From 7958fecd1f2ccbbf087fc14e839f28953126b14b Mon Sep 17 00:00:00 2001 From: eggy Date: Wed, 12 Oct 2022 14:48:09 -0400 Subject: [PATCH] math117: add up to derivatives --- docs/ce1/math117.md | 177 ++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 177 insertions(+) diff --git a/docs/ce1/math117.md b/docs/ce1/math117.md index 7c81987..6c9a54d 100644 --- a/docs/ce1/math117.md +++ b/docs/ce1/math117.md @@ -305,3 +305,180 @@ The sign of $\alpha$ should be determined via its quadrant via the signs of $a$ ∴ \alpha &= \tan^{-1}\frac{5}{3} + \pi \end{align*} $$ + +## Limits + +### Limits of sequences + +!!! definition + - A **sequence** is an infinitely long list of numbers with the **domain** of all natural numbers (may also include 0). + - A sequence that does not converge is a **diverging** sequence. + +A sequence is typically denoted via braces. + +$$\{a_n\}\text{ or } \{a_n\}^\infty_{n=0}$$ + +Sometimes sequences have formulae. + +$$\left\{\frac{5^n}{3^n}\right\}^\infty_{n=0}$$ + +The **limit** of a sequence is the number $L$ that the sequence **converges** to as $n$ increases, which can be expressed in either of the two ways below: + +$$ +a_n \to L \text{ as } n\to\infty \\ +\lim_{n\to\infty}a_n=L +$$ + +: > Specifically, a sequence $\{a_n\}$ converges to limit $L$ if, for any positive number $\epsilon$, there exists an integer $N$ such that $n>N \Rightarrow |a_n - L | < \epsilon$. + +Effectively, if there is always a term number that would lead to the distance between the sequence at that term and the limit to be less than any arbitrarily small $\epsilon$, the sequence has the claimed limit. + +!!! example + A limit can be proved to exist with the above definition. To prove $\left\{\frac{1}{\sqrt{n}}\right\}\to0$ as $n\to\infty$: + $$ + \begin{align*} + \text{Proof:} \\ + n > N &\Rightarrow \left|\frac{1}{\sqrt{n}} - 0\right| < \epsilon \\ + &\Rightarrow \frac{1}{\epsilon^2} < n + \end{align*} \\ + \ce{Let \epsilon\ be any positive number{.} If n > \frac{1}{\epsilon^2}, then \frac{1}{\sqrt{n}}-> 0 as n -> \infty{.}} + $$ + +Please see [SL Math - Analysis and Approaches 1#Limits](/g11/mhf4u7/#limits) for more information. + +The **squeeze theorem** states that if a sequence lies between two other converging sequences with the same limit, it also converges to this limit. That is, if $a_n\to L$ and $c_n\to L$ as $n\to\infty$, and $a_n\leq b_n\leq c_n$ is **always true**, $b_n\to L$. + +!!! example + $\left\{\frac{\sin n}{n}\right\}$: since $-1\leq\sin n\leq 1$, $\frac{-1}{n}\leq\frac{\sin n}{n}\leq \frac{1}{n}$. Since both other functions converge at 0, and sin(n) is always between the two, sin(n) thus also converges at 0 as n approaches infinity. + +If function $f$ is continuous and $\lim_{n\to\infty}a_n$ exists: + +$$\lim_{n\to\infty}f(a_n)=f\left(\lim_{n\to\infty}a_n\right)$$ + +On a side note: + +$$\lim_{n\to\infty}\tan^{-1} n = \frac{\pi}{2}$$ + +### Limits of functions + +The definition is largely the same as for the limit of a sequence: + +: > A function $f(x)\to L$ as $x\to a$ if, for any positive $\epsilon$, there exists a number $\delta$ such that $0<|x-a|<\delta\Rightarrow|f(x)-L|<\epsilon$. + +Again, for the limit to be true, there must be a value $x$ that makes the distance between the function and the limit less than any arbitrarily small $\epsilon$. + +The extra $0 <$ is because the behaviour for when $x=a$, which may or may not be defined, is irrelevant. + +!!! example + To prove $3x-2\to 4$ as $x\to 2$: + $$ + \ce{for any \epsilon\ > 0, there is a \delta\ > 0\ such that:} + $$ + $$ + \begin{align*} + |x-2| < \delta &\Rightarrow|(3x-2) - 4| &< \epsilon \\ + &\Leftarrow |(3x-2) -4| &< \epsilon \\ + &\Leftarrow |3x-6| &< \epsilon \\ + &\Leftarrow |x-2| &< \frac{\epsilon}{3} \\ + \delta &= \frac{\epsilon}{3} + \end{align*} + $$ + $$ + \ce{Let \epsilon\ be any positive number{.} If }|x-2|<\frac{\epsilon}{3}, \\ + \text{then }|(3x-2)-4|<\epsilon\text{. Therefore }3x-2\to 4\text{ as }x\to 2. + $$ + +!!! warning + When solving for limits, negatives have to be considered if the limit approaches a negative number: + + $$\lim_{x\to -\infty}\frac{x}{\sqrt{4x^2-3}} = \frac{1}{-\frac{1}{\sqrt{x}^2}\sqrt{4x^2-3}}$$ + +As the angle in **radians** of an arc approaches 0, it is nearly equal to the sine (vertical component). + +$$ +\lim_{\theta\to 0}\frac{\sin\theta}{\theta} = 1 +$$ + +This function is commonly used in engineering and is known as the sinc function. + +$$ +\text{sinc}(x) = \begin{cases} +\frac{\sin x}{x}&\text{ if }x\neq 0 \\ +0&\text{ if }x=0 +\end{cases} +$$ + +## Continuity + +Please see [SL Math - Analysis and Approaches 1#Limits and continuity](/g11/mhf4u7/#limits-and-continuity) for more information. + +Most common functions can be assumed to be continuous (e.g., $\sin x,\cos x, x, \sqrt{x}, \frac{1}{x}, e^x, \ln x$, etc.). + +: > $f(x)$ is continuous in an interval if for any $x$ and $y$ in the interval and any positive number $\epsilon$, there exists a number $\delta$ such that $|x-y|<\delta\Rightarrow |f(x)-f(y)| < \epsilon$. + +Effectively, if $f(x)$ can be made infinitely close to $f(y)$ by making $x$ closer to $y$, the function is continuous. + +If two functions are continuous: + +- $(f\circ g)(x)$ is continuous +- $(f\pm g)(x)$ is continuous +- $(fg)(x)$ is continuous +- $\frac{1}{f(x)}$ is continuous anywhere $f(x)\neq 0$ + +### Intermediate value theorem + +The IVT states that if a function is continuous and there is a point between two other points, its term must also be between those two other points. + +: > If $f(x)$ is continuous, if $f(a)\leq C\leq f(b)$, there must be a number $c\in[a,b]$ where $f(c)=C$. + +The theorem is used to validate using binary search to find roots (guess and check). + +### Extreme value theorem + +The EVT states that any function continuous within a **closed** interval has at least one maximum and minimum. +: > If $f(x)$ is continuous in the **closed interval** $[a, b]$, there exist numbers $c$ and $d$ in $[a,b]$ such that $f(c)\leq f(x)\leq f(d)$. + +## Derivatives + +Please see [SL Math - Analysis and Approaches 1#Rate of change](/g11/mhf4u7/#rate-of-change) and [SL Math - Analysis and Approaches#Derivatives](/g11/mhf4u7/#derivatives) for more information. + +The derivative of a function $f(x)$ at $a$ is determined by the following limit: + +$$\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$ + +If the limit does not exist, the function is **not differentiable at $a$**. + +Alternative notations for $f'(x)$ include $\dot f(x)$ and $Df$ (which is equal to $\frac{d}{dx}f(x)$). + +Please see [SL Math - Analysis and Approaches 1#Finding derivatives using first principles](/g11/mhf4u7/#finding-derivatives-using-first-principles) and [SL Math - Analysis and Approaches 1#Derivative rules](/g11/mhf4u7/#derivative-rules) for more information. + +Some examples of derivatives of inverse functions: + +- $\frac{d}{dx}f^{-1}(x) = \frac{1}{\frac{dx}{dy}}$ +- $\frac{d}{dx}\sin^{-1} x = \frac{1}{\sqrt{1-x^2}}$ +- $\frac{d}{dx}\cos^{-1} x = -\frac{1}{\sqrt{1-x^2}}$ +- $\frac{d}{dx}\tan^{-1} x = \frac{1}{1+x^2}$ +- $\frac{d}{dx}\log_a x = \frac{1}{(\ln a) x}$ +- $\frac{d}{dx}a^x = (\ln a)a^x$ + +### Implicit differentiation + +Please see [SL Math - Analysis and Approaches 1#Implicit differentiation](/g11/mhf4u7/#implicit-differentiation) for more information. + +### Mean value theorem + +The MVT states that the average slope between two points will be reached at least once between them if the function is differentiable. + +: > If $f(x)$ is continuous in $[a, b]$ and differentiable in $(a, b)$, respectively, there must be a $c\in(a,b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$. + +### L'Hôpital's rule + +As long as $\frac{f(x)}{g(x)} = \frac{0}{0}\text{ or } \frac{\infty}{\infty}$: + +$$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$ + +: > If $f(x)$ and $g(x)$ are differentiable (except maybe at $a$), and $\lim_{x\to a}f(x) = 0$ and $\lim_{x\to a}g(x) = 0$, the relation is true. + +### Related rates + +Please see [SL Math - Analysis and Approaches 1#Related rates](/g11/mhf4u7/#related-rates) for more information.