aarjaneiro/planetmath huggingface dataset:
binomial formula for negative integer powers

binomial formula for negative integer powers

For negative integer powers, the

binomial formula

can be written in terms of

binomial coefficients

like so:

(

1

-

x

)

-

n

=

∑

m

=

1

∞

(

m

+

n

-

1

n

-

1

)

⁢

x

m

Proof:

We shall prove this by

induction

on

n

. First, note that, if

n

=

1

, then

(

m

0

)

=

1

, so our

formula

reduces to

(

1

-

x

)

-

1

=

∑

m

=

1

∞

x

m

,

which is the formula for the sum of an

infinite geometric series

.

Next, suppose that the formula is valid for a certain value of

n

. Then we have

(

1

-

x

)

-

n

-

1

=

(

1

-

x

)

-

1

⁢

(

1

-

x

)

-

n

=

(

∑

k

=

0

∞

x

k

)

⁢

(

∑

m

=

0

∞

(

m

+

n

-

1

n

-

1

)

⁢

x

m

)

The

product

of sums can be rewritten as the following double sum:

∑

m

=

0

∞

∑

k

=

0

m

(

n

+

k

-

1

n

-

1

)

⁢

x

m

The easiest way to see this is by rearranging the double sum as follows and adding columns

x

0

⁢

∑

m

=

0

∞

(

m

+

n

-

1

n

-

1

)

⁢

x

m

=

(

n

-

1

n

-

1

)

+

(

n

n

-

1

)

⁢

x

+

(

n

+

1

n

-

1

)

⁢

x

2

+

(

n

+

2

n

-

1

)

⁢

x

3

+

(

n

+

3

n

-

1

)

⁢

x

4

+

⋯

x

1

⁢

∑

m

=

0

∞

(

m

+

n

-

1

n

-

1

)

⁢

x

m

=

(

n

-

1

n

-

1

)

⁢

x

+

(

n

n

-

1

)

⁢

x

2

+

(

n

+

1

n

-

1

)

⁢

x

3

+

(

n

+

2

n

-

1

)

⁢

x

4

+

⋯

x

2

⁢

∑

m

=

0

∞

(

m

+

n

-

1

n

-

1

)

⁢

x

m

=

(

n

-

1

n

-

1

)

⁢

x

2

+

(

n

n

-

1

)

⁢

x

3

+

(

n

+

1

n

-

1

)

⁢

x

4

+

⋯

x

3

⁢

∑

m

=

0

∞

(

m

+

n

-

1

n

-

1

)

⁢

x

m

=

(

n

-

1

n

-

1

)

⁢

x

3

+

(

n

n

-

1

)

⁢

x

4

+

⋯

.

.

.

.

.

.

.

.

.

.

.

.

To evaluate the finite sums, we shall use the following

identity

for binomial coefficients. (See the entry

http://planetmath.org/node/273

“binomial coefficient” for more information about this identity.)

∑

k

=

0

m

(

n

+

k

-

1

n

-

1

)

=

(

m

+

n

n

)

Inserting this result value for the finite sum back into the double sum, we obtain

(

1

-

x

)

-

n

-

1

=

∑

m

=

0

∞

(

m

+

n

n

)

⁢

x

m

.

Q.E.D.

Title

binomial formula for negative integer powers

Canonical name

BinomialFormulaForNegativeIntegerPowers

Date of creation

2013-03-22 14:57:26

Last modified on

2013-03-22 14:57:26

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Corollary

Classification

msc 26A06

Related topic

GeneralizedBinomialCoefficients

Generated on Fri Feb 9 19:42:08 2018 by

LaTeXML
Bolzano-Weierstrass theorem

Bolzano-Weierstrass theorem

Given any

bounded

sequence

(

a

n

)

of

real numbers

,
there exists a

convergent

subsequence

(

a

n

j

)

.

More generally, any sequence

(

a

n

)

in a

compact subset

of a

metric space

has a convergent subsequence.

Title

Bolzano-Weierstrass theorem

Canonical name

BolzanoWeierstrassTheorem

Date of creation

2013-03-22 12:22:21

Last modified on

2013-03-22 12:22:21

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Theorem

Classification

msc 26A06

Classification

msc 40A05

Synonym

Bolzano-Weierstraß theorem

Related topic

ConvergentSequence

Related topic

SequentiallyCompact

Related topic

AlternateStatementOfBolzanoWeierstrassTheorem

Generated on Fri Feb 9 19:42:10 2018 by

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Bolzano’s theorem

Bolzano’s theorem

A

continuous function

can not change its sign (

http://planetmath.org/SignumFunction

) without going through the zero.

This contents of Bolzano’s

theorem

may be formulated more precisely as the

Theorem

.

If a real function

f

is

continuous

on a

closed interval

I

and the values of

f

in the

end points

of

I

have opposite (

http://planetmath.org/

Positive

) signs, then there exists a zero of this

function

inside the interval.

The theorem is used when using the interval halving method for getting an approximate value of a root of an equation of the form

f

⁢

(

x

)

=

0

.

Title

Bolzano’s theorem

Canonical name

BolzanosTheorem

Date of creation

2013-03-22 15:39:06

Last modified on

2013-03-22 15:39:06

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A06

Related topic

PolynomialEquationOfOddDegree

Related topic

Evolute2

Related topic

ExampleOfConvergingIncreasingSequence

Generated on Fri Feb 9 19:42:12 2018 by

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boundedness theorem

boundedness theorem

Boundedness Theorem

.

Let

a

and

b

be

real numbers

with

a

<

b

, and let

f

be a

continuous

, real valued

function

on

[

a

,

b

]

. Then

f

is bounded above and below on

[

a

,

b

]

.

Proof.

Suppose not. Then for all

natural numbers

n

we can find some

x

n

∈

[

a

,

b

]

such that

|

f

⁢

(

x

n

)

|

>

n

. The

sequence

(

x

n

)

is

bounded

, so by the

Bolzano-Weierstrass theorem

it has a

convergent

sub sequence, say

(

x

n

i

)

. As

[

a

,

b

]

is closed

(

x

n

i

)

converges

to a value in

[

a

,

b

]

. By the

continuity

of

f

we should have that

f

⁢

(

x

n

i

)

converges, but by construction it diverges. This

contradiction

finishes the proof.

Title

boundedness theorem

Canonical name

BoundednessTheorem

Date of creation

2013-03-22 14:29:18

Last modified on

2013-03-22 14:29:18

Owner

classicleft (5752)

Last modified by

classicleft (5752)

Numerical id

6

Author

classicleft (5752)

Entry type

Theorem

Classification

msc 26A06

Generated on Fri Feb 9 19:42:13 2018 by

LaTeXML
Cauchy criterion for the existence of a limit of a function

Cauchy criterion for the existence of a limit of a function

Theorem 1

.

Let

S

be a set and

B

a

filter basis

in

S

. A

function

f

:

S

→

R

possesses limit on

B

, iff for every

ϵ

>

0

there exists

X

∈

B

such that the oscillation of

f

on

X

is less than

ϵ

.

For a proof of this

theorem

see

[

1

]

.

References

1

V., Zorich,

Mathematical Analysis

I

, pp. 132ff, First Ed., Springer-Verlag, 2004.

Title

Cauchy criterion

for the

existence

of a limit of a function

Canonical name

CauchyCriterionForTheExistenceOfALimitOfAFunction

Date of creation

2013-03-22 17:45:52

Last modified on

2013-03-22 17:45:52

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

6

Author

perucho (2192)

Entry type

Theorem

Classification

msc 26A06

Related topic

CauchyConditionForLimitOfFunction

Generated on Fri Feb 9 19:42:15 2018 by

LaTeXML
chain rule

chain rule

Let

f

,

g

be

differentiable

,
real-valued functions such that

g

is defined on an open set

I

⊆

ℝ

, and

f

is defined on

g

⁢

(

I

)

.
Then the

derivative

of the

composition

f

∘

g

is given by
the

chain rule

, which asserts that

(

f

∘

g

)

′

⁢

(

x

)

=

(

f

′

∘

g

)

⁢

(

x

)

⁢

g

′

⁢

(

x

)

,

x

∈

I

.

The chain rule has a particularly suggestive appearance in terms of
the Leibniz

formalism

. Suppose that

z

depends differentiably on

y

, and that

y

in turn depends differentiably on

x

. Then we have

d

⁢

z

d

⁢

x

=

d

⁢

z

d

⁢

y

⁢

d

⁢

y

d

⁢

x

.

The apparent cancellation of the

d

⁢

y

term is at best a formal
mnemonic, and does not constitute a

rigorous

proof of this result.
Rather, the Leibniz format is well suited to the

interpretation

of the
chain rule in terms of

related rates

. To wit:

The instantaneous rate of change of

z

relative to

x

is equal to the
rate of change of

z

relative to

y

times the rate of change of

y

relative to

x

.

Title

chain rule

Canonical name

ChainRule

Date of creation

2013-03-22 12:26:43

Last modified on

2013-03-22 12:26:43

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

12

Author

matte (1858)

Entry type

Theorem

Classification

msc 26A06

Related topic

Derivative

Related topic

ChainRuleSeveralVariables

Related topic

ExampleOnSolvingAFunctionalEquation

Related topic

GudermannianFunction

Generated on Fri Feb 9 19:42:17 2018 by

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change of variable in definite integral

change of variable in definite integral

Theorem.

Let the

real function

x

↦

f

⁢

(

x

)

be

continuous

on the

interval

[

a

,

b

]

.  We introduce via the the equation

x

=

φ

⁢

(

t

)

a new variable

t

satisfying

•

φ

⁢

(

α

)

=

a

,

φ

⁢

(

β

)

=

b

,

•

φ

and

φ

′

are continuous on the interval with endpoints

α

and

β

.

Then

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

∫

α

β

f

⁢

(

φ

⁢

(

t

)

)

⁢

φ

′

⁢

(

t

)

⁢

𝑑

t

.

Proof.

As a

continuous function

,

f

has an

antiderivative

F

.  Then the

composite function

F

∘

φ

is an antiderivative of

(

f

∘

φ

)

⋅

φ

′

, since by the

chain rule

we have

d

d

⁢

t

⁢

F

⁢

(

φ

⁢

(

t

)

)

=

F

′

⁢

(

φ

⁢

(

t

)

)

⁢

φ

′

⁢

(

t

)

=

f

⁢

(

φ

⁢

(

t

)

)

⁢

φ

′

⁢

(

t

)

.

Using the Newton–Leibniz formula (

http://planetmath.org/node/40459

) we obtain

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

F

⁢

(

b

)

-

F

⁢

(

a

)

=

F

⁢

(

φ

⁢

(

β

)

)

-

F

⁢

(

φ

⁢

(

α

)

)

=

∫

α

β

f

⁢

(

φ

⁢

(

t

)

)

⁢

φ

′

⁢

(

t

)

⁢

𝑑

t

,

Q.E.D.

Title

change of variable in

definite integral

Canonical name

ChangeOfVariableInDefiniteIntegral

Date of creation

2014-05-27 13:13:22

Last modified on

2014-05-27 13:13:22

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A06

Synonym

change of variable in

Riemann integral

Related topic

RiemannIntegral

Related topic

SubstitutionForIntegration

Related topic

FundamentalTheoremOfCalculus

Related topic

IntegralsOfEvenAndOddFunctions

Related topic

OrthogonalityOfChebyshevPolynomials

Generated on Fri Feb 9 19:42:19 2018 by

LaTeXML
complex mean-value theorem

complex mean-value theorem

Theorem

[

1

]

Suppose

Ω

is an open convex set in

ℂ

,
suppose

f

is a

holomorphic function

f

:

Ω

→

ℂ

, and
suppose

a

,

b

are distinct points in

Ω

.
Then there exist points

u

,

v

on

L

a

⁢

b

(the

straight line

connecting

a

and

b

not containing the

endpoints

), such that

ℜ

⁡

{

f

⁢

(

b

)

-

f

⁢

(

a

)

b

-

a

}

=

ℜ

⁡

{

f

′

⁢

(

u

)

}

,

ℑ

⁡

{

f

⁢

(

b

)

-

f

⁢

(

a

)

b

-

a

}

=

ℑ

⁡

{

f

′

⁢

(

v

)

}

,

where

ℜ

and

ℑ

are the real (

http://planetmath.org/RealPart

) and

imaginary parts

of
a

complex number

, respectively.

References

1

J.-Cl. Evard, F. Jafari,

A Complex Rolle’s Theorem

,
American Mathematical Monthly, Vol. 99, Issue 9, (Nov. 1992), pp. 858-861.

Title

complex mean-value theorem

Canonical name

ComplexMeanvalueTheorem

Date of creation

2013-03-22 13:49:02

Last modified on

2013-03-22 13:49:02

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Theorem

Classification

msc 26A06

Generated on Fri Feb 9 19:42:43 2018 by

LaTeXML
conditionally convergent real series

conditionally convergent real series

Theorem.

If the series

u

1

+

u

2

+

u

3

+

…

(1)

with real terms

u

i

is

conditionally convergent

, i.e.

converges

but

|

u

1

|

+

|

u

2

|

+

|

u

3

|

+

⋯

diverges,
then the both series

a

1

+

a

2

+

a

3

+

…

and

-

b

1

-

b

2

-

b

3

-

…

(2)

consisting of the

positive

and negative terms of (1) are divergent — more accurately,

lim

n

→

∞

⁡

∑

i

=

1

n

a

n

=

+

∞

and

lim

n

→

∞

⁡

∑

i

=

1

n

(

-

b

n

)

=

-

∞

.

Proof.

If both of the series (2) were

convergent

, having the sums

A

and

-

B

, then we had

0

≦

|

u

1

|

+

|

u

2

|

+

…

+

|

u

n

|

<

A

+

B

for every

n

.  This would however mean that (1) would converge absolutely, contrary to the

conditional convergence

.  If, on the other hand, one of the series (2) were convergent and the other divergent, then we can see that (1) had to diverge, contrary to what is supposed in the theorem.  In fact, if e.g.

a

1

+

a

2

+

a

3

+

…

were convergent, then the

partial sum

a

1

+

a

2

+

…

+

a

n

were below a finite bound for each

n

, whereas the

n

th

partial sum of the divergent one of (2) would tend to

-

∞

as

n

→

∞

; then should also the

n

th

partial sum of (1) tend to

-

∞

.

Title

conditionally convergent real series

Canonical name

ConditionallyConvergentRealSeries

Date of creation

2013-03-22 18:41:41

Last modified on

2013-03-22 18:41:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A06

Classification

msc 40A05

Related topic

SumOfSeriesDependsOnOrder

Generated on Fri Feb 9 19:42:46 2018 by

LaTeXML
convergent series

convergent series

A series

∑

a

n

is said to be

convergent

if the sequence of partial sums

∑

i

=

1

n

a

i

is convergent.
A series that is not convergent is said to be

divergent

.

A series

∑

a

n

is said to be

absolutely convergent

if

∑

|

a

n

|

is convergent.

When the terms of the series live in

ℝ

n

,
an

equivalent

condition for

absolute convergence

of the series
is that all possible series obtained by rearrangements
of the terms are also convergent.
(This is not true in arbitrary metric spaces.)

It can be shown that absolute convergence implies

convergence

.
A series that

converges

, but is not absolutely convergent,
is called conditionally convergent.

Let

∑

a

n

be an absolutely convergent series,
and

∑

b

n

be a conditionally convergent series.
Then any rearrangement of

∑

a

n

is convergent to the same sum.
It is a result due to Riemann that

∑

b

n

can be rearranged to converge to any sum, or not converge at all.

Title

convergent series

Canonical name

ConvergentSeries

Date of creation

2013-03-22 12:24:51

Last modified on

2013-03-22 12:24:51

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Definition

Classification

msc 26A06

Classification

msc 40A05

Related topic

Series

Related topic

HarmonicNumber

Related topic

ConvergesUniformly

Related topic

SumOfSeriesDependsOnOrder

Related topic

UncoditionalConvergence

Related topic

WeierstrassMTest

Related topic

DeterminingSeriesConvergence

Defines

absolute convergence

Defines

conditional convergence

Defines

absolutely convergent

Defines

conditionally convergent

Defines

converges absolutely

Defines

convergent

Defines

divergent

Defines

divergent series

Generated on Fri Feb 9 19:42:47 2018 by

LaTeXML
converse of Darboux’s theorem (analysis) is not true

converse of Darboux’s theorem (analysis) is not true

Darboux’

theorem

says that, if

f

:

ℝ

→

ℝ

has an

antiderivative

, than

f

has to

satisfy

the

intermediate value property

, namely, for any

a

<

b

, for any number

C

with

f

⁢

(

a

)

<

C

<

f

⁢

(

b

)

or

f

⁢

(

b

)

<

C

<

f

⁢

(

a

)

, there exists a

c

∈

(

a

,

b

)

such that

f

⁢

(

c

)

=

C

. With this theorem, we understand that if

f

does not satisfy the intermediate value property, then no

function

F

satisfies

F

′

=

f

on

ℝ

.

Now, we will give an example to show that the

converse

is not true, i.e., a function that satisfies the intermediate value property might still have no antiderivative.

Let

f

⁢

(

x

)

=

{

1

x

⁢

cos

⁡

(

ln

⁡

x

)

if

x

>

0

0

if

x

≤

0

.

First let us see that

f

satisfies the intermediate value property. Let

a

<

b

. If

0

<

a

or

b

≤

0

, the property is satisfied, since

f

is

continuous

on

(

-

∞

,

0

]

and

(

0

,

∞

)

. If

a

≤

0

<

b

, we have

f

⁢

(

a

)

=

0

and

f

⁢

(

b

)

=

(

1

/

b

)

⁢

cos

⁡

(

ln

⁡

b

)

. Let

C

be between

f

⁢

(

a

)

and

(

b

)

. Let

a

0

=

exp

⁡

(

-

2

⁢

π

⁢

k

0

+

π

)

for some

k

0

large enough such that

a

0

<

b

. Then

f

⁢

(

a

0

)

=

0

=

f

⁢

(

a

)

, and since

f

is continuous on

(

a

0

,

b

)

, we must have a

c

∈

(

a

0

,

b

)

with

f

⁢

(

c

)

=

C

.

Assume, for a

contradiction

that there exists a

differentiable function

F

such that

F

′

⁢

(

x

)

=

f

⁢

(

x

)

on

ℝ

. Then consider the function

G

⁢

(

x

)

=

sin

⁡

(

ln

⁡

x

)

which is defined on

(

0

,

∞

)

.
We have

G

′

⁢

(

x

)

=

f

⁢

(

x

)

on

(

0

,

∞

)

, and since it is a an open connected set, we must have

F

⁢

(

x

)

=

G

⁢

(

x

)

+

c

on

(

0

,

∞

)

for some

c

∈

ℝ

. But then, we have

lim sup

x

→

0

+

⁡

F

⁢

(

x

)

=

lim sup

x

→

0

+

⁡

G

⁢

(

x

)

+

c

=

1

+

c

and

lim inf

x

→

0

+

⁡

F

⁢

(

x

)

=

lim inf

x

→

0

+

⁡

G

⁢

(

x

)

+

c

=

-

1

+

c

which contradicts the differentiability of

F

at

0

.

Title

converse of Darboux’s theorem (

analysis

) is not true

Canonical name

ConverseOfDarbouxsTheoremanalysisIsNotTrue

Date of creation

2013-03-22 17:33:51

Last modified on

2013-03-22 17:33:51

Owner

Gorkem (3644)

Last modified by

Gorkem (3644)

Numerical id

6

Author

Gorkem (3644)

Entry type

Example

Classification

msc 26A06

Generated on Fri Feb 9 19:42:50 2018 by

LaTeXML
Darboux’s theorem (analysis)

Darboux’s theorem (analysis)

Let

f

:

[

a

,

b

]

→

ℝ

be a real-valued

continuous function

on

[

a

,

b

]

, which is

differentiable

on

(

a

,

b

)

, differentiable from the right at

a

, and differentiable from the left at

b

. Then

f

′

the

intermediate value theorem

: for every

t

between

f

+

′

⁢

(

a

)

and

f

-

′

⁢

(

b

)

, there is some

x

∈

[

a

,

b

]

such that

f

′

⁢

(

x

)

=

t

.

Note that when

f

is continuously differentiable (

f

∈

C

1

⁢

(

[

a

,

b

]

)

), this is trivially true

by

the intermediate value theorem. But even when

f

′

is

not

continuous

, Darboux’s

theorem

places a severe

restriction

on what it can be.

Title

Darboux’s theorem (

analysis

)

Canonical name

DarbouxsTheoremanalysis

Date of creation

2013-03-22 12:45:01

Last modified on

2013-03-22 12:45:01

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

7

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 26A06

Synonym

intermediate value property of the

derivative

Generated on Fri Feb 9 19:42:52 2018 by

LaTeXML
definite integral

definite integral

The

definite integral

with respect to

x

of some

function

f

⁢

(

x

)

over the compact

interval

[

a

,

b

]

with

a

<

b

, the

interval of integration

, is
defined to be the “area under the graph of

f

⁢

(

x

)

with respect to

x

” (if

f

⁢

(

x

)

is

negative

, then you have a negative area). The numbers

a

and

b

are called

lower

and

upper limit

respectively. It is written as:

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

.

One way to find the value of the

integral

is to take a limit of an approximation technique
as the precision increases to

infinity

.

For example, use a

Riemann sum

which approximates
the area by dividing it into

n

intervals of equal widths, and then calculating the area
of

rectangles

with the width of the interval and height dependent on the function’s value in the interval.
Let

R

n

be this approximation, which can be written as

R

n

=

∑

i

=

1

n

f

⁢

(

x

i

*

)

⁢

Δ

⁢

x

,

where

x

i

*

is some

x

inside the

i

th

interval. This process is illustrated by figure

1

.

Figure 1:

The area under the graph approximated by rectangles

Then, the integral would be

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

lim

n

→

∞

⁡

R

n

=

lim

n

→

∞

⁡

∑

i

=

1

n

f

⁢

(

x

i

*

)

⁢

Δ

⁢

x

.

This limit does not necessarily exist for every function

f

and it may depend on the particular choice of the

x

i

*

. If all those limits coincide and are finite, then the integral exists. This is true in particular for

continuous

f

.

Furthermore we define

∫

b

a

f

⁢

(

x

)

⁢

𝑑

x

=

-

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

.

We can use this

definition

to arrive at some important

properties

of definite integrals
(

a

,

b

,

c

are

constant

with respect to

x

):

∫

a

b

(

f

⁢

(

x

)

+

g

⁢

(

x

)

)

⁢

𝑑

x

=

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

+

∫

a

b

g

⁢

(

x

)

⁢

𝑑

x

;

∫

a

b

(

f

⁢

(

x

)

-

g

⁢

(

x

)

)

⁢

𝑑

x

=

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

-

∫

a

b

g

⁢

(

x

)

⁢

𝑑

x

;

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

∫

a

c

f

⁢

(

x

)

⁢

𝑑

x

+

∫

c

b

f

⁢

(

x

)

⁢

𝑑

x

;

∫

a

b

c

⁢

f

⁢

(

x

)

⁢

𝑑

x

=

c

⁢

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

.

There are other

generalizations

about integrals, but many require the

fundamental theorem of calculus

.

Title

definite integral

Canonical

name

DefiniteIntegral

Date of creation

2013-03-22 12:15:17

Last modified on

2013-03-22 12:15:17

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

16

Author

mathwizard (128)

Entry type

Definition

Classification

msc 26A06

Related topic

AreaOfPlaneRegion

Related topic

IntegralsOfEvenAndOddFunctions

Related topic

IntegralOverAPeriodInterval

Defines

interval of integration

Defines

upper limit

Defines

lower limit

Generated on Fri Feb 9 19:42:55 2018 by

LaTeXML
derivative of even/odd function (proof)

derivative of even/odd function (proof)

Suppose

f

⁢

(

x

)

=

±

f

⁢

(

-

x

)

. We need to show that

f

′

⁢

(

x

)

=

∓

f

′

⁢

(

-

x

)

.
To do this, let us define the auxiliary function

m

:

ℝ

→

ℝ

,

m

⁢

(

x

)

=

-

x

.
The condition on

f

is then

f

⁢

(

x

)

=

±

(

f

∘

m

)

⁢

(

x

)

.
Using the

chain rule

, we have that

f

′

⁢

(

x

)

=

±

(

f

∘

m

)

′

⁢

(

x

)

=

±

f

′

⁢

(

m

⁢

(

x

)

)

⁢

m

′

⁢

(

x

)

=

∓

f

′

⁢

(

-

x

)

,

and the claim follows.

□

Title

derivative

of even/

odd function

(proof)

Canonical name

DerivativeOfEvenoddFunctionproof

Date of creation

2013-03-22 13:37:57

Last modified on

2013-03-22 13:37:57

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Proof

Classification

msc 26A06

Generated on Fri Feb 9 19:42:57 2018 by

LaTeXML