AP Calc Notes: DA2 – 2 Mean Value Theorem
Mean Value Theorem (MVT): If f is continuous on [a, b], differentiable on (a, b), then there exists a number
c in (a, b) such that
f (b ) − f ( a )
f '(c) =
.
b−a
1. The instantaneous rate of change of f at x = c
is the same as the average rate of change of f on
[a, b].
y = f(x)
y
2. The slope of the tangent line at x = c is the
same as the slope of the secant line.
a
c
b
x
3. This existence is only guaranteed for
continuous, differentiable functions.
The Mean Value Theorem Song
Tune: “From the Halls of Montezuma”
If a function is continuous
On a closed set “a” to “b”
And it’s also differentiable
On the open set “a” “b”
You can always find a “c” inside
Such that f prime at point c
Is equivalent to just the slope
Of the line from a to b
Similar to the IVT (existence of y-values),
the MVT is an existence theorem for values
of the derivative.
To invoke this theorem, Harry Potter (and you) need to use the right words. There are many correct ways to
say it but unfortunately, an even larger number of incorrect ways. I suggest you model your answers
after the following example.
Q: A differentiable function for all real numbers, f(x), has values f(3) = 7 and f(5) = 10. Explain why there
3
must be a value d for 3 ≤ d ≤ 5 such that f ' ( d ) = .
2
A: Since f(x) is a differentiable function on (3,5) with
for some d in (3,5).
f ( 5 ) − f ( 3) 10 − 7 3
3
=
= , by the MVT, f ' ( d ) =
5−3
2
2
2
Ex: The function f ( x ) is continuous on [0, 20] and differentiable on (0, 20). Select values of f ( x ) are given
in the table below.
x
0
5
10
15
20
f ( x)
-1
6
16
-4
-9
a. Explain why there is a value, b for 0 < b < 20 such that f ' ( b ) = −1 .
Since _______ is a differentiable function on _______ with __________=___________ ,
by the MVT, __________for some ___in ___________.
b. Explain why there is a value, b for 0 < b < 20 such that f ' ( b ) = 2 .
Ex: Let f be a function that is differentiable for all real numbers. The table below gives the values of f and its
derivative f ' for selected points x in the closed interval -1.5 ≤ x ≤ 1.5. The second derivative of f has the
property that f " ( x ) > 0 for -1.5 ≤ x ≤ 1.5.
x
f ( x)
-1.5
-1
-1.0
-4
-0.5
-6
0
-7
0.5
-6
1.0
-4
1.5
-1
f '( x)
-7
-5
-3
0
3
5
7
Find a positive real number r having the property that there must exist a value c with 0 < c < 0.5 and f " ( c ) = r .
Give a reason for your answer.
Ex: On August 11, Farmer Flembacher’s pond had 95,000 gallons of water in it. A week later, it contained
77,500 gallons of water.
a. What was the average rate of change of V over that week?
b. Was there a time when water was being lost at 2,500 gallons a day?
Ex: Suppose that f(3) = 5 and f '(x) ≤ 4 for all values of x. Use the MVT to determine
a. an upper bound for f(8).
b. a lower bound for f(0).
Rolle’s Theorem: If f is continuous on [a, b], differentiable on (a, b) and f ( a ) = f ( b ) , then there is a point c
in (a, b) such that f '(c) = 0.
If a and b are both roots, then somewhere in
(a, b) is a point where f has a horizontal
tangent line (not necessarily a max or min.)
y = f(x)
y
a
c
b
x