aarjaneiro/planetmath huggingface dataset:
derivatives of s⁢i⁢nx and c⁢o⁢sx

derivatives of

s

⁢

i

⁢

n

x

and

c

⁢

o

⁢

s

x

Theorem 1

.

d

d

⁢

x

⁢

(

sin

⁡

x

)

=

cos

⁡

x

Proof.

d

d

⁢

x

⁢

(

sin

⁡

x

)

=

lim

h

→

0

⁡

sin

⁡

(

x

+

h

)

-

sin

⁡

x

h

=

lim

h

→

0

⁡

sin

⁡

x

⁢

cos

⁡

h

+

cos

⁡

x

⁢

sin

⁡

h

-

sin

⁡

x

h

by

addition formula for sine

=

lim

h

→

0

⁡

sin

⁡

x

⁢

(

cos

⁡

h

-

1

)

+

cos

⁡

x

⁢

sin

⁡

h

h

=

lim

h

→

0

⁡

(

sin

⁡

x

⋅

cos

⁡

h

-

1

h

+

cos

⁡

x

⋅

sin

⁡

h

h

)

=

sin

⁡

x

⁢

(

lim

h

→

0

⁡

cos

⁡

h

-

1

h

)

+

cos

⁡

x

⁢

(

lim

h

→

0

⁡

sin

⁡

h

h

)

by this entry (

http://planetmath.org/LimitRulesOfFunctions

)

=

sin

⁡

x

⋅

0

+

cos

⁡

x

⋅

1

by this theorem (

http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0

) and its corollary (

http://planetmath.org/LimitOfDisplaystyleFrac1CosXxAsXApproaches0

)

=

cos

⁡

x

∎

Theorem 2

.

d

d

⁢

x

⁢

(

cos

⁡

x

)

=

-

sin

⁡

x

Proof.

d

d

⁢

x

⁢

(

cos

⁡

x

)

=

lim

h

→

0

⁡

cos

⁡

(

x

+

h

)

-

cos

⁡

x

h

=

lim

h

→

0

⁡

cos

⁡

x

⁢

cos

⁡

h

-

sin

⁡

x

⁢

sin

⁡

h

-

cos

⁡

x

h

by

addition formula

for cosine (

http://planetmath.org/AdditionFormulaForCosine

)

=

lim

h

→

0

⁡

cos

⁡

x

⁢

(

cos

⁡

h

-

1

)

+

sin

⁡

x

⁢

sin

⁡

h

h

=

lim

h

→

0

⁡

(

cos

⁡

x

⋅

cos

⁡

h

-

1

h

-

sin

⁡

x

⋅

sin

⁡

h

h

)

=

cos

⁡

x

⁢

(

lim

h

→

0

⁡

cos

⁡

h

-

1

h

)

-

sin

⁡

x

⁢

(

lim

h

→

0

⁡

sin

⁡

h

h

)

by this entry (

http://planetmath.org/LimitRulesOfFunctions

)

=

cos

⁡

x

⋅

0

-

sin

⁡

x

⋅

1

by this theorem (

http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0

) and its corollary (

http://planetmath.org/LimitOfDisplaystyleFrac1CosXxAsXApproaches0

)

=

-

sin

⁡

x

∎

Title

derivatives

of

s

⁢

i

⁢

n

x

and

c

⁢

o

⁢

s

x

Canonical name

DerivativesOfsinXAndcosX

Date of creation

2013-03-22 16:58:51

Last modified on

2013-03-22 16:58:51

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 26A06

Classification

msc 26A09

Classification

msc 26A03

Related topic

Derivative2

Related topic

LimitOfDisplaystyleFracsinXxAsXApproaches0

Related topic

LimitOfDisplaystyleFrac1CosXxAsXApproaches0

Related topic

DerivativesOfSineAndCosine

Generated on Fri Feb 9 19:43:00 2018 by

LaTeXML
difference quotient

difference quotient

The

difference quotient

f

⁢

(

z

+

Δ

⁢

z

)

-

f

⁢

(

z

)

Δ

⁢

z

of a complex (

http://planetmath.org/ComplexFunction

) or

real function

f

is the ratio of the change of the

function

to the corresponding change of the variable.

The difference quotient is needed in defining the derivative of the function. In the case of a real function

f

the difference quotient may be interpreted as the slope of a

secant line

of the graph of the function.

Title

difference quotient

Canonical name

DifferenceQuotient

Date of creation

2013-03-22 17:30:46

Last modified on

2013-03-22 17:30:46

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Definition

Classification

msc 26A06

Classification

msc 30A99

Related topic

Derivative2

Related topic

Slope

Related topic

SecondDerivativeAsSimpleLimit

Generated on Fri Feb 9 19:43:02 2018 by

LaTeXML
direct sum of even/odd functions (example)

direct sum of even/odd functions (example)

Example.

Direct sum

of even and odd functions

Let us define the sets

F

=

{

f

|

f

is a function from

ℝ

to

ℝ

}

,

F

+

=

{

f

∈

F

|

f

(

x

)

=

f

(

-

x

)

for all

x

∈

ℝ

}

,

F

-

=

{

f

∈

F

|

f

(

x

)

=

-

f

(

-

x

)

for all

x

∈

ℝ

}

.

In other words,

F

contain all

functions

from

ℝ

to

ℝ

,

F

+

⊂

F

contain all even functions, and

F

-

⊂

F

contain all odd functions.
All of these spaces have a natural

vector space

structure

:
for functions

f

and

g

we define

f

+

g

as the function

x

↦

f

⁢

(

x

)

+

g

⁢

(

x

)

. Similarly, if

c

is
a real

constant

, then

c

⁢

f

is the
function

x

↦

c

⁢

f

⁢

(

x

)

. With these

operations

, the

zero vector

is the

mapping

x

↦

0

.

We claim that

F

is the direct sum of

F

+

and

F

-

, i.e.,
that

F

=

F

+

⊕

F

-

.

(1)

To prove this claim, let us first note that

F

±

are

vector subspaces

of

F

.
Second, given an arbitrary function

f

in

F

, we can define

f

+

⁢

(

x

)

=

1

2

⁢

(

f

⁢

(

x

)

+

f

⁢

(

-

x

)

)

,

f

-

⁢

(

x

)

=

1

2

⁢

(

f

⁢

(

x

)

-

f

⁢

(

-

x

)

)

.

Now

f

+

and

f

-

are even and odd functions and

f

=

f

+

+

f

-

.
Thus any function in

F

can be split into two

components

f

+

and

f

-

,
such that

f

+

∈

F

+

and

f

-

∈

F

-

.
To show that the sum is direct, suppose

f

is an

element

in

F

+

∩

F

-

.
Then we have that

f

⁢

(

x

)

=

-

f

⁢

(

-

x

)

=

-

f

⁢

(

x

)

, so

f

⁢

(

x

)

=

0

for all

x

, i.e.,

f

is
the zero vector in

F

. We have established

equation

1

.

Title

direct sum of even/odd functions (example)

Canonical name

DirectSumOfEvenoddFunctionsexample

Date of creation

2013-03-22 13:34:24

Last modified on

2013-03-22 13:34:24

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Example

Classification

msc 26A06

Related topic

DirectSumOfHermitianAndSkewHermitianMatrices

Related topic

ProductAndQuotientOfFunctionsSum

Generated on Fri Feb 9 19:43:04 2018 by

LaTeXML
even and odd functions

even and odd functions

Definition

Let

f

be a

function

from

ℝ

to

ℝ

.
If

f

⁢

(

-

x

)

=

f

⁢

(

x

)

for all

x

∈

ℝ

,
then

f

is an

even function

.
Similarly,
if

f

⁢

(

-

x

)

=

-

f

⁢

(

x

)

for all

x

∈

ℝ

,
then

f

is an

odd function

.

Although this entry is mainly concerned with functions

ℝ

→

ℝ

,
the definition can be generalized to other types of function.

Notes

A

real function

is even if and only if it is

symmetric about

the

y

-axis.
It is odd if and only if symmetric about the origin.

Examples

1.

The function

f

⁢

(

x

)

=

x

is odd.

2.

The function

f

⁢

(

x

)

=

|

x

|

is even.

3.

The sine and

cosine functions

are odd and even, respectively.

Properties

1.

The only function that is both even and odd
is the function defined by

f

⁢

(

x

)

=

0

for all real

x

.

2.

A sum of even functions is even, and a sum of odd functions is odd.
In fact, the even functions form a

real vector space

,
as do the odd functions.

3.

Every real function can be expressed in a unique way
as the sum of an odd function and an even function.

4.

From the above it follows that
the

vector space

of real functions is the

direct sum

of
the vector space of even functions and the vector space of odd functions.
See the entry
direct sum of even/odd functions (example) (

http://planetmath.org/DirectSumOfEvenoddFunctionsExample

).)

5.

Let

f

:

ℝ

→

ℝ

be a

differentiable function

.

(a)

If

f

is an even function, then the

derivative

f

′

is an odd function.

(b)

If

f

is an odd function, then the derivative

f

′

is an even function.

(For a proof, see the entry
derivative of even/odd function (proof) (

http://planetmath.org/DerivativeOfEvenoddFunctionProof

).)

6.

Let

f

:

ℝ

→

ℝ

be a

smooth function

.
Then there exist smooth functions

g

,

h

:

ℝ

→

ℝ

such that

f

⁢

(

x

)

=

g

⁢

(

x

2

)

+

x

⁢

h

⁢

(

x

2

)

for all

x

∈

ℝ

.
Thus, if

f

is even, we have

f

⁢

(

x

)

=

g

⁢

(

x

2

)

,
and if

f

is odd, we have

f

⁢

(

x

)

=

x

⁢

h

⁢

(

x

2

)

.
(

[

1

]

, Exercise 1.2)

7.

The

Fourier transform

of a real even function is purely real and even.
The Fourier transform of a real odd function is purely imaginary and odd.

References

1

L. Hörmander,

The

Analysis

of Linear Partial Differential

Operators

I,
(

Distribution

theory and

Fourier Analysis

)

, 2nd ed, Springer-Verlag, 1990.

Title

even and odd functions

Canonical name

EvenAndOddFunctions

Date of creation

2013-03-22 13:34:19

Last modified on

2013-03-22 13:34:19

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Definition

Classification

msc 26A06

Related topic

HermitianFunction

Defines

even function

Defines

odd function

Generated on Fri Feb 9 19:43:07 2018 by

LaTeXML
example of chain rule

example of chain rule

Suppose we wanted to differentiate

h

⁢

(

x

)

=

sin

⁡

(

x

)

.

Here,

h

⁢

(

x

)

is given by the

composition

h

⁢

(

x

)

=

f

⁢

(

g

⁢

(

x

)

)

,

where

f

⁢

(

x

)

=

x

and

g

⁢

(

x

)

=

sin

⁡

(

x

)

.

Then chain rule
says that

h

′

⁢

(

x

)

=

f

′

⁢

(

g

⁢

(

x

)

)

⁢

g

′

⁢

(

x

)

.

Since

f

′

⁢

(

x

)

=

1

2

⁢

x

,

and

g

′

⁢

(

x

)

=

cos

⁡

(

x

)

,

we have by chain rule

h

′

⁢

(

x

)

=

(

1

2

⁢

sin

⁡

x

)

⁢

cos

⁡

x

=

cos

⁡

x

2

⁢

sin

⁡

x

Using the Leibniz

formalism

, the above calculation would have the
following appearance. First we describe the functional relation as

z

=

sin

⁡

(

x

)

.

Next, we introduce an

auxiliary variable

y

, and write

z

=

y

,

y

=

sin

⁡

(

x

)

.

We then have

d

⁢

z

d

⁢

y

=

1

2

⁢

y

,

d

⁢

y

d

⁢

x

=

cos

⁡

(

x

)

,

and hence the chain rule gives

d

⁢

z

d

⁢

x

=

1

2

⁢

y

⁢

cos

⁡

(

x

)

=

1

2

⁢

cos

⁡

(

x

)

sin

⁡

(

x

)

Title

example of chain rule

Canonical

name

ExampleOfChainRule

Date of creation

2013-03-22 12:35:32

Last modified on

2013-03-22 12:35:32

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

4

Author

rmilson (146)

Entry type

Example

Classification

msc 26A06

Generated on Fri Feb 9 19:43:09 2018 by

LaTeXML
example of changing variable

example of changing variable

If one performs in the improper integral

I

:=

∫

-

∞

∞

e

k

⁢

x

1

+

e

x

d

x

(

0

<

k

<

1

)

(1)

the change of variable (

http://planetmath.org/ChangeOfVariableInDefiniteIntegral

)

x

=

-

ln

⁡

t

,

d

⁢

x

=

-

d

⁢

t

t

,

the new

lower limit

becomes

∞

and the new

upper limit

0; hence one obtains

I

=

-

∫

∞

0

e

-

k

⁢

ln

⁡

t

⁢

d

⁢

t

(

1

+

e

-

ln

⁡

t

)

⁢

t

=

∫

0

∞

t

-

k

t

+

1

⁢

𝑑

t

.

Thus one has recurred

I

to the

integral

∫

0

∞

x

-

k

x

+

1

⁢

𝑑

x

,

(2)

the value of which has been determined in the entry using

residue theorem

near

branch point

.  Accordingly, we may write the result

∫

-

∞

∞

e

k

⁢

x

1

+

e

x

⁢

𝑑

x

=

π

sin

⁡

π

⁢

k

.

Calculating the integral (1) directly is quite laborious:  one has to use Cauchy residue theorem to the integral

∮

c

e

k

⁢

z

1

+

e

z

⁢

𝑑

z

about the perimetre

c

of the rectangle

-

a

≦

Re

⁢

z

≦

a

,

0

≦

Im

⁢

z

≦

2

⁢

π

and then to let

a

→

∞

(one cannot use the same half-disk as in determining the integral (2)).  As for using the method (

http://planetmath.org/MethodsOfEvaluatingImproperIntegrals

) of

differentiation

under the

integral sign

or taking

Laplace transform

with respect to

k

yields a more complicated integral.

Title

example of changing

variable

Canonical

name

ExampleOfChangingVariable

Date of creation

2013-03-22 18:45:49

Last modified on

2013-03-22 18:45:49

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Example

Classification

msc 26A06

Related topic

UsingResidueTheoremNearBranchPoint

Related topic

MethodsOfEvaluatingImproperIntegrals

Generated on Fri Feb 9 19:43:11 2018 by

LaTeXML
example of increasing/decreasing/monotone function

example of increasing/decreasing/monotone function

The

function

f

⁢

(

x

)

=

e

x

is

strictly increasing

and hence strictly monotone. Similarly

g

⁢

(

x

)

=

e

-

x

is

strictly decreasing

and hence strictly monotone. Consider the function

h

:

[

1

,

10

]

↦

[

1

,

5

]

where

h

⁢

(

x

)

=

x

-

4

⁢

x

-

1

+

3

+

x

-

6

⁢

x

-

1

+

8

. It is not strictly monotone since it is constant on an

interval

, however it is decreasing and hence

monotone

.

Title

example of increasing/decreasing/

monotone function

Canonical name

ExampleOfIncreasingdecreasingmonotoneFunction

Date of creation

2013-03-22 13:36:08

Last modified on

2013-03-22 13:36:08

Owner

Johan (1032)

Last modified by

Johan (1032)

Numerical id

4

Author

Johan (1032)

Entry type

Example

Classification

msc 26A06

Generated on Fri Feb 9 19:43:18 2018 by

LaTeXML
example of integral mean value theorem

example of integral mean value theorem

Example

.

If

f

is a

continuous

real function

on an

interval

[

a

,

b

]

, then there exists a

ζ

∈

(

a

,

b

)

such that

∫

a

b

f

⁢

(

x

)

⁢

d

x

=

f

⁢

(

ζ

)

⁢

(

b

-

a

)

.

Proof.

Let

g

⁢

(

x

)

≡

1

. Then by the

Integral Mean Value Theorem

, there exists

ζ

∈

(

a

,

b

)

such that

∫

a

b

f

⁢

(

x

)

⁢

d

x

=

∫

a

b

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

d

x

=

f

⁢

(

ζ

)

⁢

∫

a

b

g

⁢

(

x

)

⁢

d

x

=

f

⁢

(

ζ

)

⁢

∫

a

b

1

⁢

d

x

=

f

⁢

(

ζ

)

⁢

(

b

-

a

)

as required.
∎

Title

example of integral mean value theorem

Canonical name

ExampleOfIntegralMeanValueTheorem

Date of creation

2013-03-22 18:20:24

Last modified on

2013-03-22 18:20:24

Owner

me_and (17092)

Last modified by

me_and (17092)

Numerical id

4

Author

me_and (17092)

Entry type

Example

Classification

msc 26A06

Generated on Fri Feb 9 19:43:20 2018 by

LaTeXML
example of ratio test

example of ratio test

Consider the

sequence

given by

a

n

=

x

n

(

geometric progression

) where

|

x

|

<

1

.
Then the series

∑

j

=

0

∞

a

n

converges

.  To see this, we can use the

ratio test

. We need to consider the sequence

|

a

n

+

1

/

a

n

|

. But for any

n

≥

0

we have (when

x

≠

0

)

|

a

n

+

1

a

n

|

=

|

x

n

+

1

x

n

|

=

|

x

|

<

1

,

and therefore the series converges. The ratio test and the previous argument shows that the

geometric series

diverges for

|

x

|

>

1

.

Title

example of ratio test

Canonical name

ExampleOfRatioTest

Date of creation

2013-03-22 15:03:20

Last modified on

2013-03-22 15:03:20

Owner

drini (3)

Last modified by

drini (3)

Numerical id

9

Author

drini (3)

Entry type

Example

Classification

msc 26A06

Classification

msc 40A05

Generated on Fri Feb 9 19:43:22 2018 by

LaTeXML
examples of logarithms simplifying calculations

examples of logarithms simplifying calculations

Before the days of

calculators

and

computers

, tables of

logarithms

were used to simplify what would otherwise be difficult calculations with pencil and paper, thanks to algebraic identities such as

log

⁡

a

⁢

b

=

log

⁡

a

+

log

⁡

b

and

log

⁡

(

a

b

)

=

log

⁡

a

-

log

⁡

b

. Even

log

⁡

a

b

=

log

⁡

a

b

could be more efficient than the “long” method for extracting the

square root

.

Because we usually use base 10 to represent numbers, base 10 logarithms were more helpful than

natural logarithms

in simplifying calculations. These examples will therefore all use base 10 as the base of logarithms.

First, an example of

multiplication

as might have arisen for an inland revenue officer: multiply 17.81 by 6.520. The base 10 logarithms of these numbers are 1.25066 and 0.81425, respectively. Adding the two logarithms gives 2.06491, which is the logarithm of the desired product. To get the product, one looks in the table for the logarithm closest to 0.06491 and finds that the logarithm of 1.161 is 0.06521 so the answer is 116.1 to four digits. Some

arithmetic

books from the past sometimes also included tables of

antilogarithms

, but reading the table of logarithms “backwards” accomplishes the same goal.

Now let’s consider an example of

division

that might have arisen in an astronomy class. A student wonders how many Earths would fit between the Sun and Jupiter at Jupiter’s perihelion. Jupiter’s perihelion is 740573600 kilometers. The equatorial radius of the Earth is 6378 kilometers. The first of these numbers is out of the range of even Babbage’s extensive tables of logarithms, but if we divide both numbers by a

thousand

and discard the

fractional parts

, the result should still be good enough for this application, with is thoroughly hypothetical in nature. The logarithm of 740537 is 5.86957. The logarithm of 6 is 0.778151. Subtracting gives 5.09142. Looking this up in the table of antilogarithms gives 123429, which is off from the actual answer of 116114 by more than seven thousand. Seven thousand Earths is not so much of an error when we consider that at least a hundred thousand Earths would be necessary to fill the given distance.

References

1

William Harris Johnston,

Loftus’s inland revenue officer’s manual

. Oxford: Oxford University Press (1865)

2

Raymond Puzio, PlanetMath message, June 13, 2008

3

Gerald R. Rising,

Inside Your Calculator: From Simple Programs to Significant Insights

. Hoboken, New Jersey: John Wiley & Sons (2007): Appendix P

Title

examples of logarithms simplifying calculations

Canonical

name

ExamplesOfLogarithmsSimplifyingCalculations

Date of creation

2013-03-22 18:08:47

Last modified on

2013-03-22 18:08:47

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 26A06

Classification

msc 26A09

Classification

msc 26-00

Generated on Fri Feb 9 19:43:23 2018 by

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extended mean-value theorem

extended mean-value theorem

Let

f

:

[

a

,

b

]

→

ℝ

and

g

:

[

a

,

b

]

→

ℝ

be

continuous

on

[

a

,

b

]

and

differentiable

on

(

a

,

b

)

. Then there exists some number

ξ

∈

(

a

,

b

)

satisfying:

(

f

⁢

(

b

)

-

f

⁢

(

a

)

)

⁢

g

′

⁢

(

ξ

)

=

(

g

⁢

(

b

)

-

g

⁢

(

a

)

)

⁢

f

′

⁢

(

ξ

)

.

If

g

is linear this becomes the usual

mean-value theorem

.

Title

extended mean-value theorem

Canonical name

ExtendedMeanvalueTheorem

Date of creation

2013-03-22 13:04:11

Last modified on

2013-03-22 13:04:11

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

9

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 26A06

Synonym

Cauchy’s mean value theorem

Synonym

extended mean value theorem

Synonym

generalized mean value theorem

Related topic

MeanValueTheorem

Generated on Fri Feb 9 19:43:25 2018 by

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extreme value theorem

extreme value theorem

Extreme Value Theorem

.

Let

a

and

b

be

real numbers

with

a

<

b

, and let

f

be a

continuous

, real valued

function

on

[

a

,

b

]

. Then there exists

c

,

d

∈

[

a

,

b

]

such that

f

⁢

(

c

)

≤

f

⁢

(

x

)

≤

f

⁢

(

d

)

for all

x

∈

[

a

,

b

]

.

Proof.

We show only the

existence

of

d

. By the

boundedness theorem

f

⁢

(

[

a

,

b

]

)

is bounded above; let

l

be the least

upper bound

of

f

⁢

(

[

a

,

b

]

)

. Suppose, for a

contradiction

, that there is no

d

∈

[

a

,

b

]

such that

f

⁢

(

d

)

=

l

. Then the function

g

⁢

(

x

)

=

1

l

-

f

⁢

(

x

)

is well defined and continuous on

[

a

,

b

]

. Since

l

is the least upper bound of

f

⁢

(

[

a

,

b

]

)

, for any positive real number

M

we can find

α

∈

[

a

,

b

]

such that

f

⁢

(

α

)

>

l

-

1

M

, then

M

<

1

l

-

f

⁢

(

α

)

⁢

.

So

g

is

unbounded

on

[

a

,

b

]

. But by the boundedness theorem

g

is

bounded

on

[

a

,

b

]

. This contradiction finishes the proof.

Title

extreme value theorem

Canonical name

ExtremeValueTheorem

Date of creation

2013-03-22 14:29:21

Last modified on

2013-03-22 14:29:21

Owner

classicleft (5752)

Last modified by

classicleft (5752)

Numerical id

7

Author

classicleft (5752)

Entry type

Theorem

Classification

msc 26A06

Synonym

Weierstrass extreme value theorem

Generated on Fri Feb 9 19:43:27 2018 by

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Fermat’s theorem (stationary points)

Fermat’s theorem (stationary points)

Let

f

:

(

a

,

b

)

→

ℝ

be a

continuous function

and suppose that

x

0

∈

(

a

,

b

)

is a

local extremum

of

f

. If

f

is

differentiable

in

x

0

then

f

′

⁢

(

x

0

)

=

0

.

Moreover if

f

has a

local maximum

at

a

and

f

is differentiable at

a

(the

right derivative

exists)
then

f

′

⁢

(

a

)

≤

0

; if

f

has a

local minimum

at

a

then

f

′

⁢

(

a

)

≥

0

.
If

f

is differentiable in

b

and
has a local maximum at

b

then

f

′

⁢

(

b

)

≥

0

while if it has a local minimum at

b

then

f

′

⁢

(

b

)

≤

0

.

Title

Fermat’s

theorem

(

stationary points

)

Canonical name

FermatsTheoremstationaryPoints

Date of creation

2013-03-22 13:45:05

Last modified on

2013-03-22 13:45:05

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

7

Author

paolini (1187)

Entry type

Theorem

Classification

msc 26A06

Related topic

ProofOfLeastAndReatestValueOfFunction

Related topic

LeastAndGreatestValueOfFunction

Generated on Fri Feb 9 19:43:29 2018 by

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fundamental theorem of integral calculus

fundamental theorem of integral calculus

The

derivative

of a

real function

, which has on a whole

interval

a

constant

(

http://planetmath.org/ConstantFunction

) value

c

,

vanishes

in every point of this interval:

d

d

⁢

x

⁢

c

=

0

The

converse theorem

of this is also true.  Ernst Lindelöf calls it the

fundamental theorem of integral calculus

(in Finnish

integraalilaskun peruslause

).  It can be formulated as

Theorem

.

If a real function in

continuous

and its derivative vanishes in all points of an interval, the value of this

function

does not change on this interval.

Proof.

We make the antithesis that there were on the interval two distinct points

x

1

and

x

2

with

f

⁢

(

x

1

)

≠

f

⁢

(

x

2

)

.  Then the

mean-value theorem

guarantees a point

ξ

between

x

1

and

x

2

such that

f

′

⁢

(

ξ

)

=

f

⁢

(

x

1

)

-

f

⁢

(

x

2

)

x

1

-

x

2

,

which value is distinct from zero.  This is, however, impossible by the

assumption

of the theorem.  So the antithesis is wrong and the theorem .

The contents of the theorem may be expressed also such that if two functions have the same derivative on a whole interval, then the

difference

of the functions is constant on this interval.  Accordingly, if

F

is an

antiderivative

of a function

f

, then any other antiderivative of

f

has the form

x

↦

F

⁢

(

x

)

+

C

, where

C

is a constant.

Title

fundamental theorem of integral calculus

Canonical name

FundamentalTheoremOfIntegralCalculus

Date of creation

2013-03-22 18:50:49

Last modified on

2013-03-22 18:50:49

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A06

Related topic

FundamentalTheoremOfCalculusClassicalVersion

Related topic

VanishingOfGradientInDomain

Generated on Fri Feb 9 19:43:35 2018 by

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