The Mean Value Theorem
Rolle’s Theorem: If f is continuous on [a, b] and differentiable on (a, b),
and if f(a) = f(b), then there is at least one number c in (a, b) such that
f ʹ′(c) = 0 .
Justification: The graph has no breaks
(since f is continuous) and is smooth
(since f is differentiable), so it has to
level off somewhere (at least once)
between a and b.
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Why does f need to be continuous on the closed
interval [a, b]? Why isn’t it good enough to be
continuous on (a, b)?
To answer this, draw the graph of a function that
has f(a) = f(b) but has a jump discontinuity at
x = a and a constant negative slope on the interval
(a, b).
Ex 1. Show that Rolle’s Theorem applies to f (x) = x 3 – 3x 2 + 2x – 6 on [0, 1] and
find a value of c guaranteed by the theorem.
€ differentiable on (a, b), but f(a) ≠ f(b).
Now, let f be continuous on [a, b] and
What can Rolle’s Theorem tell us?
We could calculate the secant line for this function from a to b:
and
msec =
y – f (a) = msec (x – a)
Solving for y gives: y = msec (x – a) + f (a)
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Now, define a new function g:
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g(x) = f (x) – [ msec (x – a) + f (a)]
Note that g€is ______________ on [a, b] and g is ________________ on (a, b).
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Also, g(a) = ____________________________________________________
and g(b) = _____________________________________________________
So, g(a) = g(b).
Then, by Rolle’s Theorem, there is at least one c in (a, b) such that gʹ′(c) = 0 .
But, what does gʹ′(x) equal?
Then, gʹ′(c) = 0 ⇒
gʹ′(x) = ______________________
f ʹ′(c) = msec
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This, then, gives us...
The
€ Mean Value Theorem (MVT): If f is continuous on [a, b] and
differentiable on (a, b), then there is at least one number c in (a, b) such
f (b) – f (a)
that f ʹ′(c) =
.
b–a
(Note that Rolle’s Theorem is just a specific case of the MVT.)
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Ex 2. In the time interval from t = 10 sec to t = 30 sec, Rob traveled (continuously)
a distance of 800 feet. What does the MVT tell us about this situation?
Ex 3. Find the conclusion of the MVT for g(x) = 2x – 3 on [2, 6] and find a
value of c guaranteed by the theorem.
€ “existence” theorems – they say that something
Rolle’s Theorem and the MVT are
exists, but they don’t say how to find it.
The primary use of the MVT (and Rolle’s Theorem) is in proving other facts and
theorems.
Ex 4. Prove that, if a > 0 and n is any positive integer, then the polynomial
p(x) = x 2n+1 + ax + b cannot have two real zeros.
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