Mean Value Theorem for Derivatives
If f (x) is a differentiable function over [a,b], then
at some point between a and b:
f b  f  a 
ba
 f c
Mean Value Theorem for Derivatives
If f (x) is a differentiable function over [a,b], then
at some point between a and b:
f b  f  a 
ba
 f c
Differentiable implies that the function is also continuous.
Mean Value Theorem for Derivatives
If f (x) is a differentiable function over [a,b], then
at some point between a and b:
f b  f  a 
ba
 f c
Differentiable implies that the function is also continuous.
The Mean Value Theorem only applies over a closed interval.

Mean Value Theorem for Derivatives
If f (x) is a differentiable function over [a,b], then
at some point between a and b:
f b  f  a 
ba
 f c
The Mean Value Theorem says that at some point in
the closed interval, the slope of the tangent line
equals the average slope ( i.e. the slope of the line
joining the endpoints).

Tangent is parallel to
the chord.
y
Slope of tangent:
f  c
B
Slope of chord:
f b   f  a 
ba
A
0
y  f  x
a
c
x
b

The Mean Value Theorem
For f  x  differentiable on  a,b  and continuous
on  a,b  , there exists at least one value c in  a,b 
There is some value, c,
between a and b where
the slope of the tangent
equals the slope of the
secant
such that
f c 
f b   f  a 
ba
.
a
c
b
Rolle’s Theorem
a
c
b
(A special case of the mean
value theorem)
For f  x  differentiable on  a,b  and continuous
There is some value, c,
between a and b where the
on  a,b  , if f  b   f  a   0 then there exists at
slope of the tangent is 0
least one c in  a,b  such that f   c   0.