What are the prerequisites for \(f\) and \(g\) to apply l'Hôpital's?
Note 1: ETH::A&D
Deck: ETH::A&D
Note Type: Horvath Classic
GUID:
modified
Note Type: Horvath Classic
GUID:
k#~pL>w{_$
Before
Front
Back
What are the prerequisites for \(f\) and \(g\) to apply l'Hôpital's?
- \(f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) and \(g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\)$ are differentiable
- \(\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = \infty\) they tend to infinity
- and \(\forall x \in \mathbb{R}^{+} , g'(x) \neq 0\) and \(g\) never equals to 0
This guarantees that we can take the fraction f/g
After
Front
What are the prerequisites for \(f\) and \(g\) to apply l'Hôpital's?
Back
What are the prerequisites for \(f\) and \(g\) to apply l'Hôpital's?
- \(f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) and \(g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) are differentiable
- \(\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = \infty\) they tend to infinity
- and \(\forall x \in \mathbb{R}^{+} , g'(x) \neq 0\) and \(g\) never equals to 0
This guarantees that we can take the fraction f/g
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Back | <ol><li>\(f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) and \(g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) |
<ol><li>\(f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) and \(g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) are <b>differentiable</b></li><li>\(\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = \infty\) they <b>tend to infinity</b></li><li>and \(\forall x \in \mathbb{R}^{+} , g'(x) \neq 0\) and \(g\) <b>never equals to 0</b></li></ol><div>This guarantees that we can take the fraction f/g</div> |
Note 2: ETH::A&D
Deck: ETH::A&D
Note Type: Horvath Classic
GUID:
modified
Note Type: Horvath Classic
GUID:
D%cpEp.*I3
Before
Front
L'Hôpital's Rule:
Back
L'Hôpital's Rule:
If\(\lim_{x \rightarrow \infty} \frac{f’(x)}{g’(x)} = C \in \mathbb{R}^{+}0\) or \(\lim{x \rightarrow \infty} \frac{f’(x)}{g’(x)} = \infty\), then:
\[\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \infty} \frac{f’(x)}{g’(x)}\]The ratio of the derivatives tend to the same limit
After
Front
L'Hôpital's Rule:
Back
L'Hôpital's Rule:
If \(\lim_{x \rightarrow \infty} \frac{f’(x)}{g’(x)} = C \in \mathbb{R}^{+}0\) or \(\lim{x \rightarrow \infty} \frac{f’(x)}{g’(x)} = \infty\), then:
\[\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \infty} \frac{f’(x)}{g’(x)}\]The ratio of the derivatives tend to the same limit
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Back | <div>If\(\lim_{x \rightarrow \infty} \frac{f’(x)}{g’(x)} = C \in \mathbb{R}^{+}0\) or \(\lim{x \rightarrow \infty} \frac{f’(x)}{g’(x)} = \infty\), then:</div> <div>\[\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \infty} \frac{f’(x)}{g’(x)}\]The ratio of the <b>derivatives</b> tend to the <b>same limit</b><br></div> | <div>If \(\lim_{x \rightarrow \infty} \frac{f’(x)}{g’(x)} = C \in \mathbb{R}^{+}0\) or \(\lim{x \rightarrow \infty} \frac{f’(x)}{g’(x)} = \infty\), then:</div> <div>\[\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \infty} \frac{f’(x)}{g’(x)}\]The ratio of the <b>derivatives</b> tend to the <b>same limit</b><br></div> |
Note 3: ETH::A&D
Deck: ETH::A&D
Note Type: Horvath Classic
GUID:
modified
Note Type: Horvath Classic
GUID:
HVQq>1?_s&
Before
Front
Invariant
Back
Invariant
An invariant holds before, for each iteration and after. We use it to prove an algorithm preserves a certain property.
After
Front
What is an Invariant?
Back
What is an Invariant?
An invariant holds before, for each iteration and after. We use it to prove an algorithm preserves a certain property.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | <b>Invariant</b> | <b>What is an Invariant?</b> |
Note 4: ETH::DiskMat
Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID:
modified
Note Type: Horvath Cloze
GUID:
xDDC{82KOB
Before
Front
Proof method: Proof by Contradiction
Back
Proof method: Proof by Contradiction
1.
2.
3.
After
Front
Proof method: Proof by Contradiction
1. Find a suitable statement \( T\)
1. Find a suitable statement \( T\)
2. Prove that \( T \) is false
3. Assume that \( S \) is false and prove that \( T \) is true (-> contradiction)
Back
Proof method: Proof by Contradiction
1. Find a suitable statement \( T\)
1. Find a suitable statement \( T\)
2. Prove that \( T \) is false
3. Assume that \( S \) is false and prove that \( T \) is true (-> contradiction)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | Proof method: Proof by Contradiction | Proof method: Proof by Contradiction<br><br>1. {{c1:: Find a suitable statement \( T\)}}<div>2. {{c2:: Prove that \( T \) is false}}</div><div>3. {{c3:: Assume that \( S \) is false and prove that \( T \) is true (-> contradiction)}}</div> |
| Extra |
Note 5: ETH::EProg
Deck: ETH::EProg
Note Type: Horvath Cloze
GUID:
modified
Note Type: Horvath Cloze
GUID:
z3M|&J1r.r
Before
Front
Not every EBNF language (Sprache) can be described with repetition (wiederholung).
Back
Not every EBNF language (Sprache) can be described with repetition (wiederholung).
After
Front
Not every EBNF language (Sprache) can be described with repetition (Wiederholung).
Back
Not every EBNF language (Sprache) can be described with repetition (Wiederholung).
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | Not every EBNF language (Sprache) can be described with {{c2:: repetition (Wiederholung)}}. |
Note 6: ETH::LinAlg
Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID:
modified
Note Type: Horvath Classic
GUID:
nkw:=NZ1ua
Before
Front
The scalar product of \(\textbf{v} \cdot \textbf{v}\) is (inequality)
Back
The scalar product of \(\textbf{v} \cdot \textbf{v}\) is (inequality)
\(\textbf{v} \cdot \textbf{v} \geq 0\) with equality exactly if \(\textbf{v} = \textbf{0}\).
This is because we essentially square the entries and thus can't get negatives.
This is because we essentially square the entries and thus can't get negatives.
After
Front
The scalar product of \(\textbf{v} \cdot \textbf{v}\) is \(\leq or \geq\) to what?
Back
The scalar product of \(\textbf{v} \cdot \textbf{v}\) is \(\leq or \geq\) to what?
\(\textbf{v} \cdot \textbf{v} \geq 0\) with equality exactly if \(\textbf{v} = \textbf{0}\).
This is because we essentially square the entries and thus can't get negatives.
This is because we essentially square the entries and thus can't get negatives.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | The <b>scalar product</b> of \(\textbf{v} \cdot \textbf{v}\) is |
The <b>scalar product</b> of \(\textbf{v} \cdot \textbf{v}\) is \(\leq or \geq\) to what? |