id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-4300 | Reverse Triangle Inequality/Real and Complex Fields/Corollary 1 | Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$.
Then:
:$\size {x - y} \ge \size x - \size y$
where $\size x$ denotes either the absolute value of a real number or the complex modulus of a complex number. | From the Reverse Triangle Inequality:
:$\cmod {x - y} \ge \cmod {\cmod x - \cmod y}$
By the definition of both absolute value and complex modulus:
:$\cmod {\cmod x - \cmod y} \ge 0$
As:
:$\cmod x - \cmod y = \pm \cmod {\cmod x - \cmod y}$
it follows that:
:$\cmod {\cmod x - \cmod y} \ge \cmod x - \cmod y$
Hence the res... | Let $x$ and $y$ be elements of either the [[Definition:Real Number|real numbers]] $\R$ or the [[Definition:Complex Number|complex numbers]] $\C$.
Then:
:$\size {x - y} \ge \size x - \size y$
where $\size x$ denotes either the [[Definition:Absolute Value|absolute value]] of a [[Definition:Real Number|real number]] or t... | From the [[Reverse Triangle Inequality/Real and Complex Fields/Proof 1|Reverse Triangle Inequality]]:
:$\cmod {x - y} \ge \cmod {\cmod x - \cmod y}$
By the definition of both [[Definition:Absolute Value|absolute value]] and [[Definition:Complex Modulus|complex modulus]]:
:$\cmod {\cmod x - \cmod y} \ge 0$
As:
:$\cmod... | Reverse Triangle Inequality/Real and Complex Fields/Corollary 1/Proof 1 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Corollary_1 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Corollary_1/Proof_1 | [
"Triangle Inequality"
] | [
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Absolute Value",
"Definition:Real Number",
"Definition:Complex Modulus",
"Definition:Complex Number"
] | [
"Reverse Triangle Inequality/Real and Complex Fields/Proof 1",
"Definition:Absolute Value",
"Definition:Complex Modulus"
] |
proofwiki-4301 | Reverse Triangle Inequality/Real and Complex Fields/Corollary 1 | Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$.
Then:
:$\size {x - y} \ge \size x - \size y$
where $\size x$ denotes either the absolute value of a real number or the complex modulus of a complex number. | By the Triangle Inequality:
:$\cmod {x + y} - \cmod y \le \cmod x$
Let $z = x + y$.
Then $x = z - y$ and so:
:$\cmod z - \cmod y \le \cmod {z - y}$
Renaming variables as appropriate gives:
:$\cmod {x - y} \ge \cmod x - \cmod y$
{{qed}} | Let $x$ and $y$ be elements of either the [[Definition:Real Number|real numbers]] $\R$ or the [[Definition:Complex Number|complex numbers]] $\C$.
Then:
:$\size {x - y} \ge \size x - \size y$
where $\size x$ denotes either the [[Definition:Absolute Value|absolute value]] of a [[Definition:Real Number|real number]] or t... | By the [[Triangle Inequality]]:
:$\cmod {x + y} - \cmod y \le \cmod x$
Let $z = x + y$.
Then $x = z - y$ and so:
:$\cmod z - \cmod y \le \cmod {z - y}$
Renaming variables as appropriate gives:
:$\cmod {x - y} \ge \cmod x - \cmod y$
{{qed}} | Reverse Triangle Inequality/Real and Complex Fields/Corollary 1/Proof 2 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Corollary_1 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Corollary_1/Proof_2 | [
"Triangle Inequality"
] | [
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Absolute Value",
"Definition:Real Number",
"Definition:Complex Modulus",
"Definition:Complex Number"
] | [
"Triangle Inequality"
] |
proofwiki-4302 | Reverse Triangle Inequality/Real and Complex Fields/Corollary 1 | Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$.
Then:
:$\size {x - y} \ge \size x - \size y$
where $\size x$ denotes either the absolute value of a real number or the complex modulus of a complex number. | Let $z_1$ and $z_2$ be represented by the points $A$ and $B$ respectively in the complex plane.
From Geometrical Interpretation of Complex Subtraction, we can construct the parallelogram $OACB$ where:
:$OA$ and $OB$ represent $z_1$ and $z_2$ respectively
:$BA$ represents $z_1 - z_2$.
:400px
But $OA$, $OB$ and $BA$ form... | Let $x$ and $y$ be elements of either the [[Definition:Real Number|real numbers]] $\R$ or the [[Definition:Complex Number|complex numbers]] $\C$.
Then:
:$\size {x - y} \ge \size x - \size y$
where $\size x$ denotes either the [[Definition:Absolute Value|absolute value]] of a [[Definition:Real Number|real number]] or t... | Let $z_1$ and $z_2$ be represented by the [[Definition:Point|points]] $A$ and $B$ respectively in the [[Definition:Complex Plane|complex plane]].
From [[Geometrical Interpretation of Complex Subtraction]], we can construct the [[Definition:Parallelogram|parallelogram]] $OACB$ where:
:$OA$ and $OB$ represent $z_1$ and ... | Reverse Triangle Inequality/Real and Complex Fields/Corollary 1/Proof 3 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Corollary_1 | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Real_and_Complex_Fields/Corollary_1/Proof_3 | [
"Triangle Inequality"
] | [
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Absolute Value",
"Definition:Real Number",
"Definition:Complex Modulus",
"Definition:Complex Number"
] | [
"Definition:Point",
"Definition:Complex Number/Complex Plane",
"Geometrical Interpretation of Complex Subtraction",
"Definition:Quadrilateral/Parallelogram",
"File:Complex-Reverse-Triangle-Inequality-Corollary.png",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)",
"Sum of Two Sides of Tria... |
proofwiki-4303 | Combination Theorem for Sequences/Real/Sum Rule | :$\ds \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$ | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
:$\ds \lim_{n \mathop \to \infty} y_n = m$
By definition of the limit of a real sequence, we can find $N_1$ such that:
:$\forall n > N_1: \size {x_n - l} < \dfrac \epsilon 2$
where $\size {x_n - l}$ ... | :$\ds \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$ | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
:$\ds \lim_{n \mathop \to \infty} y_n = m$
By definition of the [[Definition:Limit of Real Sequence|limit of a real sequence]], we can find $N_1$ such that:
:$\forall n > N_1: \size {x_n - l} < \d... | Combination Theorem for Sequences/Real/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Real/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Real/Sum_Rule | [
"Combination Theorems for Sequences"
] | [] | [
"Definition:Limit of Sequence/Real Numbers",
"Definition:Absolute Value",
"Triangle Inequality/Real Numbers"
] |
proofwiki-4304 | Combination Theorem for Sequences/Real/Multiple Rule | :$\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$ | Let $\epsilon > 0$.
We need to find $N$ such that:
:$\forall n > N: \size {\lambda x_n - \lambda l} < \epsilon$
If $\lambda = 0$ the result is trivial.
So, assume $\lambda \ne 0$.
Then $\size \lambda > 0$ from the definition of the absolute value of $\lambda$.
Hence $\dfrac \epsilon {\size \lambda} > 0$.
We have that $... | :$\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$ | Let $\epsilon > 0$.
We need to find $N$ such that:
:$\forall n > N: \size {\lambda x_n - \lambda l} < \epsilon$
If $\lambda = 0$ the result is trivial.
So, assume $\lambda \ne 0$.
Then $\size \lambda > 0$ from the definition of the [[Definition:Absolute Value|absolute value]] of $\lambda$.
Hence $\dfrac \epsilon ... | Combination Theorem for Sequences/Real/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Real/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Real/Multiple_Rule | [
"Combination Theorems for Sequences"
] | [] | [
"Definition:Absolute Value",
"Absolute Value Function is Completely Multiplicative"
] |
proofwiki-4305 | Combination Theorem for Sequences/Real/Combined Sum Rule | :$\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$ | From the Multiple Rule for Real Sequences, we have:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$
:$\ds \lim_{n \mathop \to \infty} \paren {\mu y_n} = \mu m$
The result now follows directly from the Sum Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \la... | :$\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$ | From the [[Multiple Rule for Real Sequences]], we have:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$
:$\ds \lim_{n \mathop \to \infty} \paren {\mu y_n} = \mu m$
The result now follows directly from the [[Sum Rule for Real Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y... | Combination Theorem for Sequences/Real/Combined Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Real/Combined_Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Real/Combined_Sum_Rule | [
"Combination Theorems for Sequences"
] | [] | [
"Combination Theorem for Sequences/Real/Multiple Rule",
"Combination Theorem for Sequences/Real/Sum Rule"
] |
proofwiki-4306 | Combination Theorem for Sequences/Real/Product Rule | :$\ds \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$ | Because $\sequence {x_n}$ converges, it is bounded by Convergent Sequence is Bounded.
Suppose $\size {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then:
{{begin-eqn}}
{{eqn | l = \size {x_n y_n - l m}
| r = \size {x_n y_n - x_n m + x_n m - l m}
| c =
}}
{{eqn | o = \le
| r = \size {x_n y_n - x_n m} + \size... | :$\ds \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$ | Because $\sequence {x_n}$ [[Definition:Convergent Real Sequence|converges]], it is bounded by [[Convergent Sequence is Bounded]].
Suppose $\size {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then:
{{begin-eqn}}
{{eqn | l = \size {x_n y_n - l m}
| r = \size {x_n y_n - x_n m + x_n m - l m}
| c =
}}
{{eqn | o = ... | Combination Theorem for Sequences/Real/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Real/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Real/Product_Rule | [
"Limits of Sequences"
] | [] | [
"Definition:Convergent Sequence/Real Numbers",
"Convergent Sequence in Metric Space is Bounded",
"Triangle Inequality/Real Numbers",
"Absolute Value Function is Completely Multiplicative",
"Convergent Sequence Minus Limit",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Squeeze Theorem/Seq... |
proofwiki-4307 | Combination Theorem for Sequences/Real/Quotient Rule | :$\ds \lim_{n \mathop \to \infty} \frac {x_n} {y_n} = \frac l m$
provided that $m \ne 0$. | As $y_n \to m$ as $n \to \infty$, it follows from Modulus of Limit that $\size {y_n} \to \size m$ as $n \to \infty$.
As $m \ne 0$, it follows from the definition of the modulus of $m$ that $\size m > 0$.
As the statement is given, it is possible that $y_n = 0$ for some $n$.
At such $n$, the terms $\dfrac {x_n} {y_n}$ a... | :$\ds \lim_{n \mathop \to \infty} \frac {x_n} {y_n} = \frac l m$
provided that $m \ne 0$. | As $y_n \to m$ as $n \to \infty$, it follows from [[Modulus of Limit]] that $\size {y_n} \to \size m$ as $n \to \infty$.
As $m \ne 0$, it follows from the definition of the [[Definition:Complex Modulus|modulus]] of $m$ that $\size m > 0$.
As the statement is given, it is possible that $y_n = 0$ for some $n$.
At suc... | Combination Theorem for Sequences/Real/Quotient Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Real/Quotient_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Real/Quotient_Rule | [
"Combination Theorems for Sequences"
] | [] | [
"Modulus of Limit",
"Definition:Complex Modulus",
"Definition:Term of Sequence",
"Sequence Converges to Within Half Limit",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Squeeze Theorem/Sequences/Real Numbers"
] |
proofwiki-4308 | Squeeze Theorem/Sequences/Real Numbers | Let $\sequence {x_n}$, $\sequence {y_n}$ and $\sequence {z_n}$ be sequences in $\R$.
Let $\sequence {y_n}$ and $\sequence {z_n}$ both be convergent to the following limit:
:$\ds \lim_{n \mathop \to \infty} y_n = l, \lim_{n \mathop \to \infty} z_n = l$
Suppose that:
:$\forall n \in \N: y_n \le x_n \le z_n$
Then:
:$x_n \... | From Negative of Absolute Value: Corollary 1:
:$\size {x - l} < \epsilon \iff l - \epsilon < x < l + \epsilon$
Let $\epsilon > 0$.
We need to prove that:
:$\exists N: \forall n > N: \size {x_n - l} < \epsilon$
As $\ds \lim_{n \mathop \to \infty} y_n = l$ we know that:
:$\exists N_1: \forall n > N_1: \size {y_n - l} < \... | Let $\sequence {x_n}$, $\sequence {y_n}$ and $\sequence {z_n}$ be [[Definition:Real Sequence|sequences in $\R$]].
Let $\sequence {y_n}$ and $\sequence {z_n}$ both be [[Definition:Convergent Real Sequence|convergent]] to the following [[Definition:Limit of Real Sequence|limit]]:
:$\ds \lim_{n \mathop \to \infty} y_n = ... | From [[Negative of Absolute Value/Corollary 1|Negative of Absolute Value: Corollary 1]]:
:$\size {x - l} < \epsilon \iff l - \epsilon < x < l + \epsilon$
Let $\epsilon > 0$.
We need to prove that:
:$\exists N: \forall n > N: \size {x_n - l} < \epsilon$
As $\ds \lim_{n \mathop \to \infty} y_n = l$ we know that:
:$\ex... | Squeeze Theorem/Sequences/Real Numbers | https://proofwiki.org/wiki/Squeeze_Theorem/Sequences/Real_Numbers | https://proofwiki.org/wiki/Squeeze_Theorem/Sequences/Real_Numbers | [
"Real Analysis",
"Squeeze Theorem"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Lim... | [
"Negative of Absolute Value/Corollary 1"
] |
proofwiki-4309 | Squeeze Theorem/Sequences/Complex Numbers | Let $\sequence {a_n}$ be a null sequence in $\R$, that is:
:$a_n \to 0$ as $n \to \infty$
Let $\sequence {z_n}$ be a sequence in $\C$.
Suppose $\sequence {a_n}$ dominates $\sequence {z_n}$.
That is:
: $\forall n \in \N: \cmod {z_n} \le a_n$
Then $\sequence {z_n}$ is a null sequence. | {{begin-eqn}}
{{eqn | q = \forall n \in \N
| l = \cmod {z_n}
| o = \le
| r = a_n
| c = {{Defof|Dominate (Analysis)}}
}}
{{eqn | q = \forall n \in \N
| l = a_n
| o = \le
| r = \size {a_n}
| c = Negative of Absolute Value
}}
{{eqn | q = \forall \epsilon \in \R_{>0}: \exists... | Let $\sequence {a_n}$ be a [[Definition:Null Sequence/Real Numbers|null sequence in $\R$]], that is:
:$a_n \to 0$ as $n \to \infty$
Let $\sequence {z_n}$ be a [[Definition:Complex Sequence|sequence in $\C$]].
Suppose $\sequence {a_n}$ [[Definition:Dominate (Analysis)|dominates]] $\sequence {z_n}$.
That is:
: $\fora... | {{begin-eqn}}
{{eqn | q = \forall n \in \N
| l = \cmod {z_n}
| o = \le
| r = a_n
| c = {{Defof|Dominate (Analysis)}}
}}
{{eqn | q = \forall n \in \N
| l = a_n
| o = \le
| r = \size {a_n}
| c = [[Negative of Absolute Value]]
}}
{{eqn | q = \forall \epsilon \in \R_{>0}: \ex... | Squeeze Theorem/Sequences/Complex Numbers | https://proofwiki.org/wiki/Squeeze_Theorem/Sequences/Complex_Numbers | https://proofwiki.org/wiki/Squeeze_Theorem/Sequences/Complex_Numbers | [
"Limits of Sequences"
] | [
"Definition:Null Sequence/Real Numbers",
"Definition:Complex Sequence",
"Definition:Dominate (Analysis)",
"Definition:Null Sequence/Complex Numbers"
] | [
"Negative of Absolute Value",
"Extended Transitivity",
"Definition:Null Sequence/Complex Numbers"
] |
proofwiki-4310 | Squeeze Theorem/Functions | Let $a$ be a point on an open real interval $I$.
Let $f$, $g$ and $h$ be real functions defined at all points of $I$ except for possibly at point $a$.
Suppose that:
:$\forall x \ne a \in I: \map g x \le \map f x \le \map h x$
:$\ds \lim_{x \mathop \to a} \map g x = \lim_{x \mathop \to a} \map h x = L$
Then:
:$\ds \lim_... | We start by proving the special case where $\forall x: \map g x = 0$ and $L = 0$, in which case:
:$\ds \lim_{x \mathop \to a} \map h x = 0$
Let $\epsilon > 0$ be a positive real number.
Then by the definition of the limit of a function:
:$\exists \delta > 0: 0 < \size {x - a} < \delta \implies \size {\map h x} < \epsil... | Let $a$ be a point on an [[Definition:Open Real Interval|open real interval]] $I$.
Let $f$, $g$ and $h$ be [[Definition:Real Function|real functions]] defined at all points of $I$ except for possibly at point $a$.
Suppose that:
:$\forall x \ne a \in I: \map g x \le \map f x \le \map h x$
:$\ds \lim_{x \mathop \to a} ... | We start by proving the special case where $\forall x: \map g x = 0$ and $L = 0$, in which case:
:$\ds \lim_{x \mathop \to a} \map h x = 0$
Let $\epsilon > 0$ be a positive [[Definition:Real Number|real number]].
Then by the definition of the [[Definition:Limit of Real Function|limit of a function]]:
:$\exists \delta... | Squeeze Theorem/Functions/Proof 1 | https://proofwiki.org/wiki/Squeeze_Theorem/Functions | https://proofwiki.org/wiki/Squeeze_Theorem/Functions/Proof_1 | [
"Squeeze Theorem for Functions",
"Limits of Real Functions",
"Squeeze Theorem"
] | [
"Definition:Real Interval/Open",
"Definition:Real Function"
] | [
"Definition:Real Number",
"Definition:Limit of Real Function",
"Definition:Ordering"
] |
proofwiki-4311 | Squeeze Theorem/Functions | Let $a$ be a point on an open real interval $I$.
Let $f$, $g$ and $h$ be real functions defined at all points of $I$ except for possibly at point $a$.
Suppose that:
:$\forall x \ne a \in I: \map g x \le \map f x \le \map h x$
:$\ds \lim_{x \mathop \to a} \map g x = \lim_{x \mathop \to a} \map h x = L$
Then:
:$\ds \lim_... | Let $f, g, h$ be real functions defined on an open interval $\openint a b$, except possibly at the point $c \in \openint a b$.
Let:
:$\ds \lim_{x \mathop \to c} \map g x = L$
:$\ds \lim_{x \mathop \to c} \map h x = L$
:$\map g x \le \map f x \le \map h x$ except perhaps at $x = c$.
Let $\sequence {x_n}$ be a sequence o... | Let $a$ be a point on an [[Definition:Open Real Interval|open real interval]] $I$.
Let $f$, $g$ and $h$ be [[Definition:Real Function|real functions]] defined at all points of $I$ except for possibly at point $a$.
Suppose that:
:$\forall x \ne a \in I: \map g x \le \map f x \le \map h x$
:$\ds \lim_{x \mathop \to a} ... | Let $f, g, h$ be [[Definition:Real Function|real functions]] defined on an [[Definition:Open Real Interval|open interval]] $\openint a b$, except possibly at the point $c \in \openint a b$.
Let:
:$\ds \lim_{x \mathop \to c} \map g x = L$
:$\ds \lim_{x \mathop \to c} \map h x = L$
:$\map g x \le \map f x \le \map h x$... | Squeeze Theorem/Functions/Proof 2 | https://proofwiki.org/wiki/Squeeze_Theorem/Functions | https://proofwiki.org/wiki/Squeeze_Theorem/Functions/Proof_2 | [
"Squeeze Theorem for Functions",
"Limits of Real Functions",
"Squeeze Theorem"
] | [
"Definition:Real Interval/Open",
"Definition:Real Function"
] | [
"Definition:Real Function",
"Definition:Real Interval/Open",
"Definition:Sequence",
"Limit of Function by Convergent Sequences",
"Squeeze Theorem/Sequences/Real Numbers",
"Limit of Function by Convergent Sequences"
] |
proofwiki-4312 | Squeeze Theorem/Functions | Let $a$ be a point on an open real interval $I$.
Let $f$, $g$ and $h$ be real functions defined at all points of $I$ except for possibly at point $a$.
Suppose that:
:$\forall x \ne a \in I: \map g x \le \map f x \le \map h x$
:$\ds \lim_{x \mathop \to a} \map g x = \lim_{x \mathop \to a} \map h x = L$
Then:
:$\ds \lim_... | By the definition of the limit of a real function, we have to prove that:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \paren {\size {x - a} < \delta \implies \size {\map f x - L} < \epsilon}$
Let $\epsilon \in \R_{>0}$ be given.
We have:
:$\ds \lim_{x \mathop \to a} \map g x = \lim_{x \mathop \to a} \ma... | Let $a$ be a point on an [[Definition:Open Real Interval|open real interval]] $I$.
Let $f$, $g$ and $h$ be [[Definition:Real Function|real functions]] defined at all points of $I$ except for possibly at point $a$.
Suppose that:
:$\forall x \ne a \in I: \map g x \le \map f x \le \map h x$
:$\ds \lim_{x \mathop \to a} ... | By the definition of the [[Definition:Limit of Real Function|limit of a real function]], we have to prove that:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \paren {\size {x - a} < \delta \implies \size {\map f x - L} < \epsilon}$
Let $\epsilon \in \R_{>0}$ be given.
We have:
:$\ds \lim_{x \mathop \to... | Squeeze Theorem/Functions/Proof 3 | https://proofwiki.org/wiki/Squeeze_Theorem/Functions | https://proofwiki.org/wiki/Squeeze_Theorem/Functions/Proof_3 | [
"Squeeze Theorem for Functions",
"Limits of Real Functions",
"Squeeze Theorem"
] | [
"Definition:Real Interval/Open",
"Definition:Real Function"
] | [
"Definition:Limit of Real Function",
"Combination Theorem for Limits of Functions/Real/Sum Rule",
"Definition:Limit of Real Function"
] |
proofwiki-4313 | Weierstrass's Theorem | There exists a real function $f: \closedint 0 1 \to \closedint 0 1$ such that:
:$(1): \quad f$ is continuous
:$(2): \quad f$ is nowhere differentiable. | Let $C \closedint 0 1$ denote the set of all real functions $f: \closedint 0 1 \to \R$ which are continuous on $\closedint 0 1$.
From Closed Real Interval is Compact Space, it follows that $\closedint 0 1$ is compact.
From Metric Space is Hausdorff, it follows that $\R$ is a Hausdorff space.
From Subspace of Hausdorff ... | There exists a [[Definition:Real Function|real function]] $f: \closedint 0 1 \to \closedint 0 1$ such that:
:$(1): \quad f$ is [[Definition:Continuous on Interval|continuous]]
:$(2): \quad f$ is nowhere [[Definition:Differentiable Real Function at Point|differentiable]]. | Let $C \closedint 0 1$ denote the [[Definition:Set|set]] of all [[Definition:Real Function|real functions]] $f: \closedint 0 1 \to \R$ which are [[Definition:Continuous on Interval|continuous]] on $\closedint 0 1$.
From [[Closed Real Interval is Compact Space]], it follows that $\closedint 0 1$ is [[Definition:Compact... | Weierstrass's Theorem | https://proofwiki.org/wiki/Weierstrass's_Theorem | https://proofwiki.org/wiki/Weierstrass's_Theorem | [
"Weierstrass's Theorem",
"Real Analysis"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Differentiable Mapping/Real Function/Point"
] | [
"Definition:Set",
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Closed Real Interval is Compact Space",
"Definition:Compact Space/Real Analysis",
"Metric Space is T2",
"Definition:T2 Space",
"T2 Property is Hereditary",
"Definition:T2 Space",
"Continuous Functions on... |
proofwiki-4314 | Combination Theorem for Limits of Functions/Real/Sum Rule | :$\ds \lim_{x \mathop \to c} \paren {\map f x + \map g x} = l + m$ | Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} \ x_n = c$
By Limit of Real Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Sum Rule ... | :$\ds \lim_{x \mathop \to c} \paren {\map f x + \map g x} = l + m$ | Let $\sequence {x_n}$ be any [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} \ x_n = c$
By [[Limit of Real Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \... | Combination Theorem for Limits of Functions/Real/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Real/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Real/Sum_Rule | [
"Combination Theorems for Limits of Functions"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences/Real Number Line",
"Combination Theorem for Sequences/Real/Sum Rule",
"Limit of Function by Convergent Sequences/Real Number Line"
] |
proofwiki-4315 | Combination Theorem for Limits of Functions/Real/Multiple Rule | :$\ds \lim_{x \mathop \to c} \lambda \map f x = \lambda l$ | Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} x_n = c$
By Limit of Real Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
By the Multiple Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \in... | :$\ds \lim_{x \mathop \to c} \lambda \map f x = \lambda l$ | Let $\sequence {x_n}$ be any [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} x_n = c$
By [[Limit of Real Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
By the [[Multip... | Combination Theorem for Limits of Functions/Real/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Real/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Real/Multiple_Rule | [
"Combination Theorems for Limits of Functions"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences/Real Number Line",
"Combination Theorem for Sequences/Real/Multiple Rule",
"Limit of Function by Convergent Sequences/Real Number Line",
"Category:Combination Theorems for Limits of Functions"
] |
proofwiki-4316 | Combination Theorem for Limits of Functions/Real/Combined Sum Rule | :$\ds \lim_{x \mathop \to c} \paren {\lambda \map f x + \mu \map g x} = \lambda l + \mu m$ | Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N^*: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} x_n = c$
By Limit of Real Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Combined Sum R... | :$\ds \lim_{x \mathop \to c} \paren {\lambda \map f x + \mu \map g x} = \lambda l + \mu m$ | Let $\sequence {x_n}$ be any [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N^*: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} x_n = c$
By [[Limit of Real Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \matho... | Combination Theorem for Limits of Functions/Real/Combined Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Real/Combined_Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Real/Combined_Sum_Rule | [
"Combination Theorems for Limits of Functions"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences/Real Number Line",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Limit of Function by Convergent Sequences/Real Number Line"
] |
proofwiki-4317 | Combination Theorem for Limits of Functions/Real/Product Rule | :$\ds \lim_{x \mathop \to c} \paren {\map f x \map g x} = l m$ | Let $\sequence {x_n}$ be a sequence of elements of $S$ such that:
:$\forall n \in \N: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} x_n = c$
By Limit of Real Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Product Rule for R... | :$\ds \lim_{x \mathop \to c} \paren {\map f x \map g x} = l m$ | Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} x_n = c$
By [[Limit of Real Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \t... | Combination Theorem for Limits of Functions/Real/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Real/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Real/Product_Rule | [
"Combination Theorems for Limits of Functions"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences/Real Number Line",
"Combination Theorem for Sequences/Real/Product Rule",
"Limit of Function by Convergent Sequences/Real Number Line"
] |
proofwiki-4318 | Combination Theorem for Limits of Functions/Real/Quotient Rule | :$\ds \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \frac l m$
provided that $m \ne 0$. | Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} x_n = c$
By Limit of Real Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Quotient Ru... | :$\ds \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \frac l m$
provided that $m \ne 0$. | Let $\sequence {x_n}$ be any [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} x_n = c$
By [[Limit of Real Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \ma... | Combination Theorem for Limits of Functions/Real/Quotient Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Real/Quotient_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Real/Quotient_Rule | [
"Combination Theorems for Limits of Functions"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences/Real Number Line",
"Combination Theorem for Sequences/Real/Quotient Rule",
"Limit of Function by Convergent Sequences/Real Number Line"
] |
proofwiki-4319 | Combination Theorem for Continuous Functions/Real/Sum Rule | :$f + g$ is continuous on $S$. | By definition of continuous:
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map f x = \map f c$
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map g x = \map g c$
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to c} \map f x = l$
:$\ds \lim_{x \mathop \to c} \map g x = m$
From the Sum Rule for Lim... | :$f + g$ is [[Definition:Continuous Real Function|continuous]] on $S$. | By definition of [[Definition:Continuous Real Function|continuous]]:
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map f x = \map f c$
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map g x = \map g c$
Let $f$ and $g$ tend to the following [[Definition:Limit of Real Function|limits]]:
:$\ds \lim_{x \mathop \to c} \m... | Combination Theorem for Continuous Functions/Real/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Real/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Real/Sum_Rule | [
"Combination Theorems for Continuous Real Functions"
] | [
"Definition:Continuous Real Function"
] | [
"Definition:Continuous Real Function",
"Definition:Limit of Real Function",
"Combination Theorem for Limits of Functions/Real/Sum Rule",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function"
] |
proofwiki-4320 | Combination Theorem for Continuous Functions/Real/Product Rule | :$f g$ is continuous on $S$ | By definition of continuous:
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map f x = \map f c$
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map g x = \map g c$
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to c} \map f x = l$
:$\ds \lim_{x \mathop \to c} \map g x = m$
From the Product Rule for... | :$f g$ is [[Definition:Continuous Real Function|continuous]] on $S$ | By definition of [[Definition:Continuous Real Function|continuous]]:
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map f x = \map f c$
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map g x = \map g c$
Let $f$ and $g$ tend to the following [[Definition:Limit of Real Function|limits]]:
:$\ds \lim_{x \mathop \to c} \m... | Combination Theorem for Continuous Functions/Real/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Real/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Real/Product_Rule | [
"Combination Theorems for Continuous Real Functions"
] | [
"Definition:Continuous Real Function"
] | [
"Definition:Continuous Real Function",
"Definition:Limit of Real Function",
"Combination Theorem for Limits of Functions/Real/Product Rule",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function"
] |
proofwiki-4321 | Combination Theorem for Continuous Functions/Real/Multiple Rule | :$\lambda f$ is continuous on $S$. | By definition of continuous, we have that
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map f x = \map f c$
Let $f$ tend to the following limit:
:$\ds \lim_{x \mathop \to c} \map f x = l$
From the Multiple Rule for Limits of Real Functions, we have that:
:$\ds \lim_{x \mathop \to c} \paren {\lambda \map f x} = \lambda... | :$\lambda f$ is [[Definition:Continuous Real Function|continuous]] on $S$. | By definition of [[Definition:Continuous Real Function|continuous]], we have that
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map f x = \map f c$
Let $f$ tend to the following [[Definition:Limit of Real Function|limit]]:
:$\ds \lim_{x \mathop \to c} \map f x = l$
From the [[Multiple Rule for Limits of Real Funct... | Combination Theorem for Continuous Functions/Real/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Real/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Real/Multiple_Rule | [
"Combination Theorems for Continuous Real Functions"
] | [
"Definition:Continuous Real Function"
] | [
"Definition:Continuous Real Function",
"Definition:Limit of Real Function",
"Combination Theorem for Limits of Functions/Real/Multiple Rule",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function",
"Category:Combination Theorems for Continuous Real Functions"
] |
proofwiki-4322 | Combination Theorem for Continuous Functions/Real/Combined Sum Rule | :$\lambda f + \mu g$ is continuous on $S$. | By definition of continuous, we have that
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map f x = \map f c$
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map g x = \map g c$
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to c} \map f x = l$
:$\ds \lim_{x \mathop \to c} \map g x = m$
From the Com... | :$\lambda f + \mu g$ is [[Definition:Continuous Real Function|continuous]] on $S$. | By definition of [[Definition:Continuous Real Function|continuous]], we have that
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map f x = \map f c$
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map g x = \map g c$
Let $f$ and $g$ tend to the following [[Definition:Limit of Real Function|limits]]:
:$\ds \lim_{x \mat... | Combination Theorem for Continuous Functions/Real/Combined Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Real/Combined_Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Real/Combined_Sum_Rule | [
"Combination Theorems for Continuous Real Functions"
] | [
"Definition:Continuous Real Function"
] | [
"Definition:Continuous Real Function",
"Definition:Limit of Real Function",
"Combination Theorem for Limits of Functions/Real/Combined Sum Rule",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function"
] |
proofwiki-4323 | Combination Theorem for Continuous Functions/Real/Quotient Rule | :$\dfrac f g$ is continuous on $S \setminus \set {x \in S: \map g x = 0}$
that is, on all the points $x$ of $S$ where $\map g x \ne 0$. | By definition of continuous:
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map f x = \map f c$
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map g x = \map g c$
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to c} \map f x = l$
:$\ds \lim_{x \mathop \to c} \map g x = m$
From the Quotient Rule fo... | :$\dfrac f g$ is [[Definition:Continuous Real Function|continuous]] on $S \setminus \set {x \in S: \map g x = 0}$
that is, on all the points $x$ of $S$ where $\map g x \ne 0$. | By definition of [[Definition:Continuous Real Function|continuous]]:
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map f x = \map f c$
:$\forall c \in S: \ds \lim_{x \mathop \to c} \map g x = \map g c$
Let $f$ and $g$ tend to the following [[Definition:Limit of Real Function|limits]]:
:$\ds \lim_{x \mathop \to c} \m... | Combination Theorem for Continuous Functions/Real/Quotient Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Real/Quotient_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Real/Quotient_Rule | [
"Combination Theorems for Continuous Real Functions"
] | [
"Definition:Continuous Real Function"
] | [
"Definition:Continuous Real Function",
"Definition:Limit of Real Function",
"Combination Theorem for Limits of Functions/Real/Quotient Rule",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function"
] |
proofwiki-4324 | Brouwer's Fixed Point Theorem/One-Dimensional Version | Let $f: \closedint a b \to \closedint a b$ be a real function which is continuous on the closed interval $\closedint a b$.
Then:
:$\exists \xi \in \closedint a b: \map f \xi = \xi$
That is, a continuous real function from a closed real interval to itself fixes some point of that interval. | By Subset of Real Numbers is Interval iff Connected, $\closedint a b$ is connected.
{{AimForCont}} there is no fixed point.
Then $\map f a > a$ and $\map f b < b$.
Let:
:$U = \set {x \in \closedint a b: \map f x > x}$
:$V = \set {x \in \closedint a b: \map f x < x}$
Then $U$ and $V$ are open in $\closedint a b$.
Becaus... | Let $f: \closedint a b \to \closedint a b$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then:
:$\exists \xi \in \closedint a b: \map f \xi = \xi$
That is, a [[Definition:Continuo... | By [[Subset of Real Numbers is Interval iff Connected]], $\closedint a b$ is [[Definition:Connected Topological Space|connected]].
{{AimForCont}} there is no [[Definition:Fixed Point|fixed point]].
Then $\map f a > a$ and $\map f b < b$.
Let:
:$U = \set {x \in \closedint a b: \map f x > x}$
:$V = \set {x \in \closed... | Brouwer's Fixed Point Theorem/One-Dimensional Version/Proof Using Connectedness | https://proofwiki.org/wiki/Brouwer's_Fixed_Point_Theorem/One-Dimensional_Version | https://proofwiki.org/wiki/Brouwer's_Fixed_Point_Theorem/One-Dimensional_Version/Proof_Using_Connectedness | [
"Brouwer's Fixed Point Theorem"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Continuous Real Function/Interval",
"Definition:Real Function",
"Definition:Real Interval/Closed",
"Definition:Fixed Point",
"Definition:Real Interval/Closed"
] | [
"Subset of Real Numbers is Interval iff Connected",
"Definition:Connected Topological Space",
"Definition:Fixed Point",
"Definition:Open Set/Real Analysis",
"Definition:Non-Empty Set",
"Definition:Connected Topological Space",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Fixed P... |
proofwiki-4325 | Brouwer's Fixed Point Theorem/One-Dimensional Version | Let $f: \closedint a b \to \closedint a b$ be a real function which is continuous on the closed interval $\closedint a b$.
Then:
:$\exists \xi \in \closedint a b: \map f \xi = \xi$
That is, a continuous real function from a closed real interval to itself fixes some point of that interval. | As the codomain of $f$ is $\closedint a b$, it follows that the image of $f$ is a subset of $\closedint a b$.
Thus:
:$\map f a \ge a$
and
:$\map f b \le b$
Let us define the real function $g: \closedint a b \to \R$ by:
:$\map g x = \map f x - x$
Then by the Combined Sum Rule for Continuous Real Functions, $\map g x$ is... | Let $f: \closedint a b \to \closedint a b$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then:
:$\exists \xi \in \closedint a b: \map f \xi = \xi$
That is, a [[Definition:Continuo... | As the [[Definition:Codomain of Mapping|codomain]] of $f$ is $\closedint a b$, it follows that the [[Image is Subset of Codomain|image of $f$ is a subset of $\closedint a b$]].
Thus:
:$\map f a \ge a$
and
:$\map f b \le b$
Let us define the [[Definition:Real Function|real function]] $g: \closedint a b \to \R$ by:
:$\... | Brouwer's Fixed Point Theorem/One-Dimensional Version/Proof by Intermediate Value Theorem | https://proofwiki.org/wiki/Brouwer's_Fixed_Point_Theorem/One-Dimensional_Version | https://proofwiki.org/wiki/Brouwer's_Fixed_Point_Theorem/One-Dimensional_Version/Proof_by_Intermediate_Value_Theorem | [
"Brouwer's Fixed Point Theorem"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Continuous Real Function/Interval",
"Definition:Real Function",
"Definition:Real Interval/Closed",
"Definition:Fixed Point",
"Definition:Real Interval/Closed"
] | [
"Definition:Codomain (Set Theory)/Mapping",
"Image is Subset of Codomain",
"Definition:Real Function",
"Combination Theorem for Continuous Functions/Real/Combined Sum Rule",
"Definition:Continuous Real Function/Interval",
"Intermediate Value Theorem"
] |
proofwiki-4326 | Brouwer's Fixed Point Theorem/Smooth Mapping | A smooth mapping $f$ of the closed unit ball $\overline B^n \subset \R^n$ into itself has a fixed point:
:$\forall f \in \map {C^\infty} {\overline B^n \to \overline B^n}: \exists x \in \overline B^n: \map f x = x$ | Suppose there exists such a mapping $f$ of the unit ball to itself without fixed points.
Since $\map f x \ne x$, the two points $x$ and $\map f x$ are distinct and there is a unique ray from $x$ to $\map f x$ on which they both lie.
Call this line $L$ and let $\map h x = \partial \overline B^n \cap L$.
If $x \in \parti... | A [[Definition:Smooth Mapping|smooth mapping]] $f$ of the [[Definition:Closed Ball|closed]] [[Definition:Unit Ball|unit ball]] $\overline B^n \subset \R^n$ into itself has a [[Definition:Fixed Point|fixed point]]:
:$\forall f \in \map {C^\infty} {\overline B^n \to \overline B^n}: \exists x \in \overline B^n: \map f x ... | Suppose there exists such a [[Definition:Mapping|mapping]] $f$ of the unit ball to itself without fixed points.
Since $\map f x \ne x$, the two points $x$ and $\map f x$ are [[Definition:Distinct|distinct]] and there is a unique [[Definition:Ray (Geometry)|ray]] from $x$ to $\map f x$ on which they both lie.
Call thi... | Brouwer's Fixed Point Theorem/Smooth Mapping | https://proofwiki.org/wiki/Brouwer's_Fixed_Point_Theorem/Smooth_Mapping | https://proofwiki.org/wiki/Brouwer's_Fixed_Point_Theorem/Smooth_Mapping | [
"Brouwer's Fixed Point Theorem"
] | [
"Definition:Smooth Mapping",
"Definition:Closed Ball",
"Definition:Unit Ball",
"Definition:Fixed Point"
] | [
"Definition:Mapping",
"Definition:Distinct",
"Definition:Line/Infinite Half-Line",
"Solution to Quadratic Equation",
"Retraction Theorem",
"Category:Brouwer's Fixed Point Theorem"
] |
proofwiki-4327 | Power Rule for Derivatives/Natural Number Index | Let $n \in \N$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n-1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $\map f x = x^n$ for $x \in \R, n \in \N$.
By the definition of the derivative:
:$\ds \dfrac \d {\d x} \map f x = \lim_{h \mathop \to 0} \dfrac {\map f {x + h} - \map f x} h = \lim_{h \mathop \to 0} \dfrac {\paren {x + h}^n - x^n} h$
Using the Binomial Theorem this simplifies to:
{{begin-eqn}}
{{eqn | o =
| ... | Let $n \in \N$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n-1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $\map f x = x^n$ for $x \in \R, n \in \N$.
By the definition of the [[Definition:Derivative|derivative]]:
:$\ds \dfrac \d {\d x} \map f x = \lim_{h \mathop \to 0} \dfrac {\map f {x + h} - \map f x} h = \lim_{h \mathop \to 0} \dfrac {\paren {x + h}^n - x^n} h$
Using the [[Binomial Theorem/Integral Index|Binomial ... | Power Rule for Derivatives/Natural Number Index/Proof by Binomial Theorem | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Natural_Number_Index | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Natural_Number_Index/Proof_by_Binomial_Theorem | [
"Power Rule for Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Derivative",
"Binomial Theorem/Integral Index",
"Binomial Coefficient with One"
] |
proofwiki-4328 | Power Rule for Derivatives/Natural Number Index | Let $n \in \N$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n-1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $\map f x = x^n$ for $x \in \R, n \in \N$.
Let $a \in \R$.
By definition of the derivative:
:$\ds \map {f'} a = \lim_{x \mathop \to a} \frac {\map f x - \map f a} {x - a} = \lim_{x \mathop \to a} \frac {x^n - a^n} {x - a}$
=== Case $\text I$ ===
For $n = 0$ it is possible to do:
{{begin-eqn}}
{{eqn | l = \map {f'} ... | Let $n \in \N$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n-1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $\map f x = x^n$ for $x \in \R, n \in \N$.
Let $a \in \R$.
By definition of the [[Definition:Derivative|derivative]]:
:$\ds \map {f'} a = \lim_{x \mathop \to a} \frac {\map f x - \map f a} {x - a} = \lim_{x \mathop \to a} \frac {x^n - a^n} {x - a}$
=== Case $\text I$ ===
For $n = 0$ it is possible to do:
{{be... | Power Rule for Derivatives/Natural Number Index/Proof by Difference of Two Powers | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Natural_Number_Index | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Natural_Number_Index/Proof_by_Difference_of_Two_Powers | [
"Power Rule for Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Derivative",
"Derivative of Identity Function/Real",
"Definition:Commutative Ring",
"Difference of Two Powers",
"Real Polynomial Function is Continuous"
] |
proofwiki-4329 | Power Rule for Derivatives/Natural Number Index | Let $n \in \N$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n-1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | We will use the notation $D \map f x = \map {f'} x$ as it is convenient.
Let $n = 0$.
Then:
:$\forall x \in \R: x^n = 1$
Thus $\map f x$ is the constant function $\map {f_1} x$ on $\R$.
Thus from Derivative of Constant, $D \map f x = \map D {x^0} = 0 x^{-1}$, except where $x = 0$.
So the result holds for $n = 0$.
Let $... | Let $n \in \N$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n-1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | We will use the notation $D \map f x = \map {f'} x$ as it is convenient.
Let $n = 0$.
Then:
:$\forall x \in \R: x^n = 1$
Thus $\map f x$ is the [[Definition:Constant Mapping|constant function]] $\map {f_1} x$ on $\R$.
Thus from [[Derivative of Constant]], $D \map f x = \map D {x^0} = 0 x^{-1}$, except where $x = ... | Power Rule for Derivatives/Natural Number Index/Proof by Induction | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Natural_Number_Index | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Natural_Number_Index/Proof_by_Induction | [
"Power Rule for Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Constant Mapping",
"Derivative of Constant",
"Derivative of Identity Function",
"Product Rule for Derivatives",
"Principle of Mathematical Induction"
] |
proofwiki-4330 | Power Rule for Derivatives/Integer Index | Let $n \in \Z$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | When $n \ge 0$ we use the result for Natural Number Index.
Now let $n \in \Z: n < 0$.
Then let $m = -n$ and so $m > 0$.
Thus $x^n = \dfrac 1 {x^m}$.
{{begin-eqn}}
{{eqn | l = \map D {x^n}
| r = \map D {\frac 1 {x^m} }
| c =
}}
{{eqn | r = \frac {x^m \cdot 0 - 1 \cdot m x^{m - 1} } {x^{2 m} }
| c = Qu... | Let $n \in \Z$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | When $n \ge 0$ we use the result for [[Power Rule for Derivatives/Natural Number Index|Natural Number Index]].
Now let $n \in \Z: n < 0$.
Then let $m = -n$ and so $m > 0$.
Thus $x^n = \dfrac 1 {x^m}$.
{{begin-eqn}}
{{eqn | l = \map D {x^n}
| r = \map D {\frac 1 {x^m} }
| c =
}}
{{eqn | r = \frac {x^m \... | Power Rule for Derivatives/Integer Index | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Integer_Index | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Integer_Index | [
"Power Rule for Derivatives"
] | [
"Definition:Real Function"
] | [
"Power Rule for Derivatives/Natural Number Index",
"Quotient Rule for Derivatives"
] |
proofwiki-4331 | Power Rule for Derivatives/Fractional Index | Let $n \in \N_{>0}$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^{1 / n}$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $n \in \N_{>0}$.
Thus, let $\map f x = x^{1 / n}$.
From the definition of the power to a rational number, or alternatively from the definition of the root of a number, $\map f x$ is defined when $x \ge 0$.
(However, see the special case where $x = 0$.)
From Continuity of Root Function, $\map f x$ is continuous over... | Let $n \in \N_{>0}$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^{1 / n}$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $n \in \N_{>0}$.
Thus, let $\map f x = x^{1 / n}$.
From the definition of the [[Definition:Rational Power|power to a rational number]], or alternatively from the definition of the [[Definition:Root of Number|root]] of a [[Definition:Number|number]], $\map f x$ is defined when $x \ge 0$.
(However, see the [[Defin... | Power Rule for Derivatives/Fractional Index/Proof 1 | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Fractional_Index | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Fractional_Index/Proof_1 | [
"Power Rule for Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Power (Algebra)/Rational Number",
"Definition:Root of Number",
"Definition:Number",
"Definition:Power (Algebra)/Power of Zero",
"Continuity of Root Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Continuous Real Function/Right-Continu... |
proofwiki-4332 | Power Rule for Derivatives/Fractional Index | Let $n \in \N_{>0}$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^{1 / n}$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $n \in \N_{>0}$.
Thus, let $\map f x = y = x^{1/n}$.
Thus $\map {f^{-1} } y = x = y^n$ from the definition of root.
So:
{{begin-eqn}}
{{eqn | l = D x^{1/n}
| r = \frac 1 {D y^n}
| c = Derivative of Inverse Function
}}
{{eqn | r = \frac 1 {n y^{n - 1} }
| c = Power Rule for Derivatives: Integer Ind... | Let $n \in \N_{>0}$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^{1 / n}$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $n \in \N_{>0}$.
Thus, let $\map f x = y = x^{1/n}$.
Thus $\map {f^{-1} } y = x = y^n$ from the definition of [[Definition:Root of Number|root]].
So:
{{begin-eqn}}
{{eqn | l = D x^{1/n}
| r = \frac 1 {D y^n}
| c = [[Derivative of Inverse Function]]
}}
{{eqn | r = \frac 1 {n y^{n - 1} }
| c = [... | Power Rule for Derivatives/Fractional Index/Proof 2 | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Fractional_Index | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Fractional_Index/Proof_2 | [
"Power Rule for Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Root of Number",
"Derivative of Inverse Function",
"Power Rule for Derivatives/Integer Index"
] |
proofwiki-4333 | Power Rule for Derivatives/Rational Index | Let $n \in \Q$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $n \in \Q$, such that $n = \dfrac p q$ where $p, q \in \Z, q \ne 0$.
Then we have:
{{begin-eqn}}
{{eqn | l = \map D {x^n}
| r = \map D {x^{p / q} }
| c =
}}
{{eqn | r = \map D {\paren {x^p}^{1 / q} }
| c =
}}
{{eqn | r = \frac 1 q \paren {x^p}^{1 / q} x^{-p} p x^{p - 1}
| c = Chain Rule fo... | Let $n \in \Q$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $n \in \Q$, such that $n = \dfrac p q$ where $p, q \in \Z, q \ne 0$.
Then we have:
{{begin-eqn}}
{{eqn | l = \map D {x^n}
| r = \map D {x^{p / q} }
| c =
}}
{{eqn | r = \map D {\paren {x^p}^{1 / q} }
| c =
}}
{{eqn | r = \frac 1 q \paren {x^p}^{1 / q} x^{-p} p x^{p - 1}
| c = [[Chain Ru... | Power Rule for Derivatives/Rational Index | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Rational_Index | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Rational_Index | [
"Power Rule for Derivatives"
] | [
"Definition:Real Function"
] | [
"Derivative of Composite Function"
] |
proofwiki-4334 | Power Rule for Derivatives/Real Number Index | Let $n \in \R$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n-1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | We are going to prove that $\map {f'} x = n x^{n - 1}$ holds for all real $n$.
To do this, we compute the limit $\ds \lim_{h \mathop \to 0} \frac {\paren {x + h}^n - x^n} h$:
{{begin-eqn}}
{{eqn | l = \frac {\paren {x + h}^n - x^n} h
| r = \frac {x^n} h \paren {\paren {1 + \frac h x}^n - 1}
| c =
}}
{{eqn ... | Let $n \in \R$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n-1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | We are going to prove that $\map {f'} x = n x^{n - 1}$ holds for all [[Definition:Real Number|real]] $n$.
To do this, we compute the limit $\ds \lim_{h \mathop \to 0} \frac {\paren {x + h}^n - x^n} h$:
{{begin-eqn}}
{{eqn | l = \frac {\paren {x + h}^n - x^n} h
| r = \frac {x^n} h \paren {\paren {1 + \frac h x}... | Power Rule for Derivatives/Real Number Index/Proof 1 | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Real_Number_Index | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Real_Number_Index/Proof_1 | [
"Power Rule for Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Real Number",
"Derivative of Exponential at Zero",
"Derivative of Logarithm at One"
] |
proofwiki-4335 | Power Rule for Derivatives/Real Number Index | Let $n \in \R$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n-1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Note this proof does not hold for $x = 0$.
Let $y$ = $\map f x$.
Then $y = x^n$.
Then:
{{begin-eqn}}
{{eqn | l = y
| r = x^n
}}
{{eqn | ll= \leadsto
| l = \size y
| r = \size {x^n}
| c = taking the absolute value of both sides
}}
{{eqn | r = \size x^n
| c = Absolute Value of Power
}}
{{eqn... | Let $n \in \R$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n-1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Note this proof does not hold for $x = 0$.
Let $y$ = $\map f x$.
Then $y = x^n$.
Then:
{{begin-eqn}}
{{eqn | l = y
| r = x^n
}}
{{eqn | ll= \leadsto
| l = \size y
| r = \size {x^n}
| c = taking the [[Definition:Absolute Value|absolute value]] of both sides
}}
{{eqn | r = \size x^n
| c ... | Power Rule for Derivatives/Real Number Index/Proof 2 | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Real_Number_Index | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Real_Number_Index/Proof_2 | [
"Power Rule for Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Absolute Value",
"Absolute Value of Power",
"Definition:Natural Logarithm",
"Logarithm of Power",
"Derivative of Composite Function",
"Derivative of Constant Multiple",
"Exponent Combination Laws/Quotient of Powers"
] |
proofwiki-4336 | Well-Ordering Minimal Elements are Unique | Let $\struct {S,\preceq}$ be a well-ordered set.
Then every non-empty subset of $S$ has a unique minimal element. | The proof consists of a uniqueness and an existence part.
Let $S'$ be a non-empty subset of $S$. | Let $\struct {S,\preceq}$ be a [[Definition:Well-Ordered Set|well-ordered set]].
Then every [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$ has a [[Definition:Unique|unique]] [[Definition:Minimal Element|minimal element]]. | The proof consists of a [[Definition:Unique|uniqueness]] and an existence part.
Let $S'$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$. | Well-Ordering Minimal Elements are Unique | https://proofwiki.org/wiki/Well-Ordering_Minimal_Elements_are_Unique | https://proofwiki.org/wiki/Well-Ordering_Minimal_Elements_are_Unique | [
" Well-Orderings"
] | [
"Definition:Well-Ordered Set",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Unique",
"Definition:Minimal/Element"
] | [
"Definition:Unique",
"Definition:Non-Empty Set",
"Definition:Subset"
] |
proofwiki-4337 | Fundamental Theorem of Calculus/First Part | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.
Let $F$ be a real function which is defined on $\closedint a b$ by:
:$\ds \map F x = \int_a^x \map f t \rd t$
Then $F$ is a primitive of $f$ on $\closedint a b$. | To show that $F$ is a primitive of $f$ on $\closedint a b$, we need to establish the following:
:$F$ is continuous on $\closedint a b$
:$F$ is differentiable on the open interval $\openint a b$
:$\forall x \in \closedint a b: \map {F'} x = \map f x$.
=== Proof that $F$ is Continuous ===
We have that $f$ is continuous o... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $F$ be a [[Definition:Real Function|real function]] which is defined on $\closedint a b$ by:
:$\ds \map F x = \int_a^x \map ... | To show that $F$ is a [[Definition:Primitive (Calculus)|primitive]] of $f$ on $\closedint a b$, we need to establish the following:
:$F$ is [[Definition:Continuous on Interval|continuous]] on $\closedint a b$
:$F$ is [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open i... | Fundamental Theorem of Calculus/First Part/Proof 1 | https://proofwiki.org/wiki/Fundamental_Theorem_of_Calculus/First_Part | https://proofwiki.org/wiki/Fundamental_Theorem_of_Calculus/First_Part/Proof_1 | [
"Fundamental Theorem of Calculus"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Real Function",
"Definition:Primitive (Calculus)"
] | [
"Definition:Primitive (Calculus)",
"Definition:Continuous Real Function/Interval",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Continuous Real Function/Interval",
"Continuous Image of Closed Real Interval is Closed Real Interval",
"Definition:... |
proofwiki-4338 | Fundamental Theorem of Calculus/First Part | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.
Let $F$ be a real function which is defined on $\closedint a b$ by:
:$\ds \map F x = \int_a^x \map f t \rd t$
Then $F$ is a primitive of $f$ on $\closedint a b$. | {{begin-eqn}}
{{eqn | l = \dfrac \d {\d x} \map F x
| r = \lim_{\Delta x \mathop \to 0} \frac 1 {\Delta x} \paren {\int_a^{x + \Delta x} \map f t \rd t - \int_a^x \map f t \rd t}
| c = {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \lim_{\Delta x \mathop \to 0} \frac 1 {\Delta x} \paren {\int... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $F$ be a [[Definition:Real Function|real function]] which is defined on $\closedint a b$ by:
:$\ds \map F x = \int_a^x \map ... | {{begin-eqn}}
{{eqn | l = \dfrac \d {\d x} \map F x
| r = \lim_{\Delta x \mathop \to 0} \frac 1 {\Delta x} \paren {\int_a^{x + \Delta x} \map f t \rd t - \int_a^x \map f t \rd t}
| c = {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \lim_{\Delta x \mathop \to 0} \frac 1 {\Delta x} \paren {\int... | Fundamental Theorem of Calculus/First Part/Proof 2 | https://proofwiki.org/wiki/Fundamental_Theorem_of_Calculus/First_Part | https://proofwiki.org/wiki/Fundamental_Theorem_of_Calculus/First_Part/Proof_2 | [
"Fundamental Theorem of Calculus"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Real Function",
"Definition:Primitive (Calculus)"
] | [
"Definition:Definite Integral",
"Sum of Integrals on Adjacent Intervals for Continuous Functions",
"Definition:Real Interval/Closed",
"Definition:By Hypothesis",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Continuous Real Function/Interval",
"Definiti... |
proofwiki-4339 | Fundamental Theorem of Calculus/First Part | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.
Let $F$ be a real function which is defined on $\closedint a b$ by:
:$\ds \map F x = \int_a^x \map f t \rd t$
Then $F$ is a primitive of $f$ on $\closedint a b$. | By Topological Manifold/Examples/Real Cartesian Space, the closed real interval is a manifold.
We have that $F$ is a smooth $0$-form with compact support on a smooth $1$-dimensional oriented manifold $\closedint a b$.
We have that the boundary of $\closedint a b$ is $\partial \closedint a b$.
We denote $\d F$ to be th... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $F$ be a [[Definition:Real Function|real function]] which is defined on $\closedint a b$ by:
:$\ds \map F x = \int_a^x \map ... | By [[Topological Manifold/Examples/Real Cartesian Space]], the [[Definition:Closed Real Interval|closed real interval]] is a [[Definition:Topological Manifold|manifold]].
We have that $F$ is a [[Definition:Smooth Real Function|smooth]] $0$-[[Definition:Differential Form|form]] with [[Definition:Compact Topological Sp... | Fundamental Theorem of Calculus/First Part/Proof 3 | https://proofwiki.org/wiki/Fundamental_Theorem_of_Calculus/First_Part | https://proofwiki.org/wiki/Fundamental_Theorem_of_Calculus/First_Part/Proof_3 | [
"Fundamental Theorem of Calculus"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Real Function",
"Definition:Primitive (Calculus)"
] | [
"Topological Manifold/Examples/Real Cartesian Space",
"Definition:Real Interval/Closed",
"Definition:Topological Manifold",
"Definition:Smooth Real Function",
"Definition:Differential Form",
"Definition:Compact Topological Space",
"Definition:Support of Mapping to Algebraic Structure/Real-Valued Functio... |
proofwiki-4340 | Fundamental Theorem of Calculus/Second Part | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.
Then:
:$(1): \quad f$ has a primitive on $\closedint a b$
:$(2): \quad$ If $F$ is any primitive of $f$ on $\closedint a b$, then:
:::$\ds \int_a^b \map f t \rd t = \map F b - \map F a = \bigintlimits {\map F t} a b$ | Let $G$ be defined on $\closedint a b$ by:
:$\ds \map G x = \int_a^x \map f t \rd t$
We have:
:$\ds \map G a = \int_a^a \map f t \rd t = 0$ from Integral on Zero Interval
:$\ds \map G b = \int_a^b \map f t \rd t$ from the definition of $G$ above.
Therefore, we can compute:
{{begin-eqn}}
{{eqn | l = \int_a^b \map f t \r... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous Real Function on Closed Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then:
:$(1): \quad f$ has a [[Definition:Primitive (Calculus)|primitive]] on $\closedint a b$
:$(2): \quad$... | Let $G$ be defined on $\closedint a b$ by:
:$\ds \map G x = \int_a^x \map f t \rd t$
We have:
:$\ds \map G a = \int_a^a \map f t \rd t = 0$ from [[Integral on Zero Interval]]
:$\ds \map G b = \int_a^b \map f t \rd t$ from the definition of $G$ above.
Therefore, we can compute:
{{begin-eqn}}
{{eqn | l = \int_a^b \map ... | Fundamental Theorem of Calculus/Second Part/Proof 1 | https://proofwiki.org/wiki/Fundamental_Theorem_of_Calculus/Second_Part | https://proofwiki.org/wiki/Fundamental_Theorem_of_Calculus/Second_Part/Proof_1 | [
"Fundamental Theorem of Calculus"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Primitive (Calculus)",
"Definition:Primitive (Calculus)"
] | [
"Definite Integral on Zero Interval",
"Sum of Integrals on Adjacent Intervals for Continuous Functions",
"Fundamental Theorem of Calculus/First Part",
"Definition:Primitive (Calculus)",
"Primitives which Differ by Constant",
"Definition:Primitive (Calculus)"
] |
proofwiki-4341 | Fundamental Theorem of Calculus/Second Part | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.
Then:
:$(1): \quad f$ has a primitive on $\closedint a b$
:$(2): \quad$ If $F$ is any primitive of $f$ on $\closedint a b$, then:
:::$\ds \int_a^b \map f t \rd t = \map F b - \map F a = \bigintlimits {\map F t} a b$ | As $f$ is continuous, by the first part of the theorem, it has a primitive. Call it $F$.
$\closedint a b$ can be divided into any number of closed subintervals of the form $\closedint {x_{k - 1} } {x_k}$ where:
:$a = x_0 < x_1 \cdots < x_{k-1} < x_k = b$
Fix such a finite subdivision of the interval $\closedint a b$; c... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous Real Function on Closed Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then:
:$(1): \quad f$ has a [[Definition:Primitive (Calculus)|primitive]] on $\closedint a b$
:$(2): \quad$... | As $f$ is [[Definition:Continuous on Interval|continuous]], by the [[Fundamental Theorem of Calculus/First Part|first part]] of the theorem, it has a [[Definition:Primitive (Calculus)|primitive]]. Call it $F$.
$\closedint a b$ can be divided into any number of [[Definition:Closed Real Interval|closed subintervals]] of... | Fundamental Theorem of Calculus/Second Part/Proof 2 | https://proofwiki.org/wiki/Fundamental_Theorem_of_Calculus/Second_Part | https://proofwiki.org/wiki/Fundamental_Theorem_of_Calculus/Second_Part/Proof_2 | [
"Fundamental Theorem of Calculus"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Primitive (Calculus)",
"Definition:Primitive (Calculus)"
] | [
"Definition:Continuous Real Function/Interval",
"Fundamental Theorem of Calculus/First Part",
"Definition:Primitive (Calculus)",
"Definition:Real Interval/Closed",
"Definition:Subdivision of Interval/Finite",
"Definition:Real Interval/Closed",
"Telescoping Series/Example 2",
"Definition:Differentiable... |
proofwiki-4342 | Natural Logarithm of 1 is 0 | :$\ln 1 = 0$ | We use the definition of the natural logarithm as an integral:
:$\ds \ln x = \int_1^x \frac {\d t} t$
From Integral on Zero Interval:
:$\ds \ln 1 = \int_1^1 \frac {\d t} t = 0$
{{qed}} | :$\ln 1 = 0$ | We use the definition of the [[Definition:Natural Logarithm|natural logarithm]] as an [[Definition:Natural Logarithm/Positive Real/Definition 1|integral]]:
:$\ds \ln x = \int_1^x \frac {\d t} t$
From [[Integral on Zero Interval]]:
:$\ds \ln 1 = \int_1^1 \frac {\d t} t = 0$
{{qed}} | Natural Logarithm of 1 is 0/Proof 1 | https://proofwiki.org/wiki/Natural_Logarithm_of_1_is_0 | https://proofwiki.org/wiki/Natural_Logarithm_of_1_is_0/Proof_1 | [
"Examples of Natural Logarithms",
"Natural Logarithm of 1 is 0"
] | [] | [
"Definition:Natural Logarithm",
"Definition:Natural Logarithm/Positive Real/Definition 1",
"Definite Integral on Zero Interval"
] |
proofwiki-4343 | Natural Logarithm of 1 is 0 | :$\ln 1 = 0$ | We use the definition of the natural logarithm as the inverse of the exponential:
:$\ln x = y \iff e^y = x$
Then:
{{begin-eqn}}
{{eqn | l = e^0
| r = 1
| c = Exponential of Zero
}}
{{eqn | ll= \leadstoandfrom
| l = \ln 1
| r = 0
}}
{{end-eqn}}
{{qed}} | :$\ln 1 = 0$ | We use the definition of the [[Definition:Natural Logarithm|natural logarithm]] as the [[Definition:Natural Logarithm/Positive Real/Definition 2|inverse of the exponential]]:
:$\ln x = y \iff e^y = x$
Then:
{{begin-eqn}}
{{eqn | l = e^0
| r = 1
| c = [[Exponential of Zero]]
}}
{{eqn | ll= \leadstoandfrom... | Natural Logarithm of 1 is 0/Proof 2 | https://proofwiki.org/wiki/Natural_Logarithm_of_1_is_0 | https://proofwiki.org/wiki/Natural_Logarithm_of_1_is_0/Proof_2 | [
"Examples of Natural Logarithms",
"Natural Logarithm of 1 is 0"
] | [] | [
"Definition:Natural Logarithm",
"Definition:Natural Logarithm/Positive Real/Definition 2",
"Exponential of Zero"
] |
proofwiki-4344 | Natural Logarithm of 1 is 0 | :$\ln 1 = 0$ | We use the definition of the natural logarithm as the limit of a sequence:
:$\ds \ln x = \lim_{n \mathop \to \infty} n \paren {\sqrt [n] x - 1}$
Then:
{{begin-eqn}}
{{eqn | l = \ln 1
| r = \lim_{n \mathop \to \infty} n \paren {\sqrt [n] 1 - 1}
}}
{{eqn | r = \lim_{n \mathop \to \infty} n \times 0
}}
{{eqn | r = \... | :$\ln 1 = 0$ | We use the definition of the [[Definition:Natural Logarithm|natural logarithm]] as the [[Definition:Natural Logarithm/Positive Real/Definition 3|limit of a sequence]]:
:$\ds \ln x = \lim_{n \mathop \to \infty} n \paren {\sqrt [n] x - 1}$
Then:
{{begin-eqn}}
{{eqn | l = \ln 1
| r = \lim_{n \mathop \to \infty} n \... | Natural Logarithm of 1 is 0/Proof 3 | https://proofwiki.org/wiki/Natural_Logarithm_of_1_is_0 | https://proofwiki.org/wiki/Natural_Logarithm_of_1_is_0/Proof_3 | [
"Examples of Natural Logarithms",
"Natural Logarithm of 1 is 0"
] | [] | [
"Definition:Natural Logarithm",
"Definition:Natural Logarithm/Positive Real/Definition 3"
] |
proofwiki-4345 | Derivative of Natural Logarithm Function | Let $\ln x$ be the natural logarithm function.
Then:
:$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$ | {{begin-eqn}}
{{eqn | l = \ln x
| o = :=
| r = \int_1^x \dfrac 1 t \rd t
| c = {{Defof|Natural Logarithm|subdef = Positive Real|index = 1}}
}}
{{eqn | l = \frac \d {\d x} \ln x
| r = \frac \d {\d x} \int_1^x \dfrac 1 t \rd t
}}
{{eqn | r = \frac 1 x
| c = Fundamental Theorem of Calculus
}}... | Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm function]].
Then:
:$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$ | {{begin-eqn}}
{{eqn | l = \ln x
| o = :=
| r = \int_1^x \dfrac 1 t \rd t
| c = {{Defof|Natural Logarithm|subdef = Positive Real|index = 1}}
}}
{{eqn | l = \frac \d {\d x} \ln x
| r = \frac \d {\d x} \int_1^x \dfrac 1 t \rd t
}}
{{eqn | r = \frac 1 x
| c = [[Fundamental Theorem of Calculus/... | Derivative of Natural Logarithm Function/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_Function | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_Function/Proof_1 | [
"Derivative of Natural Logarithm Function",
"Derivatives involving Logarithm Functions",
"Natural Logarithms"
] | [
"Definition:Natural Logarithm"
] | [
"Fundamental Theorem of Calculus/First Part/Corollary"
] |
proofwiki-4346 | Derivative of Natural Logarithm Function | Let $\ln x$ be the natural logarithm function.
Then:
:$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$ | This proof assumes the definition of the natural logarithm as the inverse of the exponential function, where the exponential function is defined as the limit of a sequence:
:$e^x := \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$
It also assumes the Laws of Logarithms.
{{begin-eqn}}
{{eqn | l = \map {\frac \... | Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm function]].
Then:
:$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$ | This proof assumes the definition of the natural logarithm as the inverse of the exponential function, where the exponential function is defined as the [[Definition:Exponential Function/Real/Limit of Sequence|limit of a sequence]]:
:$e^x := \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$
It also assumes th... | Derivative of Natural Logarithm Function/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_Function | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_Function/Proof_2 | [
"Derivative of Natural Logarithm Function",
"Derivatives involving Logarithm Functions",
"Natural Logarithms"
] | [
"Definition:Natural Logarithm"
] | [
"Definition:Exponential Function/Real/Limit of Sequence",
"Laws of Logarithms",
"Difference of Logarithms",
"Logarithm of Power/Natural Logarithm",
"Logarithm of Power/Natural Logarithm",
"Limit of Composite Function",
"Definition:Exponential Function/Real/Limit of Sequence",
"Real Natural Logarithm F... |
proofwiki-4347 | Derivative of Natural Logarithm Function | Let $\ln x$ be the natural logarithm function.
Then:
:$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$ | This proof assumes the definition of the natural logarithm as the inverse of the exponential function as defined by differential equation:
:$y = \dfrac {\d y} {\d x}$
:$y = e^x \iff \ln y = x$
{{begin-eqn}}
{{eqn | l = \frac {\d y} {\d x}
| r = y
| c = {{Defof|Exponential Function/Real|subdef = Differential... | Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm function]].
Then:
:$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$ | This proof assumes the definition of the [[Definition:Natural Logarithm|natural logarithm]] as the inverse of the [[Definition:Exponential Function/Real/Differential Equation|exponential function as defined by differential equation]]:
:$y = \dfrac {\d y} {\d x}$
:$y = e^x \iff \ln y = x$
{{begin-eqn}}
{{eqn | l = \f... | Derivative of Natural Logarithm Function/Proof 3 | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_Function | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_Function/Proof_3 | [
"Derivative of Natural Logarithm Function",
"Derivatives involving Logarithm Functions",
"Natural Logarithms"
] | [
"Definition:Natural Logarithm"
] | [
"Definition:Natural Logarithm",
"Definition:Exponential Function/Real/Differential Equation",
"Solution to Separable Differential Equation",
"Integral of Constant",
"Definition:Primitive (Calculus)",
"Definition:Exponential Function/Real/Differential Equation",
"Definition:Initial Condition"
] |
proofwiki-4348 | Derivative of Natural Logarithm Function | Let $\ln x$ be the natural logarithm function.
Then:
:$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$ | This proof assumes the definition of the natural logarithm as the limit of a sequence of real functions.
Let $\sequence {f_n}$ be the sequence of mappings $f_n: \R_{>0} \to \R$ defined as:
:$\map {f_n} x = n \paren {\sqrt [n] x - 1}$
Fix $x_0 \in \R_{>0}$.
Pick $k \in \N : x_0 \in J := \closedint {\dfrac 1 k} k$.
From ... | Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm function]].
Then:
:$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$ | This proof assumes the definition of the [[Definition:Natural Logarithm|natural logarithm]] as the [[Definition:Limit of Real Sequence|limit]] of a [[Definition:Sequence|sequence]] of [[Definition:Real Function|real functions]].
Let $\sequence {f_n}$ be the [[Definition:Sequence|sequence]] of mappings $f_n: \R_{>0} \t... | Derivative of Natural Logarithm Function/Proof 4 | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_Function | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_Function/Proof_4 | [
"Derivative of Natural Logarithm Function",
"Derivatives involving Logarithm Functions",
"Natural Logarithms"
] | [
"Definition:Natural Logarithm"
] | [
"Definition:Natural Logarithm",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Sequence",
"Definition:Real Function",
"Definition:Sequence",
"Definition:Real Interval/Bounded",
"Derivative of Nth Root",
"Combination Theorem for Sequences",
"Definition:Differentiable Mapping/Real Function",... |
proofwiki-4349 | Logarithm is Strictly Increasing | :$\ln x: x > 0$ is strictly increasing. | From Derivative of Natural Logarithm Function $D \ln x = \dfrac 1 x$, which is strictly positive on $x > 0$.
From Derivative of Monotone Function it follows that $\ln x$ is strictly increasing on $x > 0$.
{{qed}} | :$\ln x: x > 0$ is [[Definition:Strictly Increasing Real Function|strictly increasing]]. | From [[Derivative of Natural Logarithm Function]] $D \ln x = \dfrac 1 x$, which is [[Definition:Strictly Positive|strictly positive]] on $x > 0$.
From [[Derivative of Monotone Function]] it follows that $\ln x$ is [[Definition:Strictly Increasing Real Function|strictly increasing]] on $x > 0$.
{{qed}} | Logarithm is Strictly Increasing | https://proofwiki.org/wiki/Logarithm_is_Strictly_Increasing | https://proofwiki.org/wiki/Logarithm_is_Strictly_Increasing | [
"Logarithms",
"Examples of Strictly Increasing Real Functions"
] | [
"Definition:Strictly Increasing/Real Function"
] | [
"Derivative of Natural Logarithm Function",
"Definition:Strictly Positive",
"Derivative of Monotone Function",
"Definition:Strictly Increasing/Real Function"
] |
proofwiki-4350 | Logarithm Tends to Infinity | :$\ln x \to +\infty$ as $x \to +\infty$ | From Natural Logarithm of 2 is Greater than One Half:
:$\ln 2 \ge \dfrac 1 2$
From the definition of infinite limit at infinity, our assertion is:
:$\forall M \in \R_{>0} : \exists N > 0 : x > N \implies \ln x > M$.
As $x \to +\infty$, we will restrict our attention to sufficiently large $M$.
From Logarithm is Strictly... | :$\ln x \to +\infty$ as $x \to +\infty$ | From [[Natural Logarithm of 2 is Greater than One Half]]:
:$\ln 2 \ge \dfrac 1 2$
From the definition of [[Definition:Infinite Limit at Infinity|infinite limit at infinity]], our assertion is:
:$\forall M \in \R_{>0} : \exists N > 0 : x > N \implies \ln x > M$.
As $x \to +\infty$, we will restrict our attention to [... | Logarithm Tends to Infinity/Proof 1 | https://proofwiki.org/wiki/Logarithm_Tends_to_Infinity | https://proofwiki.org/wiki/Logarithm_Tends_to_Infinity/Proof_1 | [
"Logarithms",
"Logarithm Tends to Infinity"
] | [] | [
"Natural Logarithm of 2 is Greater than One Half",
"Definition:Limit of Real Function/Limit at Infinity/Positive/Increasing Without Bound",
"Definition:Sufficiently Large",
"Logarithm is Strictly Increasing",
"Definition:Strictly Increasing/Real Function",
"Definition:Sufficiently Large",
"Laws of Logar... |
proofwiki-4351 | Logarithm Tends to Infinity | :$\ln x \to +\infty$ as $x \to +\infty$ | From the definition of the natural logarithm:
{{begin-eqn}}
{{eqn | l = \ln x
| r = \int_1^x \dfrac 1 t \rd t
}}
{{end-eqn}}
The result follows from Integral of Reciprocal is Divergent.
{{qed}} | :$\ln x \to +\infty$ as $x \to +\infty$ | From the definition of the [[Definition:Real Natural Logarithm|natural logarithm]]:
{{begin-eqn}}
{{eqn | l = \ln x
| r = \int_1^x \dfrac 1 t \rd t
}}
{{end-eqn}}
The result follows from [[Integral of Reciprocal is Divergent]].
{{qed}} | Logarithm Tends to Infinity/Proof 2 | https://proofwiki.org/wiki/Logarithm_Tends_to_Infinity | https://proofwiki.org/wiki/Logarithm_Tends_to_Infinity/Proof_2 | [
"Logarithms",
"Logarithm Tends to Infinity"
] | [] | [
"Definition:Natural Logarithm/Positive Real",
"Integral of Reciprocal is Divergent"
] |
proofwiki-4352 | Logarithm Tends to Negative Infinity | :$\ln x \to -\infty$ as $x \to 0^+$ | From the definition of natural logarithm:
{{begin-eqn}}
{{eqn | l = \ln x
| r = \int_1^x \dfrac 1 t \ \mathrm dt
}}
{{end-eqn}}
The result follows from Integral of Reciprocal is Divergent.
{{qed}} | :$\ln x \to -\infty$ as $x \to 0^+$ | From the definition of [[Definition:Real Natural Logarithm|natural logarithm]]:
{{begin-eqn}}
{{eqn | l = \ln x
| r = \int_1^x \dfrac 1 t \ \mathrm dt
}}
{{end-eqn}}
The result follows from [[Integral of Reciprocal is Divergent]].
{{qed}} | Logarithm Tends to Negative Infinity | https://proofwiki.org/wiki/Logarithm_Tends_to_Negative_Infinity | https://proofwiki.org/wiki/Logarithm_Tends_to_Negative_Infinity | [
"Logarithms"
] | [] | [
"Definition:Natural Logarithm/Positive Real",
"Integral of Reciprocal is Divergent"
] |
proofwiki-4353 | Second Derivative of Natural Logarithm Function | Let $\ln x$ be the natural logarithm function.
Then:
:$\map {\dfrac {\d^2} {\d x^2} } {\ln x} = -\dfrac 1 {x^2}$ | From Derivative of Natural Logarithm Function:
:$\dfrac \d {\d x} \ln x = \dfrac 1 x$
From the Power Rule for Derivatives: Integer Index:
:$\dfrac {\d^2} {\d x^2} \ln x = \dfrac \d {\d x} \dfrac 1 x = -\dfrac 1 {x^2}$
{{qed}} | Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm function]].
Then:
:$\map {\dfrac {\d^2} {\d x^2} } {\ln x} = -\dfrac 1 {x^2}$ | From [[Derivative of Natural Logarithm Function]]:
:$\dfrac \d {\d x} \ln x = \dfrac 1 x$
From the [[Power Rule for Derivatives/Integer Index|Power Rule for Derivatives: Integer Index]]:
:$\dfrac {\d^2} {\d x^2} \ln x = \dfrac \d {\d x} \dfrac 1 x = -\dfrac 1 {x^2}$
{{qed}} | Second Derivative of Natural Logarithm Function | https://proofwiki.org/wiki/Second_Derivative_of_Natural_Logarithm_Function | https://proofwiki.org/wiki/Second_Derivative_of_Natural_Logarithm_Function | [
"Derivatives involving Logarithm Functions",
"Natural Logarithms"
] | [
"Definition:Natural Logarithm"
] | [
"Derivative of Natural Logarithm Function",
"Power Rule for Derivatives/Integer Index"
] |
proofwiki-4354 | Exponential is Strictly Increasing | :The function $\map f x = \exp x$ is strictly increasing. | By definition, the exponential function is the inverse of the natural logarithm function.
From Logarithm is Strictly Increasing, $\ln x$ is strictly increasing.
The result follows from Inverse of Strictly Monotone Function.
{{qed}} | :The [[Definition:Real Function|function]] $\map f x = \exp x$ is [[Definition:Strictly Increasing Real Function|strictly increasing]]. | By definition, the [[Definition:Real Exponential Function|exponential function]] is the [[Definition:Inverse Mapping|inverse]] of the [[Definition:Natural Logarithm|natural logarithm function]].
From [[Logarithm is Strictly Increasing]], $\ln x$ is [[Definition:Strictly Increasing Real Function|strictly increasing]].
... | Exponential is Strictly Increasing/Proof 1 | https://proofwiki.org/wiki/Exponential_is_Strictly_Increasing | https://proofwiki.org/wiki/Exponential_is_Strictly_Increasing/Proof_1 | [
"Exponential is Strictly Increasing",
"Exponential Function",
"Examples of Strictly Increasing Real Functions"
] | [
"Definition:Real Function",
"Definition:Strictly Increasing/Real Function"
] | [
"Definition:Exponential Function/Real",
"Definition:Inverse Mapping",
"Definition:Natural Logarithm",
"Logarithm is Strictly Increasing",
"Definition:Strictly Increasing/Real Function",
"Inverse of Strictly Monotone Function"
] |
proofwiki-4355 | Exponential is Strictly Increasing | :The function $\map f x = \exp x$ is strictly increasing. | For all $x \in \R$:
{{begin-eqn}}
{{eqn | l = D_x \exp x
| r = \exp x
| c = Derivative of Exponential Function
}}
{{eqn | o = >
| r = 0
| c = Exponential of Real Number is Strictly Positive
}}
{{end-eqn}}
Hence the result, from Derivative of Monotone Function.
{{qed}} | :The [[Definition:Real Function|function]] $\map f x = \exp x$ is [[Definition:Strictly Increasing Real Function|strictly increasing]]. | For all $x \in \R$:
{{begin-eqn}}
{{eqn | l = D_x \exp x
| r = \exp x
| c = [[Derivative of Exponential Function]]
}}
{{eqn | o = >
| r = 0
| c = [[Exponential of Real Number is Strictly Positive]]
}}
{{end-eqn}}
Hence the result, from [[Derivative of Monotone Function]].
{{qed}} | Exponential is Strictly Increasing/Proof 2 | https://proofwiki.org/wiki/Exponential_is_Strictly_Increasing | https://proofwiki.org/wiki/Exponential_is_Strictly_Increasing/Proof_2 | [
"Exponential is Strictly Increasing",
"Exponential Function",
"Examples of Strictly Increasing Real Functions"
] | [
"Definition:Real Function",
"Definition:Strictly Increasing/Real Function"
] | [
"Derivative of Exponential Function",
"Exponential of Real Number is Strictly Positive",
"Derivative of Monotone Function"
] |
proofwiki-4356 | Exponential Tends to Zero and Infinity | :$\exp x \to +\infty$ as $x \to +\infty$
:$\exp x \to 0$ as $x \to -\infty$
Thus the exponential function has domain $\R$ and image $\openint 0 \infty$. | === The exponential function approaches positive infinity as x approaches positive infinity ===
Let $M$ be a strictly positive real number.
Let $N$ be $\ln M$.
$N$ is real because $M > 0$.
From Exponential is Strictly Increasing:
:$\forall x: x > N \implies \exp x > \exp N = M$
Therefore:
:$\forall M \in \R_{>0} : \exi... | :$\exp x \to +\infty$ as $x \to +\infty$
:$\exp x \to 0$ as $x \to -\infty$
Thus the [[Definition:Real Exponential Function|exponential function]] has [[Definition:Domain of Mapping|domain]] $\R$ and [[Definition:Image of Mapping|image]] $\openint 0 \infty$. | === The exponential function approaches positive infinity as x approaches positive infinity ===
Let $M$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $N$ be $\ln M$.
$N$ is real because $M > 0$.
From [[Exponential is Strictly Increasing]]:
:$\forall x: x > N \implies \exp x > ... | Exponential Tends to Zero and Infinity | https://proofwiki.org/wiki/Exponential_Tends_to_Zero_and_Infinity | https://proofwiki.org/wiki/Exponential_Tends_to_Zero_and_Infinity | [
"Exponential Function"
] | [
"Definition:Exponential Function/Real",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Definition:Strictly Positive/Real Number",
"Exponential is Strictly Increasing",
"Definition:Limit of Real Function/Limit at Infinity/Positive/Increasing Without Bound",
"Definition:Strictly Positive/Real Number",
"Exponential is Strictly Increasing",
"Exponential is Strictly Increasing"
] |
proofwiki-4357 | Exponential of Natural Logarithm | : $\forall x > 0: \map \exp {\ln x} = x$
: $\forall x \in \R: \map \ln {\exp x} = x$ | From the definition of the exponential function:
:$e^y = x \iff \ln x = y$
Raising both sides of the equation $\ln x = y$ to the power of $e$:
{{begin-eqn}}
{{eqn | l = e^{\ln x}
| r = e^y
| c =
}}
{{eqn | r = x
| c =
}}
{{end-eqn}}
{{qed}}
Category:Exponential Function
Category:Natural Logarithms
l... | : $\forall x > 0: \map \exp {\ln x} = x$
: $\forall x \in \R: \map \ln {\exp x} = x$ | From the definition of the [[Definition:Real Exponential Function|exponential function]]:
:$e^y = x \iff \ln x = y$
Raising both sides of the equation $\ln x = y$ to the [[Definition:Power to Real Number|power]] of $e$:
{{begin-eqn}}
{{eqn | l = e^{\ln x}
| r = e^y
| c =
}}
{{eqn | r = x
| c =
}}
... | Exponential of Natural Logarithm | https://proofwiki.org/wiki/Exponential_of_Natural_Logarithm | https://proofwiki.org/wiki/Exponential_of_Natural_Logarithm | [
"Exponential Function",
"Natural Logarithms"
] | [] | [
"Definition:Exponential Function/Real",
"Definition:Power (Algebra)/Real Number",
"Category:Exponential Function",
"Category:Natural Logarithms"
] |
proofwiki-4358 | Exponent Combination Laws/Product of Powers | :$a^x a^y = a^{x + y}$ | {{begin-eqn}}
{{eqn | l = a^{x + y}
| r = \map \exp {\paren {x + y} \ln a}
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {x \ln a + y \ln a}
| c =
}}
{{eqn | r = \map \exp {x \ln a} \, \map \exp {y \ln a}
| c = Exponential of Sum
}}
{{eqn | r = a^x a^y
| c = {{Defof|Power ... | :$a^x a^y = a^{x + y}$ | {{begin-eqn}}
{{eqn | l = a^{x + y}
| r = \map \exp {\paren {x + y} \ln a}
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {x \ln a + y \ln a}
| c =
}}
{{eqn | r = \map \exp {x \ln a} \, \map \exp {y \ln a}
| c = [[Exponential of Sum]]
}}
{{eqn | r = a^x a^y
| c = {{Defof|Po... | Exponent Combination Laws/Product of Powers/Proof 1 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Product_of_Powers | https://proofwiki.org/wiki/Exponent_Combination_Laws/Product_of_Powers/Proof_1 | [
"Exponent Combination Laws"
] | [] | [
"Exponential of Sum"
] |
proofwiki-4359 | Exponent Combination Laws/Product of Powers | :$a^x a^y = a^{x + y}$ | Let $x, y \in \R$.
From Rational Sequence Decreasing to Real Number, there exist rational sequences $\sequence {x_n}$ and $\sequence {y_n}$ converging to $x$ and $y$, respectively.
Then, since Power Function on Strictly Positive Base is Continuous: Real Power:
{{begin-eqn}}
{{eqn | l = a^{x + y}
| r = a^{\ds \par... | :$a^x a^y = a^{x + y}$ | Let $x, y \in \R$.
From [[Rational Sequence Decreasing to Real Number]], there exist [[Definition:Rational Sequence|rational sequences]] $\sequence {x_n}$ and $\sequence {y_n}$ [[Definition:Convergent Real Sequence|converging]] to $x$ and $y$, respectively.
Then, since [[Power Function on Strictly Positive Base is ... | Exponent Combination Laws/Product of Powers/Proof 2 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Product_of_Powers | https://proofwiki.org/wiki/Exponent_Combination_Laws/Product_of_Powers/Proof_2 | [
"Exponent Combination Laws"
] | [] | [
"Rational Sequence Decreasing to Real Number",
"Definition:Rational Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Power Function on Strictly Positive Base is Continuous/Real Power",
"Combination Theorem for Sequences/Real/Sum Rule",
"Sequential Continuity is Equivalent to Continuity in the Re... |
proofwiki-4360 | Exponent Combination Laws/Power of Product | :$\paren {a b}^x = a^x b^x$ | {{begin-eqn}}
{{eqn | l = \paren {a b}^x
| r = \map \exp {x \map \ln {a b} }
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {x \ln a + x \ln b}
| c = Sum of Logarithms
}}
{{eqn | r = \map \exp {x \ln a} \map \exp {x \ln b}
| c = Exponential of Sum
}}
{{eqn | r = a^x b^x
| c ... | :$\paren {a b}^x = a^x b^x$ | {{begin-eqn}}
{{eqn | l = \paren {a b}^x
| r = \map \exp {x \map \ln {a b} }
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {x \ln a + x \ln b}
| c = [[Sum of Logarithms]]
}}
{{eqn | r = \map \exp {x \ln a} \map \exp {x \ln b}
| c = [[Exponential of Sum]]
}}
{{eqn | r = a^x b^x
... | Exponent Combination Laws/Power of Product | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Product | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Product | [
"Exponent Combination Laws"
] | [] | [
"Sum of Logarithms",
"Exponential of Sum"
] |
proofwiki-4361 | Exponent Combination Laws/Negative Power | : $a^{-x} = \dfrac 1 {a^x}$ | {{begin-eqn}}
{{eqn | l = a^{-x}
| r = \map \exp {-x \ln a}
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \paren {\map \exp {x \ln a} }^{-1}
| c = Exponential of Product
}}
{{eqn | r = \frac 1 {\map \exp {x \ln a} }
| c =
}}
{{eqn | r = \frac 1 {a^x}
| c = {{Defof|Power to Real Numb... | : $a^{-x} = \dfrac 1 {a^x}$ | {{begin-eqn}}
{{eqn | l = a^{-x}
| r = \map \exp {-x \ln a}
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \paren {\map \exp {x \ln a} }^{-1}
| c = [[Exponential of Product]]
}}
{{eqn | r = \frac 1 {\map \exp {x \ln a} }
| c =
}}
{{eqn | r = \frac 1 {a^x}
| c = {{Defof|Power to Real ... | Exponent Combination Laws/Negative Power | https://proofwiki.org/wiki/Exponent_Combination_Laws/Negative_Power | https://proofwiki.org/wiki/Exponent_Combination_Laws/Negative_Power | [
"Exponent Combination Laws"
] | [] | [
"Exponential of Product"
] |
proofwiki-4362 | Exponent Combination Laws/Power of Power | Let $x, y \in \R$ be real numbers.
Let $a^x$ be defined as $a$ to the power of $x$.
Then:
:$\paren {a^x}^y = a^{x y}$ | {{begin-eqn}}
{{eqn | l = a^{x y}
| r = \map \exp {x y \ln a}
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {y \, \map \ln {a^x} }
| c = Logarithms of Powers
}}
{{eqn | r = \paren {a^x}^y
| c = {{Defof|Power to Real Number}}
}}
{{end-eqn}}
{{qed}} | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $a^x$ be defined as [[Definition:Power to Real Number|$a$ to the power of $x$]].
Then:
:$\paren {a^x}^y = a^{x y}$ | {{begin-eqn}}
{{eqn | l = a^{x y}
| r = \map \exp {x y \ln a}
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {y \, \map \ln {a^x} }
| c = [[Logarithms of Powers]]
}}
{{eqn | r = \paren {a^x}^y
| c = {{Defof|Power to Real Number}}
}}
{{end-eqn}}
{{qed}} | Exponent Combination Laws/Power of Power/Proof 1 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Power | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Power/Proof_1 | [
"Exponent Combination Laws"
] | [
"Definition:Real Number",
"Definition:Power (Algebra)/Real Number"
] | [
"Logarithm of Power"
] |
proofwiki-4363 | Exponent Combination Laws/Power of Power | Let $x, y \in \R$ be real numbers.
Let $a^x$ be defined as $a$ to the power of $x$.
Then:
:$\paren {a^x}^y = a^{x y}$ | We will show that:
:$\forall \epsilon \in \R_{>0}: \size {a^{x y} - \paren {a^x}^y} < \epsilon$
{{WLOG}}, suppose that $x < y$.
Consider $I := \closedint x y$.
Let $I_\Q = I \cap \Q$.
Let $M = \max \set {\size x, \size y}$
Fix $\epsilon \in \R_{>0}$.
From Real Polynomial Function is Continuous:
:$\exists \delta' \in \R... | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $a^x$ be defined as [[Definition:Power to Real Number|$a$ to the power of $x$]].
Then:
:$\paren {a^x}^y = a^{x y}$ | We will show that:
:$\forall \epsilon \in \R_{>0}: \size {a^{x y} - \paren {a^x}^y} < \epsilon$
{{WLOG}}, suppose that $x < y$.
Consider $I := \closedint x y$.
Let $I_\Q = I \cap \Q$.
Let $M = \max \set {\size x, \size y}$
Fix $\epsilon \in \R_{>0}$.
From [[Real Polynomial Function is Continuous]]:
:$\exists \del... | Exponent Combination Laws/Power of Power/Proof 2 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Power | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Power/Proof_2 | [
"Exponent Combination Laws"
] | [
"Definition:Real Number",
"Definition:Power (Algebra)/Real Number"
] | [
"Real Polynomial Function is Continuous",
"Power Function on Strictly Positive Base is Continuous",
"Absolute Value Function is Completely Multiplicative",
"Absolute Value Function is Completely Multiplicative",
"Real Number Ordering is Compatible with Multiplication",
"Closure of Rational Interval is Clo... |
proofwiki-4364 | Exponent Combination Laws/Quotient of Powers | :$\dfrac{a^x} {a^y} = a^{x - y}$ | {{begin-eqn}}
{{eqn | l = \frac {a^x} {a^y}
| r = a^x \paren {\frac 1 {a^y} }
| c =
}}
{{eqn | r = \paren {a^x} \paren {a^{-y} }
| c = Exponent Combination Laws: Negative Power
}}
{{eqn | r = a^{x - y}
| c = Product of Powers
}}
{{end-eqn}}
{{qed}} | :$\dfrac{a^x} {a^y} = a^{x - y}$ | {{begin-eqn}}
{{eqn | l = \frac {a^x} {a^y}
| r = a^x \paren {\frac 1 {a^y} }
| c =
}}
{{eqn | r = \paren {a^x} \paren {a^{-y} }
| c = [[Exponent Combination Laws/Negative Power|Exponent Combination Laws: Negative Power]]
}}
{{eqn | r = a^{x - y}
| c = [[Product of Powers]]
}}
{{end-eqn}}
{{qed... | Exponent Combination Laws/Quotient of Powers | https://proofwiki.org/wiki/Exponent_Combination_Laws/Quotient_of_Powers | https://proofwiki.org/wiki/Exponent_Combination_Laws/Quotient_of_Powers | [
"Exponential Function"
] | [] | [
"Exponent Combination Laws/Negative Power",
"Exponent Combination Laws/Product of Powers"
] |
proofwiki-4365 | Exponent Combination Laws/Power of Quotient | :$\paren {\dfrac a b}^x = \dfrac {a^x} {b^x}$ | {{begin-eqn}}
{{eqn | l = \paren {\frac a b}^x
| r = \map \exp {x \, \map \ln {\frac a b} }
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {x \ln a - x \ln b}
| c = Sum of Logarithms
}}
{{eqn | r = \frac {\map \exp {x \ln a} } {\map \exp {x \ln b} }
| c = Exponential of Sum
}}
{{e... | :$\paren {\dfrac a b}^x = \dfrac {a^x} {b^x}$ | {{begin-eqn}}
{{eqn | l = \paren {\frac a b}^x
| r = \map \exp {x \, \map \ln {\frac a b} }
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {x \ln a - x \ln b}
| c = [[Sum of Logarithms]]
}}
{{eqn | r = \frac {\map \exp {x \ln a} } {\map \exp {x \ln b} }
| c = [[Exponential of Sum]... | Exponent Combination Laws/Power of Quotient | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Quotient | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Quotient | [
"Exponent Combination Laws"
] | [] | [
"Sum of Logarithms",
"Exponential of Sum",
"Category:Exponent Combination Laws"
] |
proofwiki-4366 | Exponent Combination Laws/Negative Power of Quotient | :$\paren {\dfrac a b}^{-x} = \paren {\dfrac b a}^x$ | {{begin-eqn}}
{{eqn | l = \paren {\frac a b} ^{-x}
| r = \paren {\frac 1 {\paren {\frac a b} } }^x
| c = Exponent Combination Laws: Negative Power
}}
{{eqn | r = \paren {\frac b a}^x
| c =
}}
{{end-eqn}}
{{qed}}
Category:Exponent Combination Laws
obg84cper7fdgxo5j5vlbmc7opho5sd | :$\paren {\dfrac a b}^{-x} = \paren {\dfrac b a}^x$ | {{begin-eqn}}
{{eqn | l = \paren {\frac a b} ^{-x}
| r = \paren {\frac 1 {\paren {\frac a b} } }^x
| c = [[Exponent Combination Laws/Negative Power|Exponent Combination Laws: Negative Power]]
}}
{{eqn | r = \paren {\frac b a}^x
| c =
}}
{{end-eqn}}
{{qed}}
[[Category:Exponent Combination Laws]]
obg8... | Exponent Combination Laws/Negative Power of Quotient | https://proofwiki.org/wiki/Exponent_Combination_Laws/Negative_Power_of_Quotient | https://proofwiki.org/wiki/Exponent_Combination_Laws/Negative_Power_of_Quotient | [
"Exponent Combination Laws"
] | [] | [
"Exponent Combination Laws/Negative Power",
"Category:Exponent Combination Laws"
] |
proofwiki-4367 | Derivative of General Exponential Function | Let $a \in \R_{>0}$.
Let $a^x$ be $a$ to the power of $x$.
Then:
:$\map {\dfrac \d {\d x} } {a^x} = a^x \ln a$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {a^x}
| r = \map {\frac \d {\d x} } {e^{x \ln a} }
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \ln a e^{x \ln a}
| c = Derivative of $e^{a x}$
}}
{{eqn | r = a^x \ln a
| c =
}}
{{end-eqn}}
{{qed}} | Let $a \in \R_{>0}$.
Let $a^x$ be $a$ to the [[Definition:Power to Real Number|power]] of $x$.
Then:
:$\map {\dfrac \d {\d x} } {a^x} = a^x \ln a$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {a^x}
| r = \map {\frac \d {\d x} } {e^{x \ln a} }
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \ln a e^{x \ln a}
| c = [[Derivative of Exponential of a x|Derivative of $e^{a x}$]]
}}
{{eqn | r = a^x \ln a
| c =
}}
{{end-eqn}}
{{qed}} | Derivative of General Exponential Function/Proof 1 | https://proofwiki.org/wiki/Derivative_of_General_Exponential_Function | https://proofwiki.org/wiki/Derivative_of_General_Exponential_Function/Proof_1 | [
"Derivative of General Exponential Function",
"Derivatives involving Exponential Function"
] | [
"Definition:Power (Algebra)/Real Number"
] | [
"Derivative of Exponential Function/Corollary 1"
] |
proofwiki-4368 | Derivative of General Exponential Function | Let $a \in \R_{>0}$.
Let $a^x$ be $a$ to the power of $x$.
Then:
:$\map {\dfrac \d {\d x} } {a^x} = a^x \ln a$ | {{begin-eqn}}
{{eqn | l = \lim_{h \mathop \to 0} \frac {a^{x + h} - a^x} h
| r = a^x \lim_{h \mathop \to 0} \frac {a^h - 1} h
| c = Product of Powers
}}
{{eqn | r = a^x \lim_{h \mathop \to 0} \frac {\map \exp {h \ln a} - 1} h
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = a^x \lim_{h \mathop \to ... | Let $a \in \R_{>0}$.
Let $a^x$ be $a$ to the [[Definition:Power to Real Number|power]] of $x$.
Then:
:$\map {\dfrac \d {\d x} } {a^x} = a^x \ln a$ | {{begin-eqn}}
{{eqn | l = \lim_{h \mathop \to 0} \frac {a^{x + h} - a^x} h
| r = a^x \lim_{h \mathop \to 0} \frac {a^h - 1} h
| c = [[Product of Powers]]
}}
{{eqn | r = a^x \lim_{h \mathop \to 0} \frac {\map \exp {h \ln a} - 1} h
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = a^x \lim_{h \mathop ... | Derivative of General Exponential Function/Proof 2 | https://proofwiki.org/wiki/Derivative_of_General_Exponential_Function | https://proofwiki.org/wiki/Derivative_of_General_Exponential_Function/Proof_2 | [
"Derivative of General Exponential Function",
"Derivatives involving Exponential Function"
] | [
"Definition:Power (Algebra)/Real Number"
] | [
"Exponent Combination Laws/Product of Powers",
"Derivative of Exponential at Zero"
] |
proofwiki-4369 | Derivative of General Exponential Function | Let $a \in \R_{>0}$.
Let $a^x$ be $a$ to the power of $x$.
Then:
:$\map {\dfrac \d {\d x} } {a^x} = a^x \ln a$ | Let $y = a^x$.
Then:
{{begin-eqn}}
{{eqn | l = \ln y
| r = x \ln a
| c = Logarithm of Power
}}
{{eqn | ll= \leadsto
| l = \dfrac 1 y \dfrac {\d y} {\d x}
| r = \ln a
| c = Derivative of Identity Function: Corollary
}}
{{eqn | ll= \leadsto
| l = \dfrac 1 {a^x} \dfrac {\d y} {\d x}
... | Let $a \in \R_{>0}$.
Let $a^x$ be $a$ to the [[Definition:Power to Real Number|power]] of $x$.
Then:
:$\map {\dfrac \d {\d x} } {a^x} = a^x \ln a$ | Let $y = a^x$.
Then:
{{begin-eqn}}
{{eqn | l = \ln y
| r = x \ln a
| c = [[Logarithm of Power]]
}}
{{eqn | ll= \leadsto
| l = \dfrac 1 y \dfrac {\d y} {\d x}
| r = \ln a
| c = [[Derivative of Identity Function/Corollary|Derivative of Identity Function: Corollary]]
}}
{{eqn | ll= \leadsto
... | Derivative of General Exponential Function/Proof 3 | https://proofwiki.org/wiki/Derivative_of_General_Exponential_Function | https://proofwiki.org/wiki/Derivative_of_General_Exponential_Function/Proof_3 | [
"Derivative of General Exponential Function",
"Derivatives involving Exponential Function"
] | [
"Definition:Power (Algebra)/Real Number"
] | [
"Logarithm of Power",
"Derivative of Identity Function/Corollary"
] |
proofwiki-4370 | Binomial Theorem/General Binomial Theorem | Let $\alpha \in \R$ be a real number.
Let $x \in \R$ be a real number such that $\size x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {1 + x}^\alpha
| r = \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n
| c =
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \dbinom \alpha n x^n
| c =
}... | Let $R$ be the radius of convergence of the power series:
:$\ds \map f x = \sum_{n \mathop = 0}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {n!} x^n$
Then:
{{begin-eqn}}
{{eqn | l = \frac 1 R
| r = \lim_{n \mathop \to \infty} \frac {\size {\alpha \paren {\alpha - 1} \dotsm \paren {\a... | Let $\alpha \in \R$ be a [[Definition:Real Number|real number]].
Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $\size x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {1 + x}^\alpha
| r = \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n
| c =
}}
{{eqn | r = \sum_... | Let $R$ be the [[Definition:Radius of Convergence|radius of convergence]] of the [[Definition:Power Series|power series]]:
:$\ds \map f x = \sum_{n \mathop = 0}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {n!} x^n$
Then:
{{begin-eqn}}
{{eqn | l = \frac 1 R
| r = \lim_{n \mathop \to... | Binomial Theorem/General Binomial Theorem/Proof 1 | https://proofwiki.org/wiki/Binomial_Theorem/General_Binomial_Theorem | https://proofwiki.org/wiki/Binomial_Theorem/General_Binomial_Theorem/Proof_1 | [
"Binomial Theorem"
] | [
"Definition:Real Number",
"Definition:Real Number",
"Definition:Falling Factorial",
"Definition:Binomial Coefficient/Real Numbers"
] | [
"Definition:Radius of Convergence",
"Definition:Power Series",
"Radius of Convergence from Limit of Sequence",
"Power Series is Differentiable on Interval of Convergence"
] |
proofwiki-4371 | Binomial Theorem/General Binomial Theorem | Let $\alpha \in \R$ be a real number.
Let $x \in \R$ be a real number such that $\size x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {1 + x}^\alpha
| r = \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n
| c =
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \dbinom \alpha n x^n
| c =
}... | From Sum over k of r-kt choose k by r over r-kt by z^k:
:$\ds \sum_n \dbinom {\alpha - n t} k \dfrac \alpha {\alpha - n t} z^n = x^\alpha$
where:
:$z = x^{t + 1} - x^t$
:$x = 1$ for $z = 0$.
Setting $t = 0$:
{{begin-eqn}}
{{eqn | l = \sum_k \dbinom {\alpha - n \times 0} n \dfrac \alpha {\alpha - n \times 0} z^n
|... | Let $\alpha \in \R$ be a [[Definition:Real Number|real number]].
Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $\size x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {1 + x}^\alpha
| r = \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n
| c =
}}
{{eqn | r = \sum_... | From [[Sum over k of r-kt choose k by r over r-kt by z^k]]:
:$\ds \sum_n \dbinom {\alpha - n t} k \dfrac \alpha {\alpha - n t} z^n = x^\alpha$
where:
:$z = x^{t + 1} - x^t$
:$x = 1$ for $z = 0$.
Setting $t = 0$:
{{begin-eqn}}
{{eqn | l = \sum_k \dbinom {\alpha - n \times 0} n \dfrac \alpha {\alpha - n \times 0} z^n... | Binomial Theorem/General Binomial Theorem/Proof 2 | https://proofwiki.org/wiki/Binomial_Theorem/General_Binomial_Theorem | https://proofwiki.org/wiki/Binomial_Theorem/General_Binomial_Theorem/Proof_2 | [
"Binomial Theorem"
] | [
"Definition:Real Number",
"Definition:Real Number",
"Definition:Falling Factorial",
"Definition:Binomial Coefficient/Real Numbers"
] | [
"Sum over k of r-kt choose k by r over r-kt by z^k"
] |
proofwiki-4372 | Binomial Theorem/General Binomial Theorem | Let $\alpha \in \R$ be a real number.
Let $x \in \R$ be a real number such that $\size x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {1 + x}^\alpha
| r = \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n
| c =
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \dbinom \alpha n x^n
| c =
}... | The series is the Maclaurin series expansion of the function $\map f x = \paren {1 + x}^\alpha$.
The derivatives of $f$ are:
{{begin-eqn}}
{{eqn | l = \map {f^{\paren 0} } x
| r = \paren {1 + x}^\alpha
| c =
}}
{{eqn | l = \map {f^{\paren 1} } x
| r = \alpha \paren {1 + x}^{\alpha - 1}
| c =
}... | Let $\alpha \in \R$ be a [[Definition:Real Number|real number]].
Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $\size x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {1 + x}^\alpha
| r = \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n
| c =
}}
{{eqn | r = \sum_... | The series is the [[Definition:Maclaurin Series|Maclaurin series expansion]] of the function $\map f x = \paren {1 + x}^\alpha$.
The [[Definition:Derivative|derivatives]] of $f$ are:
{{begin-eqn}}
{{eqn | l = \map {f^{\paren 0} } x
| r = \paren {1 + x}^\alpha
| c =
}}
{{eqn | l = \map {f^{\paren 1} } x
... | Binomial Theorem/General Binomial Theorem/Proof 3 | https://proofwiki.org/wiki/Binomial_Theorem/General_Binomial_Theorem | https://proofwiki.org/wiki/Binomial_Theorem/General_Binomial_Theorem/Proof_3 | [
"Binomial Theorem"
] | [
"Definition:Real Number",
"Definition:Real Number",
"Definition:Falling Factorial",
"Definition:Binomial Coefficient/Real Numbers"
] | [
"Definition:Maclaurin Series",
"Definition:Derivative",
"Definition:Maclaurin Series"
] |
proofwiki-4373 | Real Sine Function is Continuous | The real sine function $\sin: \R \to \R$ is continuous on $\R$. | Recall the definition of the sine function:
:$\ds \forall x \in \R: \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$
Thus $\sin x$ is expressible in the form of a power series.
From Sine Function is Absolutely Convergent, we ... | The [[Definition:Real Sine Function|real sine function]] $\sin: \R \to \R$ is [[Definition:Continuous on Interval|continuous]] on $\R$. | Recall the definition of the [[Definition:Real Sine Function|sine function]]:
:$\ds \forall x \in \R: \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$
Thus $\sin x$ is expressible in the form of a [[Definition:Power Series|... | Real Sine Function is Continuous | https://proofwiki.org/wiki/Real_Sine_Function_is_Continuous | https://proofwiki.org/wiki/Real_Sine_Function_is_Continuous | [
"Sine Function"
] | [
"Definition:Sine/Real Function",
"Definition:Continuous Real Function/Interval"
] | [
"Definition:Sine/Real Function",
"Definition:Power Series",
"Sine Function is Absolutely Convergent",
"Definition:Interval of Convergence",
"Power Series is Differentiable on Interval of Convergence",
"Definition:Continuous Real Function/Interval",
"Category:Sine Function"
] |
proofwiki-4374 | Sine Function is Absolutely Convergent | The real sine function $\sin: \R \to \R$ is absolutely convergent. | Recall the definition of the sine function:
:$\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$
For:
:$\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$
to be absolutely convergent we ... | The [[Definition:Real Sine Function|real sine function]] $\sin: \R \to \R$ is [[Definition:Absolutely Convergent Series|absolutely convergent]]. | Recall the definition of the [[Definition:Real Sine Function|sine function]]:
:$\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$
For:
:$\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + ... | Sine Function is Absolutely Convergent | https://proofwiki.org/wiki/Sine_Function_is_Absolutely_Convergent | https://proofwiki.org/wiki/Sine_Function_is_Absolutely_Convergent | [
"Sine Function",
"Absolute Convergence"
] | [
"Definition:Sine/Real Function",
"Definition:Absolutely Convergent Series"
] | [
"Definition:Sine/Real Function",
"Definition:Absolutely Convergent Series",
"Power Series Expansion for Exponential Function",
"Definition:Convergent Series",
"Squeeze Theorem"
] |
proofwiki-4375 | Sine Function is Absolutely Convergent | The real sine function $\sin: \R \to \R$ is absolutely convergent. | The definition of the complex sine function is:
:$\ds \forall z \in \C: \sin z = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}$
By definition of absolutely convergent complex series, we must show that the power series
:$\ds \sum_{n \mathop = 0}^\infty \size {\paren {-1}^n \frac {z^{... | The [[Definition:Real Sine Function|real sine function]] $\sin: \R \to \R$ is [[Definition:Absolutely Convergent Series|absolutely convergent]]. | The definition of the [[Definition:Complex Sine Function|complex sine function]] is:
:$\ds \forall z \in \C: \sin z = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}$
By [[Definition:Absolutely Convergent Complex Series|definition of absolutely convergent complex series]], we must s... | Sine Function is Absolutely Convergent/Complex Case/Proof 1 | https://proofwiki.org/wiki/Sine_Function_is_Absolutely_Convergent | https://proofwiki.org/wiki/Sine_Function_is_Absolutely_Convergent/Complex_Case/Proof_1 | [
"Sine Function",
"Absolute Convergence"
] | [
"Definition:Sine/Real Function",
"Definition:Absolutely Convergent Series"
] | [
"Definition:Sine/Complex Function",
"Definition:Absolutely Convergent Series/Complex Numbers",
"Definition:Power Series/Complex Domain",
"Squeeze Theorem/Sequences/Complex Numbers",
"Power Series Expansion for Exponential Function",
"Squeeze Theorem/Sequences/Complex Numbers"
] |
proofwiki-4376 | Sine Function is Absolutely Convergent | The real sine function $\sin: \R \to \R$ is absolutely convergent. | Radius of Convergence of Power Series Expansion for Sine Function shows that the radius of convergence of the complex sine function is infinite.
Then Existence of Radius of Convergence of Complex Power Series shows that the complex sine function is absolutely convergent.
{{qed}} | The [[Definition:Real Sine Function|real sine function]] $\sin: \R \to \R$ is [[Definition:Absolutely Convergent Series|absolutely convergent]]. | [[Radius of Convergence of Power Series Expansion for Sine Function]] shows that the [[Definition:Radius of Convergence of Complex Power Series|radius of convergence]] of the [[Definition:Complex Sine Function|complex sine function]] is infinite.
Then [[Existence of Radius of Convergence of Complex Power Series/Absolu... | Sine Function is Absolutely Convergent/Complex Case/Proof 2 | https://proofwiki.org/wiki/Sine_Function_is_Absolutely_Convergent | https://proofwiki.org/wiki/Sine_Function_is_Absolutely_Convergent/Complex_Case/Proof_2 | [
"Sine Function",
"Absolute Convergence"
] | [
"Definition:Sine/Real Function",
"Definition:Absolutely Convergent Series"
] | [
"Radius of Convergence of Power Series Expansion for Sine Function",
"Definition:Radius of Convergence/Complex Domain",
"Definition:Sine/Complex Function",
"Existence of Radius of Convergence of Complex Power Series/Absolute Convergence",
"Definition:Sine/Complex Function",
"Definition:Absolutely Converge... |
proofwiki-4377 | Sine of Zero is Zero | :$\sin 0 = 0$ | Recall the definition of the sine function:
:$\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$
Thus:
:$\ds \sin 0 = 0 - \frac {0^3} {3!} + \frac {0^5} {5!} - \cdots = 0$
{{qed}} | :$\sin 0 = 0$ | Recall the definition of the [[Definition:Sine Function|sine function]]:
:$\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$
Thus:
:$\ds \sin 0 = 0 - \frac {0^3} {3!} + \frac {0^5} {5!} - \cdots = 0$
{{qed}} | Sine of Zero is Zero | https://proofwiki.org/wiki/Sine_of_Zero_is_Zero | https://proofwiki.org/wiki/Sine_of_Zero_is_Zero | [
"Sine Function"
] | [] | [
"Definition:Sine"
] |
proofwiki-4378 | Sine Function is Odd | :$\map \sin {-z} = -\sin z$
That is, the sine function is odd. | {{begin-eqn}}
{{eqn | l = \map \sinh {-x}
| r = \frac {e^{-x} - e^{-\paren {-x} } } 2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac {e^{-x} - e^x} 2
}}
{{eqn | r = -\frac {e^x - e^{-x} } 2
}}
{{eqn | r = -\sinh x
}}
{{end-eqn}}
{{qed}} | :$\map \sin {-z} = -\sin z$
That is, the [[Definition:Complex Sine Function|sine function]] is [[Definition:Odd Function|odd]]. | {{begin-eqn}}
{{eqn | l = \map \sinh {-x}
| r = \frac {e^{-x} - e^{-\paren {-x} } } 2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac {e^{-x} - e^x} 2
}}
{{eqn | r = -\frac {e^x - e^{-x} } 2
}}
{{eqn | r = -\sinh x
}}
{{end-eqn}}
{{qed}} | Hyperbolic Sine Function is Odd/Proof 1 | https://proofwiki.org/wiki/Sine_Function_is_Odd | https://proofwiki.org/wiki/Hyperbolic_Sine_Function_is_Odd/Proof_1 | [
"Sine Function is Odd",
"Sine Function",
"Examples of Odd Functions"
] | [
"Definition:Sine/Complex Function",
"Definition:Odd Function"
] | [] |
proofwiki-4379 | Sine Function is Odd | :$\map \sin {-z} = -\sin z$
That is, the sine function is odd. | {{begin-eqn}}
{{eqn | l = \map \sinh {-x}
| r = -i \, \map \sin {-i x}
| c = Hyperbolic Sine in terms of Sine
}}
{{eqn | r = i \, \map \sin {i x}
| c = Sine Function is Odd
}}
{{eqn | r = -\sinh x
| c = Hyperbolic Sine in terms of Sine
}}
{{end-eqn}}
{{qed}} | :$\map \sin {-z} = -\sin z$
That is, the [[Definition:Complex Sine Function|sine function]] is [[Definition:Odd Function|odd]]. | {{begin-eqn}}
{{eqn | l = \map \sinh {-x}
| r = -i \, \map \sin {-i x}
| c = [[Hyperbolic Sine in terms of Sine]]
}}
{{eqn | r = i \, \map \sin {i x}
| c = [[Sine Function is Odd]]
}}
{{eqn | r = -\sinh x
| c = [[Hyperbolic Sine in terms of Sine]]
}}
{{end-eqn}}
{{qed}} | Hyperbolic Sine Function is Odd/Proof 2 | https://proofwiki.org/wiki/Sine_Function_is_Odd | https://proofwiki.org/wiki/Hyperbolic_Sine_Function_is_Odd/Proof_2 | [
"Sine Function is Odd",
"Sine Function",
"Examples of Odd Functions"
] | [
"Definition:Sine/Complex Function",
"Definition:Odd Function"
] | [
"Hyperbolic Sine in terms of Sine",
"Sine Function is Odd",
"Hyperbolic Sine in terms of Sine"
] |
proofwiki-4380 | Sine Function is Odd | :$\map \sin {-z} = -\sin z$
That is, the sine function is odd. | Recall the definition of the sine function:
{{begin-eqn}}
{{eqn | l = \sin z
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}
}}
{{eqn | r = z - \frac {z^3} {3!} + \frac {z^5} {5!} - \cdots
}}
{{end-eqn}}
From Sign of Odd Power, we have that:
:$\forall n \in \N: -\paren {z^... | :$\map \sin {-z} = -\sin z$
That is, the [[Definition:Complex Sine Function|sine function]] is [[Definition:Odd Function|odd]]. | Recall the definition of the [[Definition:Complex Sine Function|sine function]]:
{{begin-eqn}}
{{eqn | l = \sin z
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}
}}
{{eqn | r = z - \frac {z^3} {3!} + \frac {z^5} {5!} - \cdots
}}
{{end-eqn}}
From [[Sign of Odd Power]], w... | Sine Function is Odd/Proof 1 | https://proofwiki.org/wiki/Sine_Function_is_Odd | https://proofwiki.org/wiki/Sine_Function_is_Odd/Proof_1 | [
"Sine Function is Odd",
"Sine Function",
"Examples of Odd Functions"
] | [
"Definition:Sine/Complex Function",
"Definition:Odd Function"
] | [
"Definition:Sine/Complex Function",
"Sign of Odd Power"
] |
proofwiki-4381 | Sine Function is Odd | :$\map \sin {-z} = -\sin z$
That is, the sine function is odd. | Using the Sine of Difference formula:
{{begin-eqn}}
{{eqn | l = \map \sin {-b}
| r = \map \sin {0 - b}
}}
{{eqn | r = \sin 0 \cos b - \cos 0 \sin b
}}
{{eqn | r = -\sin b
}}
{{end-eqn}}
{{qed}} | :$\map \sin {-z} = -\sin z$
That is, the [[Definition:Complex Sine Function|sine function]] is [[Definition:Odd Function|odd]]. | Using the [[Sine of Difference]] formula:
{{begin-eqn}}
{{eqn | l = \map \sin {-b}
| r = \map \sin {0 - b}
}}
{{eqn | r = \sin 0 \cos b - \cos 0 \sin b
}}
{{eqn | r = -\sin b
}}
{{end-eqn}}
{{qed}} | Sine Function is Odd/Proof 2 | https://proofwiki.org/wiki/Sine_Function_is_Odd | https://proofwiki.org/wiki/Sine_Function_is_Odd/Proof_2 | [
"Sine Function is Odd",
"Sine Function",
"Examples of Odd Functions"
] | [
"Definition:Sine/Complex Function",
"Definition:Odd Function"
] | [
"Sine of Difference"
] |
proofwiki-4382 | Cosine Function is Continuous | :$\cos x$ is continuous on $\R$. | Recall the definition of the cosine function:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$
Thus $\cos x$ is expressible in the form of a power series.
From Cosine Function is Absolutely Convergent, we have that the interval... | :$\cos x$ is [[Definition:Continuous on Interval|continuous]] on $\R$. | Recall the definition of the [[Definition:Real Cosine Function|cosine function]]:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$
Thus $\cos x$ is expressible in the form of a [[Definition:Power Series|power series]].
From ... | Cosine Function is Continuous | https://proofwiki.org/wiki/Cosine_Function_is_Continuous | https://proofwiki.org/wiki/Cosine_Function_is_Continuous | [
"Cosine Function"
] | [
"Definition:Continuous Real Function/Interval"
] | [
"Definition:Cosine/Real Function",
"Definition:Power Series",
"Cosine Function is Absolutely Convergent",
"Definition:Interval of Convergence",
"Power Series is Differentiable on Interval of Convergence",
"Definition:Continuous Real Function/Interval",
"Category:Cosine Function"
] |
proofwiki-4383 | Cosine Function is Absolutely Convergent | :$\cos x$ is absolutely convergent for all $x \in \R$. | Recall the definition of the cosine function:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$
For:
:$\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$
to be absolutely convergent, we want:
:$\ds ... | :$\cos x$ is [[Definition:Absolutely Convergent Series|absolutely convergent]] for all $x \in \R$. | Recall the definition of the [[Definition:Real Cosine Function|cosine function]]:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$
For:
:$\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$
to be... | Cosine Function is Absolutely Convergent | https://proofwiki.org/wiki/Cosine_Function_is_Absolutely_Convergent | https://proofwiki.org/wiki/Cosine_Function_is_Absolutely_Convergent | [
"Cosine Function is Absolutely Convergent",
"Cosine Function"
] | [
"Definition:Absolutely Convergent Series"
] | [
"Definition:Cosine/Real Function",
"Definition:Absolutely Convergent Series",
"Definition:Convergent Series/Number Field",
"Power Series Expansion for Exponential Function",
"Definition:Convergent Series",
"Definition:Sequence",
"Definition:Increasing/Sequence/Real Sequence",
"Monotone Convergence The... |
proofwiki-4384 | Cosine Function is Absolutely Convergent | :$\cos x$ is absolutely convergent for all $x \in \R$. | The definition of the complex cosine function is:
:$\ds \cos z = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}$
By definition of absolutely convergent complex series, we must show that the power series
:$\ds \sum_{n \mathop = 0}^\infty \size {\paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} }... | :$\cos x$ is [[Definition:Absolutely Convergent Series|absolutely convergent]] for all $x \in \R$. | The definition of the [[Definition:Complex Cosine Function|complex cosine function]] is:
:$\ds \cos z = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}$
By definition of [[Definition:Absolutely Convergent Complex Series|absolutely convergent complex series]], we must show that the [[Definit... | Cosine Function is Absolutely Convergent/Complex Case/Proof 1 | https://proofwiki.org/wiki/Cosine_Function_is_Absolutely_Convergent | https://proofwiki.org/wiki/Cosine_Function_is_Absolutely_Convergent/Complex_Case/Proof_1 | [
"Cosine Function is Absolutely Convergent",
"Cosine Function"
] | [
"Definition:Absolutely Convergent Series"
] | [
"Definition:Cosine/Complex Function",
"Definition:Absolutely Convergent Series/Complex Numbers",
"Definition:Power Series/Complex Domain",
"Squeeze Theorem/Sequences/Complex Numbers",
"Power Series Expansion for Exponential Function",
"Squeeze Theorem/Sequences/Complex Numbers"
] |
proofwiki-4385 | Cosine Function is Absolutely Convergent | :$\cos x$ is absolutely convergent for all $x \in \R$. | Radius of Convergence of Power Series Expansion for Cosine Function shows that the radius of convergence of the complex cosine function is infinite.
Then Existence of Radius of Convergence of Complex Power Series shows that the complex cosine function is absolutely convergent.
{{qed}} | :$\cos x$ is [[Definition:Absolutely Convergent Series|absolutely convergent]] for all $x \in \R$. | [[Radius of Convergence of Power Series Expansion for Cosine Function]] shows that the [[Definition:Radius of Convergence of Complex Power Series|radius of convergence]] of the [[Definition:Complex Cosine Function|complex cosine function]] is infinite.
Then [[Existence of Radius of Convergence of Complex Power Series/... | Cosine Function is Absolutely Convergent/Complex Case/Proof 2 | https://proofwiki.org/wiki/Cosine_Function_is_Absolutely_Convergent | https://proofwiki.org/wiki/Cosine_Function_is_Absolutely_Convergent/Complex_Case/Proof_2 | [
"Cosine Function is Absolutely Convergent",
"Cosine Function"
] | [
"Definition:Absolutely Convergent Series"
] | [
"Radius of Convergence of Power Series Expansion for Cosine Function",
"Definition:Radius of Convergence/Complex Domain",
"Definition:Cosine/Complex Function",
"Existence of Radius of Convergence of Complex Power Series/Absolute Convergence",
"Definition:Absolutely Convergent Series/Complex Numbers"
] |
proofwiki-4386 | Cosine of Zero is One | :$\cos 0 = 1$ | Recall the definition of the cosine function:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$
Thus:
:$\cos 0 = 1 - \dfrac {0^2} {2!} + \dfrac {0^4} {4!} - \cdots = 1$
{{qed}} | :$\cos 0 = 1$ | Recall the definition of the [[Definition:Complex Cosine Function|cosine function]]:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$
Thus:
:$\cos 0 = 1 - \dfrac {0^2} {2!} + \dfrac {0^4} {4!} - \cdots = 1$
{{qed}} | Cosine of Zero is One | https://proofwiki.org/wiki/Cosine_of_Zero_is_One | https://proofwiki.org/wiki/Cosine_of_Zero_is_One | [
"Cosine Function"
] | [] | [
"Definition:Cosine/Complex Function"
] |
proofwiki-4387 | Cosine of Integer Multiple of Pi | :$\forall n \in \Z: \cos n \pi = \paren {-1}^n$ | Recall the definition of the cosine function:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$
From Cosine of Zero is One, we have that:
:$\cos 0 = 1$
This takes care of the case $n = 0$.
From Sine and Cosine are Periodic on Re... | :$\forall n \in \Z: \cos n \pi = \paren {-1}^n$ | Recall the definition of the [[Definition:Real Cosine Function|cosine function]]:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$
From [[Cosine of Zero is One]], we have that:
:$\cos 0 = 1$
This takes care of the case $n ... | Cosine of Integer Multiple of Pi | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Pi | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Pi | [
"Cosine Function"
] | [] | [
"Definition:Cosine/Real Function",
"Cosine of Zero is One",
"Sine and Cosine are Periodic on Reals",
"Sine and Cosine are Periodic on Reals",
"Definition:Even Integer",
"Definition:Odd Integer"
] |
proofwiki-4388 | Cosine Function is Even | :$\map \cos {-z} = \cos z$
That is, the cosine function is even. | Recall the definition of the cosine function:
{{begin-eqn}}
{{eqn | l = \cos z
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}
}}
{{eqn | r = 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \cdots
}}
{{end-eqn}}
From Even Power is Non-Negative:
:$\forall n \in \N: z^{2 n} = \paren {-z}^... | :$\map \cos {-z} = \cos z$
That is, the [[Definition:Complex Cosine Function|cosine function]] is [[Definition:Even Function|even]]. | Recall the definition of the [[Definition:Real Cosine Function|cosine function]]:
{{begin-eqn}}
{{eqn | l = \cos z
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}
}}
{{eqn | r = 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \cdots
}}
{{end-eqn}}
From [[Even Power is Non-Negative]]:... | Cosine Function is Even/Proof 1 | https://proofwiki.org/wiki/Cosine_Function_is_Even | https://proofwiki.org/wiki/Cosine_Function_is_Even/Proof_1 | [
"Cosine Function is Even",
"Cosine Function",
"Examples of Even Functions"
] | [
"Definition:Cosine/Complex Function",
"Definition:Even Function"
] | [
"Definition:Cosine/Real Function",
"Even Power is Non-Negative"
] |
proofwiki-4389 | Cosine Function is Even | :$\map \cos {-z} = \cos z$
That is, the cosine function is even. | {{begin-eqn}}
{{eqn | l = \map \cos {-z}
| r = \frac {e^{i \paren {-z} } + e^{-i \paren {-z} } } 2
| c = Euler's Cosine Identity
}}
{{eqn | r = \frac {e^{i z} + e^{-i z} } 2
| c = simplifying
}}
{{eqn | r = \cos z
| c = Euler's Cosine Identity
}}
{{end-eqn}}
{{qed}} | :$\map \cos {-z} = \cos z$
That is, the [[Definition:Complex Cosine Function|cosine function]] is [[Definition:Even Function|even]]. | {{begin-eqn}}
{{eqn | l = \map \cos {-z}
| r = \frac {e^{i \paren {-z} } + e^{-i \paren {-z} } } 2
| c = [[Euler's Cosine Identity]]
}}
{{eqn | r = \frac {e^{i z} + e^{-i z} } 2
| c = simplifying
}}
{{eqn | r = \cos z
| c = [[Euler's Cosine Identity]]
}}
{{end-eqn}}
{{qed}} | Cosine Function is Even/Proof 2 | https://proofwiki.org/wiki/Cosine_Function_is_Even | https://proofwiki.org/wiki/Cosine_Function_is_Even/Proof_2 | [
"Cosine Function is Even",
"Cosine Function",
"Examples of Even Functions"
] | [
"Definition:Cosine/Complex Function",
"Definition:Even Function"
] | [
"Euler's Cosine Identity",
"Euler's Cosine Identity"
] |
proofwiki-4390 | Cosine Function is Even | :$\map \cos {-z} = \cos z$
That is, the cosine function is even. | {{begin-eqn}}
{{eqn | l = \map \cos {-z}
| r = \map \cos {0 - z}
}}
{{eqn | r = \cos 0 \cos z + \sin 0 \sin z
| c = Cosine of Difference
}}
{{eqn | r = \cos z
}}
{{end-eqn}}
{{qed}} | :$\map \cos {-z} = \cos z$
That is, the [[Definition:Complex Cosine Function|cosine function]] is [[Definition:Even Function|even]]. | {{begin-eqn}}
{{eqn | l = \map \cos {-z}
| r = \map \cos {0 - z}
}}
{{eqn | r = \cos 0 \cos z + \sin 0 \sin z
| c = [[Cosine of Difference]]
}}
{{eqn | r = \cos z
}}
{{end-eqn}}
{{qed}} | Cosine Function is Even/Proof 3 | https://proofwiki.org/wiki/Cosine_Function_is_Even | https://proofwiki.org/wiki/Cosine_Function_is_Even/Proof_3 | [
"Cosine Function is Even",
"Cosine Function",
"Examples of Even Functions"
] | [
"Definition:Cosine/Complex Function",
"Definition:Even Function"
] | [
"Cosine of Difference"
] |
proofwiki-4391 | Cosine Function is Even | :$\map \cos {-z} = \cos z$
That is, the cosine function is even. | {{begin-eqn}}
{{eqn | l = \map \cosh {-x}
| r = \frac {e^{-x} + e^{-\paren {-x} } } 2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{-x} + e^x} 2
}}
{{eqn | r = \frac {e^x + e^{-x} } 2
}}
{{eqn | r = \cosh x
}}
{{end-eqn}}
{{qed}} | :$\map \cos {-z} = \cos z$
That is, the [[Definition:Complex Cosine Function|cosine function]] is [[Definition:Even Function|even]]. | {{begin-eqn}}
{{eqn | l = \map \cosh {-x}
| r = \frac {e^{-x} + e^{-\paren {-x} } } 2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{-x} + e^x} 2
}}
{{eqn | r = \frac {e^x + e^{-x} } 2
}}
{{eqn | r = \cosh x
}}
{{end-eqn}}
{{qed}} | Hyperbolic Cosine Function is Even/Proof 1 | https://proofwiki.org/wiki/Cosine_Function_is_Even | https://proofwiki.org/wiki/Hyperbolic_Cosine_Function_is_Even/Proof_1 | [
"Cosine Function is Even",
"Cosine Function",
"Examples of Even Functions"
] | [
"Definition:Cosine/Complex Function",
"Definition:Even Function"
] | [] |
proofwiki-4392 | Cosine Function is Even | :$\map \cos {-z} = \cos z$
That is, the cosine function is even. | {{begin-eqn}}
{{eqn | l = \map \cosh {-x}
| r = \map \cos {-i x}
| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | r = \map \cos {i x}
| c = Cosine Function is Even
}}
{{eqn | r = \cosh x
| c = Hyperbolic Cosine in terms of Cosine
}}
{{end-eqn}}
{{qed}} | :$\map \cos {-z} = \cos z$
That is, the [[Definition:Complex Cosine Function|cosine function]] is [[Definition:Even Function|even]]. | {{begin-eqn}}
{{eqn | l = \map \cosh {-x}
| r = \map \cos {-i x}
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{eqn | r = \map \cos {i x}
| c = [[Cosine Function is Even]]
}}
{{eqn | r = \cosh x
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{end-eqn}}
{{qed}} | Hyperbolic Cosine Function is Even/Proof 2 | https://proofwiki.org/wiki/Cosine_Function_is_Even | https://proofwiki.org/wiki/Hyperbolic_Cosine_Function_is_Even/Proof_2 | [
"Cosine Function is Even",
"Cosine Function",
"Examples of Even Functions"
] | [
"Definition:Cosine/Complex Function",
"Definition:Even Function"
] | [
"Hyperbolic Cosine in terms of Cosine",
"Cosine Function is Even",
"Hyperbolic Cosine in terms of Cosine"
] |
proofwiki-4393 | Image of Interval by Derivative | Let $f$ be a real function that is everywhere differentiable.
Let $I \subseteq \Dom f$ be a real interval.
Then:
:$f' \sqbrk I$ is a real interval
where $f'$ denotes the derivative of $f$ | Let $x_1, x_2 \in f' \sqbrk I: x_1 < x_2$.
Let $\xi \in \openint {x_1} {x_2}$.
We need to show that $\xi \in f' \sqbrk I$.
Let $a, b \in I : \map {f'} a = x_1 \land \map {f'} b = x_2$.
{{WLOG}}, assume $a < b$. The case $b < a$ is handled similarly.
Let $\map g x = \map f x - \xi x$.
Then:
:$\map {g'} x = \map {f'} x -... | Let $f$ be a [[Definition:Real Function|real function]] that is everywhere [[Definition:Differentiable Real Function|differentiable]].
Let $I \subseteq \Dom f$ be a [[Definition:Real Interval|real interval]].
Then:
:$f' \sqbrk I$ is a [[Definition:Real Interval|real interval]]
where $f'$ denotes the [[Definition:De... | Let $x_1, x_2 \in f' \sqbrk I: x_1 < x_2$.
Let $\xi \in \openint {x_1} {x_2}$.
We need to show that $\xi \in f' \sqbrk I$.
Let $a, b \in I : \map {f'} a = x_1 \land \map {f'} b = x_2$.
{{WLOG}}, assume $a < b$. The case $b < a$ is handled similarly.
Let $\map g x = \map f x - \xi x$.
Then:
:$\map {g'} x = \map ... | Image of Interval by Derivative | https://proofwiki.org/wiki/Image_of_Interval_by_Derivative | https://proofwiki.org/wiki/Image_of_Interval_by_Derivative | [
"Real Intervals",
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function",
"Definition:Real Interval",
"Definition:Real Interval",
"Definition:Derivative"
] | [
"Differentiable Function is Continuous",
"Definition:Continuous Real Function",
"Restriction of Continuous Mapping is Continuous",
"Definition:Continuous Real Function",
"Definition:Deleted Neighborhood",
"Behaviour of Function Near Limit",
"Interior Extremum Theorem"
] |
proofwiki-4394 | Intermediate Value Theorem for Derivatives | Let $I$ be an open interval.
Let $f : I \to \R$ be everywhere differentiable.
Then $f'$ satisfies the Intermediate Value Property. | Since $\forall \set {a, b \in I: a < b}: \openint a b \subseteq I$, the result follows from Image of Interval by Derivative.
{{explain|Now I'm looking at this, I think we need to flesh this out a bit. Anyone up for it?}} | Let $I$ be an [[Definition: Open Real Interval|open interval]].
Let $f : I \to \R$ be everywhere [[Definition:Differentiable Real Function|differentiable]].
Then $f'$ satisfies the [[Definition:Intermediate Value Property|Intermediate Value Property]]. | Since $\forall \set {a, b \in I: a < b}: \openint a b \subseteq I$, the result follows from [[Image of Interval by Derivative]].
{{explain|Now I'm looking at this, I think we need to flesh this out a bit. Anyone up for it?}} | Intermediate Value Theorem for Derivatives | https://proofwiki.org/wiki/Intermediate_Value_Theorem_for_Derivatives | https://proofwiki.org/wiki/Intermediate_Value_Theorem_for_Derivatives | [
"Intermediate Value Theorem",
"Real Analysis"
] | [
"Definition: Open Real Interval",
"Definition:Differentiable Mapping/Real Function",
"Definition:Darboux Function"
] | [
"Image of Interval by Derivative"
] |
proofwiki-4395 | Sine of Half-Integer Multiple of Pi | :$\forall n \in \Z: \map \sin {n + \dfrac 1 2} \pi = \paren {-1}^n$ | From the discussion of Sine and Cosine are Periodic on Reals:
:$\map \sin {x + \dfrac \pi 2} = \cos x$
The result then follows directly from the Cosine of Multiple of Pi.
{{qed}}
Category:Sine Function
mpm9bskwo8z0lvs9quo23dcsouumuhz | :$\forall n \in \Z: \map \sin {n + \dfrac 1 2} \pi = \paren {-1}^n$ | From the discussion of [[Sine and Cosine are Periodic on Reals]]:
:$\map \sin {x + \dfrac \pi 2} = \cos x$
The result then follows directly from the [[Cosine of Multiple of Pi]].
{{qed}}
[[Category:Sine Function]]
mpm9bskwo8z0lvs9quo23dcsouumuhz | Sine of Half-Integer Multiple of Pi | https://proofwiki.org/wiki/Sine_of_Half-Integer_Multiple_of_Pi | https://proofwiki.org/wiki/Sine_of_Half-Integer_Multiple_of_Pi | [
"Sine Function"
] | [] | [
"Sine and Cosine are Periodic on Reals",
"Cosine of Integer Multiple of Pi",
"Category:Sine Function"
] |
proofwiki-4396 | Sine of Integer Multiple of Pi | :$\forall n \in \Z: \sin n \pi = 0$ | This is established in Zeroes of Sine and Cosine.
{{qed}} | :$\forall n \in \Z: \sin n \pi = 0$ | This is established in [[Zeroes of Sine and Cosine]].
{{qed}} | Sine of Integer Multiple of Pi | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Pi | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Pi | [
"Sine Function"
] | [] | [
"Zeroes of Sine and Cosine"
] |
proofwiki-4397 | Cosine of Half-Integer Multiple of Pi | :$\forall n \in \Z: \map \cos {n + \dfrac 1 2} \pi = 0$ | This is established in Zeroes of Sine and Cosine.
{{qed}}
Category:Cosine Function
rfuhggw0gmxdy0pt571mvoo0kbkotq4 | :$\forall n \in \Z: \map \cos {n + \dfrac 1 2} \pi = 0$ | This is established in [[Zeroes of Sine and Cosine]].
{{qed}}
[[Category:Cosine Function]]
rfuhggw0gmxdy0pt571mvoo0kbkotq4 | Cosine of Half-Integer Multiple of Pi | https://proofwiki.org/wiki/Cosine_of_Half-Integer_Multiple_of_Pi | https://proofwiki.org/wiki/Cosine_of_Half-Integer_Multiple_of_Pi | [
"Cosine Function"
] | [] | [
"Zeroes of Sine and Cosine",
"Category:Cosine Function"
] |
proofwiki-4398 | Rational Square Root of Integer is Integer | Let $n$ be an integer.
Suppose that $\sqrt n$ is a rational number.
Then $\sqrt n$ is an integer. | Suppose that $\sqrt n = \dfrac a b$, with $a, b$ coprime integers and $b > 0$.
Then we would have:
:$n = \dfrac {a^2} {b^2}$
That is:
:$n b^2 = a^2$
From Number divides Number iff Square divides Square:
:$b^2 \divides a^2 \implies b \divides a$
However, since $a \perp b$ and $b \divides a$, this means that necessarily ... | Let $n$ be an [[Definition:Integer|integer]].
Suppose that $\sqrt n$ is a [[Definition:Rational Number|rational number]].
Then $\sqrt n$ is an [[Definition:Integer|integer]]. | Suppose that $\sqrt n = \dfrac a b$, with $a, b$ [[Definition:Coprime Integers|coprime integers]] and $b > 0$.
Then we would have:
:$n = \dfrac {a^2} {b^2}$
That is:
:$n b^2 = a^2$
From [[Number divides Number iff Square divides Square]]:
:$b^2 \divides a^2 \implies b \divides a$
However, since $a \perp b$ and $b ... | Rational Square Root of Integer is Integer | https://proofwiki.org/wiki/Rational_Square_Root_of_Integer_is_Integer | https://proofwiki.org/wiki/Rational_Square_Root_of_Integer_is_Integer | [
"Number Theory",
"Integers",
"Rational Numbers"
] | [
"Definition:Integer",
"Definition:Rational Number",
"Definition:Integer"
] | [
"Definition:Coprime/Integers",
"Number divides Number iff Square divides Square",
"Definition:Integer"
] |
proofwiki-4399 | Definite Integral of Even Function | Let $f$ be an even function with a primitive on the closedinterval $\closedint {-a} a$, where $a > 0$.
Then:
:$\ds \int_{-a}^a \map f x \rd x = 2 \int_0^a \map f x \rd x$ | Let $F$ be a primitive for $f$ on the interval $\closedint {-a} a$.
Then, by Sum of Integrals on Adjacent Intervals for Integrable Functions, we have:
{{begin-eqn}}
{{eqn | l = \int_{-a}^a \map f x \rd x
| r = \int_{-a}^0 \map f x \rd x + \int_0^a \map f x \rd x
| c =
}}
{{end-eqn}}
Therefore, it suffices t... | Let $f$ be an [[Definition:Even Function|even function]] with a [[Definition:Primitive (Calculus)|primitive]] on the [[Definition:Closed Real Interval|closedinterval]] $\closedint {-a} a$, where $a > 0$.
Then:
:$\ds \int_{-a}^a \map f x \rd x = 2 \int_0^a \map f x \rd x$ | Let $F$ be a [[Definition:Primitive (Calculus)|primitive]] for $f$ on the interval $\closedint {-a} a$.
Then, by [[Sum of Integrals on Adjacent Intervals for Integrable Functions]], we have:
{{begin-eqn}}
{{eqn | l = \int_{-a}^a \map f x \rd x
| r = \int_{-a}^0 \map f x \rd x + \int_0^a \map f x \rd x
| c... | Definite Integral of Even Function | https://proofwiki.org/wiki/Definite_Integral_of_Even_Function | https://proofwiki.org/wiki/Definite_Integral_of_Even_Function | [
"Integral Calculus",
"Even Functions",
"Definite Integrals"
] | [
"Definition:Even Function",
"Definition:Primitive (Calculus)",
"Definition:Real Interval/Closed"
] | [
"Definition:Primitive (Calculus)",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Derivative of Identity Function",
"Derivative of Constant Multiple",
"Integration by Substitution",
"Definition:Even Function"
] |
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