MATH1131 Calculus
5.1: Title

MATH1131 Calculus
5.2: Introduction
We have seen in Chapter 3 that continuous functions have
various important properties (the Intermediate Value Theorem,
existence of maximum and minimum points). Differentiable
functions have these properties (because they are continuous),
and further useful properties besides.
As in Chapter 3, I suggest studying the results of this chapter
in the following way:
•
remember and understand the diagram;
•
understand how the diagram is “translated” into words
and mathematical symbolism;
•
try to understand why each assumption in the theorem
is necessary, by imagining what could happen to the
diagram if the assumption were not true;
•
make sure that you are able to apply the theorem to
specific problems.

MATH1131 Calculus
5.3: MVT
Theorem
.
The Mean Value Theorem
. Let
f
be continuous
on the interval [
a, b
] and differentiable on (
a, b
). Then there
exists a value of
c
in (
a, b
) such that
f
(
b
)
−
f
(
a
)
b
−
a
=
f
′
(
c
)
.
Example
.
Note that the quotient on the left hand side
of the equation is the
average
rate of change of
f
over the
whole interval [
a, b
], while
f
′
(
c
) is the
instantaneous
rate of
change at
c
. Thus, the Mean Value Theorem says that for
any (suitable) function, there must be a point at which the
instantaneous rate of change is equal to the average (mean)
rate of change over a whole interval.
0
x
y
a
b
c
1
c
2
f
(
a
)
f
(
b
)
Geometrically, there is a point (in this example, two points)
where the tangent to the curve is parallel to the secant over
the whole interval.
Physical interpretation
. A train travels exactly 100 km in
exactly 1 hour. Assuming that it travels continuously and
smoothly (that is, there are no jerks and no sudden starts or
stops), there must be some point at which the speed is
exactly
100 km/h.

MATH1131 Calculus
5.4: Problems
The result of the Mean Value Theorem need not hold if
f
is not continuous on the closed interval [
a, b
], or if
f
is not
differentiable at every point in (
a, b
).
0
x
y
a
b
f
(
a
)
f
(
b
)
0
x
y
a
b
x
0
f
(
a
)
f
(
b
)
Note, however, that the conditions of the Mean Value
Theorem do not require the function to be differentiable at the
endpoints
a
and
b
. You can see from the next example that if,
for example,
f
is not differentiable at
b
it makes no difference
to the conclusion of the Theorem.
0
x
y
a
b