limit at some time.
Note that the Theorem 3.2.4 is an existence theorem. It states that a special value \(c\)exists, but it does not give any indication about how to find it. It turns out that when we need the Theorem 3.2.4, existence is all we need.
Example3.2.10.Using the Mean Value Theorem.
Consider \(f(x) = x^3+5x+5\) on \([-3,3]\text{.}\) Find \(c\) in \([-3,3]\) that satisfies the Theorem 3.2.4.
Solution1.
The average rate of change of \(f\) on \([-3,3]\) is:
We have found two values \(c\) in \([-3,3]\) where the instantaneous rate of change is equal to the average rate of change; the Theorem 3.2.4 guaranteed at least one. In Figure 3.2.11, \(f\) is graphed with a line representing the average rate of change; the lines tangent to \(f\) at \(x=\pm \sqrt{3}\) are also given. Note how these lines are parallel (i.e., have the same slope) to the secant line.
This graph depicts the function \(f(x) = x^3 + 5x + 5\) over the interval \([-3, 3]\text{.}\) The function is shown as a solid blue curve, and there is a black dashed line that represents the average rate of change of the function over the given interval.
Two red dashed lines are drawn on the graph, indicating the tangent lines to the curve at the points where the function’s derivative equals the average rate of change. These points, marked on the x-axis, are where the slope of the tangent is the same as the slope of the secant line (the black dashed line), which is the essence of the Mean Value Theorem. The exact points on the x-axis where these tangents touch the curve correspond to the solutions where \(f'(x) = 3x^2 + 5\) is set equal to 14, the average rate of change.
Figure3.2.11.Demonstrating the Mean Value Theorem in Example 3.2.10
Solution2.Video solution
While the Theorem 3.2.4 has practical use (for instance, the speed monitoring application mentioned before), it is mostly used to advance other theory. We will use it in the next section to relate the shape of a graph to its derivative.
Before ending this section, we give two important consequences of the Mean Value Theorem. Each of these consequences has important applications to mathematical theory, and can be easily understood in the context of the position and velocity of objects in motion.
First, we recall that the derivative of any constant function is zero. Is the converse true? That is, are constant functions the only ones whose derivative is zero? The Mean Value Theorem says yes. This officially establishes our intuition about objects in (or, actually, not in) motion: if the velocity of an object is \(0\text{,}\) then the object’s position is unchanged; it is constant. Second, if two functions \(f\) and \(g\) have the same derivative, what does this tell us about \(f\) and \(g\text{?}\) The Mean Value Theorem implies that these functions must only differ by a constant; that is, \(f(x)=g(x)+C\text{,}\) for some constant \(C\text{.}\)
This has an application to motion that is not intuitive to some. Suppose two objects start moving while 5 ft apart, and always move with the same velocity. Then the two objects will always be 5 ft apart. (If two pennies are dropped from the 30th and 31st stories of a tall building at the same time, they will always be 1 story apart as they fall.)
Theorem3.2.12.Consequences of the Mean Value Theorem.
Let \(f\text{,}\)\(g\text{,}\) and \(h\) be differentiable (and therefore continuous) functions on an in terval \(I\text{.}\)
If \(\fp(x)=0\) for all \(x\) in the interval \(I\text{,}\) then \(f\) is a constant function on \(I\text{.}\)
If \(g'(x)=g'(x)\) for all \(x\) in \(I\text{,}\) then there is a constant \(C\) such that \(g(x)=h(x)+C\) for all \(x\) in \(I\text{.}\)
Proof.
Choose any two points \(a\) and \(b\) in the interval \(I\text{.}\) By the Mean Value Theorem, we must have
for some \(c\) between \(a\) and \(b\text{.}\) But \(\fp(c)=0\text{,}\) so \(f(b)-f(a)=0\text{,}\) or \(f(a)=f(b)\text{.}\) Since \(a\) and \(b\) were any two points, this tells us that \(f\) must have the same value at every point; that is, \(f\) must be constant.
Suppose \(g'(x)=h'(x)\) for each point \(x\) in \(I\text{,}\) and consider the function \(f(x)=g(x)-h(x)\text{.}\) By the difference rule for derivatives, we have
By the previous result, this means that \(f(x)\) is a constant function. That is, \(f(x)=C\) for each \(x\) in \(I\text{,}\) giving us \(g(x)-h(x)=C\text{,}\) or \(g(x)=h(x)+C\text{.}\)
Figure3.2.13.Showing that a function with zero derivative is constant
Using this result, we can establish another result which will be useful when we study antiderivatives: if two functions have the same derivative, then they differ by a constant.
Figure3.2.14.Showing that two functions with the equal derivatives differ by a constant
We end this section with one more proof involving the Mean Value Theorem; this time, establishing a property of the sine function.
Figure3.2.15.Demonstrating a property of the sine function
ExercisesExercises
Terms and Concepts
1.
Explain in your own words what the Mean Value Theorem states.
2.
Explain in your own words what Rolle’s Theorem states.
Problems
Exercise Group.
A function \(f(x)\) and interval \([a,b]\) are given. Check if Rolle’s Theorem can be applied to \(f\) on \([a,b]\text{;}\) if so, find \(c\) in \((a,b)\) such that \(\fp(c)=0\text{.}\)
3.
\(f(x) = 6\) on \([-1,1]\)
4.
\(f(x) = 6x\) on \([-1,1]\)
5.
\(f(x) = x^2+x-6\) on \([-3,2]\)
6.
\(f(x) = x^2+x-2\) on \([-3,2]\)
7.
\(f(x) = x^2+x\) on \([-2,2]\)
8.
\(f(x) = \sin(x)\) on \([\pi/6,5\pi/6]\)
9.
\(f(x) = \cos(x)\) on \([0,\pi]\)
10.
\(f(x) = \frac{1}{x^2-2x+1}\) on \([0,2]\)
Exercise Group.
A function \(f(x)\) and interval \([a,b]\) are given. Check if The Mean Value Theorem of Differentiation can be applied to \(f\) on \([a,b]\text{;}\) if so, find \(c\) in \((a,b)\) guaranteed by the Mean Value Theorem.
