Graphics Reference
In-Depth Information
9
Di
erentiation and completeness
Mean Value Theorems, Taylor's Theorem
ff
Concurrent reading:
Burkill ch. 4 and §5.8, Courant and John ch. 5.
Further reading:
Spivak chs 11, 19.
Both Rolle's Theorem and the Mean Value Theorem are geometrically
transparent. Each claims, with slightly more generality in the case of the
Mean Value Theorem, that for the graph of a differentiable function,
there is always a tangent parallel to a chord. It is something of a
surprise to find that such intuitive results can lead to such powerful
conclusions: namely, de l'Hoˆ pital's rule and the existence of power
series convergent to a wide family of functions.
Rolle's Theorem
1 Sketch the graphs of some differentiable functions
f
:[
a
,
b
]
R,
for which
f
(
a
)
f
(
b
).
Can you find a point
c
, with
a
c
b
, such that
f
(
c
)
0, for each
function
f
which you have sketched?
What words could you useto dscribethepoint
c
in relation to the
function
f
?
2 Must
any
differentiable function
f
:[
a
,
b
]
R
havea maximum and a minimum valu? Why?
Must
any
such function, for which
f
(
a
)
f
(
b
), havea maximum
and
a minimum valuein the
open
interval (
a
,
b
)?
