Advanced Calculus (I)
WEN-CHINGLIEN
Department of Mathematics National Cheng Kung University
4.1 The Derivative
Definition
A real function f is said to be differentiable at a point a ∈R if and only if f is defined on some open interval I
containing a and
(1) f0(a) = lim
h→0
f (a + h) − f (a) h
exists. In this case f0(a) is called the derivative of f at a.
Notation:
Dxf = dfdx =f(1)=f0
4.1 The Derivative
Definition
A real function f is said to be differentiable at a point a ∈R if and only if f is defined on some open interval I
containing a and
(1) f0(a) = lim
h→0
f (a + h) − f (a) h
exists. In this case f0(a) is called the derivative of f at a.
Notation:
Dxf = dfdx =f(1)=f0
4.1 The Derivative
Definition
A real function f is said to be differentiable at a point a ∈R if and only if f is defined on some open interval I
containing a and
(1) f0(a) = lim
h→0
f (a + h) − f (a) h
exists. In this case f0(a) is called the derivative of f at a.
Notation:
Dxf = dfdx =f(1)=f0
Theorem
A real function f is differentiable at some point a ∈R if and only if there exists an open interval I and a function
F : I →R such that a ∈ I, f is defined on I, F is continuous at a, and
(3) f (x ) = F (x )(x − a) + f (a) holds for all x ∈ I, in which case F (a) = f0(a).
Theorem
A real function f is differentiable at some point a ∈R if and only if there exists an open interval I and a function
F : I →R such that a ∈ I, f is defined on I, F is continuous at a, and
(3) f (x ) = F (x )(x − a) + f (a) holds for all x ∈ I, in which case F (a) = f0(a).
Theorem
Let f :R → R. Then f is differentiable at a if and only if there is a function T of the form T (x ) := mx such that
(4) lim
h→0
|f (a + h) − f (a) − T (h)|
|h| =0.
Theorem
Let f :R → R. Then f is differentiable at a if and only if there is a function T of the form T (x ) := mx such that
(4) lim
h→0
|f (a + h) − f (a) − T (h)|
|h| =0.
Theorem
If f is differentiable at a, then f is continuous at a.
Theorem
If f is differentiable at a, then f is continuous at a.
Definition
Let I be a nondegenerate interval.
(i)
A function f : I →R is said to be differentiable on I if and only if
f0I(a) = lim
x →a x ∈I
f (x ) − f (a) x − a exists and is finite for every a ∈ I.
(ii)
f is said to be continuously differentiable on I if and only if f0I exists and is continuous on I.
Definition
Let I be a nondegenerate interval.
(i)
A function f : I →R is said to be differentiable on I if and only if
f0I(a) = lim
x →a x ∈I
f (x ) − f (a) x − a exists and is finite for every a ∈ I.
(ii)
f is said to be continuously differentiable on I if and only if f0I exists and is continuous on I.
Definition
Let I be a nondegenerate interval.
(i)
A function f : I →R is said to be differentiable on I if and only if
f0I(a) = lim
x →a x ∈I
f (x ) − f (a) x − a exists and is finite for every a ∈ I.
(ii)
f is said to be continuously differentiable on I if and only if f0I exists and is continuous on I.
Definition
Let I be a nondegenerate interval.
(i)
A function f : I →R is said to be differentiable on I if and only if
f0I(a) = lim
x →a x ∈I
f (x ) − f (a) x − a exists and is finite for every a ∈ I.
(ii)
f is said to be continuously differentiable on I if and only if f0I exists and is continuous on I.
Definition
Let I be a nondegenerate interval.
(i)
A function f : I →R is said to be differentiable on I if and only if
f0I(a) = lim
x →a x ∈I
f (x ) − f (a) x − a exists and is finite for every a ∈ I.
(ii)
f is said to be continuously differentiable on I if and only if f0I exists and is continuous on I.
Definition
Let I be a nondegenerate interval.
(i)
A function f : I →R is said to be differentiable on I if and only if
f0I(a) = lim
x →a x ∈I
f (x ) − f (a) x − a exists and is finite for every a ∈ I.
(ii)
f is said to be continuously differentiable on I if and only if f0I exists and is continuous on I.
Example:
The function
f (x ) = (
x2sin(1
x) x 6= 0
0 x = 0
is differentiable onR but not continuously differentiable on any interival that contains the origin.
Example:
The function
f (x ) = (
x2sin(1
x) x 6= 0
0 x = 0
is differentiable onR but not continuously differentiable on any interival that contains the origin.
Remark:
f (x ) = |x | is differentiable on [0, 1] and on [−1, 0] but not on [−1, 1].
Remark:
f (x ) = |x | is differentiable on [0, 1] and on [−1, 0] but not on [−1, 1].
Proof:
Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with
f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0. However,
f0[0,1](0) = lim
h→0+
|h|
h =1 and f0[−1,0](0) = lim
h→0−
|h|
h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2
Proof:
Since f (x ) = x when x > 0 and = −x when x < 0,it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with
f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5,f is not differentiable at x = 0. However,
f0[0,1](0) = lim
h→0+
|h|
h =1 and f0[−1,0](0) = lim
h→0−
|h|
h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2
Proof:
Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with
f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0.However,
f0[0,1](0) = lim
h→0+
|h|
h =1 and f0[−1,0](0) = lim
h→0−
|h|
h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2
Proof:
Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with
f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5,f is not differentiable at x = 0. However,
f0[0,1](0) = lim
h→0+
|h|
h =1 and f0[−1,0](0) = lim
h→0−
|h|
h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2
Proof:
Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with
f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0.However,
f0[0,1](0) = lim
h→0+
|h|
h =1 and f0[−1,0](0) = lim
h→0−
|h|
h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2
Proof:
Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with
f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0. However,
f0[0,1](0) = lim
h→0+
|h|
h =1 and f0[−1,0](0) = lim
h→0−
|h|
h =1 Therefore,f is differentiable on [0, 1] and on [−1, 0]. 2
Proof:
Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with
f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0. However,
f0[0,1](0) = lim
h→0+
|h|
h =1 and f0[−1,0](0) = lim
h→0−
|h|
h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2
Proof:
Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with
f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0. However,
f0[0,1](0) = lim
h→0+
|h|
h =1 and f0[−1,0](0) = lim
h→0−
|h|
h =1 Therefore,f is differentiable on [0, 1] and on [−1, 0]. 2
Proof:
Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with
f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0. However,
f0[0,1](0) = lim
h→0+
|h|
h =1 and f0[−1,0](0) = lim
h→0−
|h|
h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2
