ANALYSIS I
Differentiation, Rolle's Thm.,
Mean Value Thm.
Handout #8 - 3/25/96
Defn.
A function f is said to be differentiable at
x0 if
|
lim
h® 0 |
|
f(x0 + h) - f(x0)
h |
|
|
exists. In this case the limit is called the derivative
of f at x0 and is denoted f¢(x0).
Note.
-
This definition is equivalent to the requirement that the
following limit exist:
|
lim
x® x0 |
|
f(x) - f(x0)
x-x0 |
= f¢(x0). |
|
-
This, in turn, is equivalent to the following statement about
how fast f(x) converges to f(x0) as x®
x0:
there exists a function h
such that limx® x0 h(x)
= 0 and
| (*)
f(x) - f(x0) = (x - x0) ( f¢(x0)
+ h(x) ). |
|
Examples:
-
If f(x) : = x2 , then f¢(x)
= 2x.
-
If g(x): = |x|,
then g¢(0) does not exist.
-
If h(x): = x|x|,
then h¢(x) exists and equals 2|x|.
Theorem.
If f is differentiable at x0, then f is continuous at x0.
Pf. Use (*) and let x®
x0. [¯]
Theorem.
(Basic rules of differentiation: sums, products,
quotients) Suppose that f and g are differentiable at x0, then
-
(f+g)¢(x0) = f¢(x0)
+ g¢(x0).
-
(fg)¢(x0) = f¢(x0)
g(x0) + f(x0) g¢(x0).
-
(f/g)¢(x0) = (g(x0)f¢(x0)
- f(x0) g¢(x0) )/
g(x0)2, if g(x0) ¹
0.
Theorem.
(Chain rule) If f is differentiable at x0
and g is differentiable at y0 : = f(x0), then h :
= g°f is differentiable at x0
and
| h¢(x0)
= g¢(f(x0)) f¢(x0) |
|
Pf. Use (*) for f at x0 and
for g at y0: = f(x0):
|
|
|
|
|
g(y) - g(y0)
x-x0 |
= |
y-y0
x-x0 |
(g¢(y0)
+ h2(y)) |
|
|
|
|
|
f(x)-f(x0)
x-x0 |
(g¢(y0)
+ h2(y)) |
|
|
|
|
| (f¢(x0)
+ h1(x))(g¢(y0)
+ h2(y)) |
|
|
|
|
where y: = f(x). The proof is completed by using
this equation, letting xn® x0,
and noticing that yn® y0
where yn: = f(xn). [¯]
Theorem.
(Rolle's Theorem) Suppose that f
is differentiable on (a,b), is continuous on [a,b], and vanishes at the
endpoints, then there exists x0 strictly between a and b such
that f¢(x0) = 0.
Pf. If f is
constant, then any point can be selected for x0. Otherwise,
we may assume WLOG that f has positive values.
By the Extreme Value Theorem, let x0 be such that f(x)
£f(x0)
for all a £ x £
b. First, let xn¯ x0,
then since x0 gives a max, we have
| 0 ³ |
f(xn)
- f(x0)
xn-x0 |
®
f¢(x0) |
|
and so, by the Squeeze Theorem, f¢(x0)
£
0. Similarly, f¢(x0)
³
0. [¯]
Note.
Within the proof we actually established the critical
point procedure of calculus: local max and min can only
occur at critical points.
Corollary.
(Mean Value Theorem) Suppose that f is differentiable
on (a,b) and is continuous on [a,b], then there exists x0 strictly
between a and b such that
| f¢(x0)
= |
f(b) - f(a)
b-a |
. |
|
Pf. Let
| f(x) :
= f(x) - |
é
ê
ë |
|
f(b)-f(a)
b-a |
(x-a) + f(a) |
ù
ú
û |
|
|
and apply Rolle's theorem. [¯]
Defn.
F is called an anti-derivative
of f if F is differentiable and F¢(x) =
f(x)
Corollary.
If both F and G are anti-derivatives of f, then they differ by a constant,
i.e. there exists a constant c such that F(x) - G(x) = c, for all x Î
dom(f).
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version 1.2.