Advanced Calculus (I)

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:

dx =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:

dx =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:

dx =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

Thank you.