From cef225ae95a1b412d6939856d4901516e6606957 Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Mon, 27 Dec 2021 19:30:41 +0000 Subject: [PATCH] add notes on mmme 1026 differentiation --- .../1026_maths_for_engineering/calculus.md | 273 ++++++++++++++++++ ...ot-2021-12-27T14:33:20,836330991+00:00.png | Bin 0 -> 54058 bytes ...ot-2021-12-27T15:30:26,494800477+00:00.png | Bin 0 -> 12207 bytes ...ot-2021-12-27T15:38:11,125781274+00:00.png | Bin 0 -> 11957 bytes ...ot-2021-12-27T15:38:29,395666506+00:00.png | Bin 0 -> 12296 bytes 5 files changed, 273 insertions(+) create mode 100644 uni/mmme/1026_maths_for_engineering/images/vimscrot-2021-12-27T14:33:20,836330991+00:00.png create mode 100644 uni/mmme/1026_maths_for_engineering/images/vimscrot-2021-12-27T15:30:26,494800477+00:00.png create mode 100644 uni/mmme/1026_maths_for_engineering/images/vimscrot-2021-12-27T15:38:11,125781274+00:00.png create mode 100644 uni/mmme/1026_maths_for_engineering/images/vimscrot-2021-12-27T15:38:29,395666506+00:00.png diff --git a/uni/mmme/1026_maths_for_engineering/calculus.md b/uni/mmme/1026_maths_for_engineering/calculus.md index 3d9e464..cf4694a 100755 --- a/uni/mmme/1026_maths_for_engineering/calculus.md +++ b/uni/mmme/1026_maths_for_engineering/calculus.md @@ -172,3 +172,276 @@ To draw the curve of an implicit function you have to rewrite it in the form $y There may be more than one $y$ value for each $x$ value. + +# Differentation + +The derivative of the function $f(x)$ is denoted by: + +$$f'(x) \text{ or } \frac{\mathrm{d}}{\mathrm dx} f(x)$$ + +Geometrically, the derivative is the gradient of the curve $y = f(x)$. + +It is a measure of the rate of change of $f(x)$ as $x$ varies. + +For example, velocity, $v$, is the rate of change of displacement, $s$, with respect to time, $t$, +or: + +$$v = \frac{\mathrm ds}{dt}$$ + + +
+ + +#### Formal Definition + + + +![](./images/vimscrot-2021-12-27T14:33:20,836330991+00:00.png) + +As $h\rightarrow 0$, the clospe of the cord $\rightarrow$ slope of the tangent, or: + +$$f'(x_0) = \lim_{h\rightarrow0}\frac{f(x_0+h) - f(x_0)}{h}$$ + +whenever this limit exists. + +
+ +## Rules for Differentiation + +### Powers + +$$\frac{\mathrm d}{\mathrm dx} x^n = nx^{-1}$$ + +### Trigonometric Functions + +$$\frac{\mathrm d}{\mathrm dx} \sin x = \cos x$$ +$$\frac{\mathrm d}{\mathrm dx} \cos x = \sin x$$ + +### Exponential Functions + +$$\frac{\mathrm d}{\mathrm dx} e^{kx} = ke^{kx}$$ + +$$\frac{\mathrm d}{\mathrm dx} \ln kx^n = \frac n x$$ + +where $n$ and $k$ are constant. + +### Linearity + +$$\frac{\mathrm d}{\mathrm dx} (f + g) = \frac{\mathrm d}{\mathrm dx} f + \frac{\mathrm d}{\mathrm dx} g$$ + +### Product Rule + +$$\frac{\mathrm d}{\mathrm dx} (fg) = \frac{\mathrm df}{\mathrm dx}g + \frac{\mathrm dg}{\mathrm dx}f$$ + +### Quotient Rule + +$$ \frac{\mathrm d}{\mathrm dx} \frac f g = \frac 1 {g^2} \left( \frac{\mathrm df}{\mathrm dx} g - f \frac{\mathrm dg}{\mathrm dx} \right) $$ + +$$ \left( \frac f g \right)' = \frac 1 {g^2} (gf' - fg')$$ + +### Chain Rule + +Let + +$$f(x) = F(u(x))$$ + +$$ \frac{\mathrm df}{\mathrm dx} = \frac{\mathrm{d}F}{\mathrm du} \frac{\mathrm du}{\mathrm dx} $$ + +
+ + +#### Example 1 + +Differentiate $f(x) = \cos{x^2}$. + + + +Let $u(x) = x^2$, $F(u) = \cos u$ + +$$ \frac{\mathrm df}{\mathrm dx} = -\sin u \cdot 2x = 2x\sin{x^2} $$ + +
+ +## L'Hôpital's Rule + +l'Hôpital's rule provides a systematic way of dealing with limits of functions like +$\frac{\sin x} x$. + +Suppose + +$$\lim_{x\rightarrow{a}} f(x) = 0$$ + +and + +$$\lim_{x\rightarrow{a}} g(x) = 0$$ + +and we want $\lim_{x\rightarrow{a}} \frac{f(x)}{g(x)}$. + +If + +$$\lim_{x\rightarrow{a}} \frac{f'(x)}{g'(x)} = L $$ + +where any $L$ is any real number or $\pm \infty$, then + +$$\lim_{x\rightarrow{a}} \frac{f(x)}{g(x)} = L$$ + +You can keep applying the rule until you get a sensible answer. + +# Graphs + +## Stationary Points + +An important application of calculus is to find where a function is a maximum or minimum. + +![](./images/vimscrot-2021-12-27T15:30:26,494800477+00:00.png) + +when these occur the gradient of the tangent to the curve, $f'(x) = 0$. +The condition $f'(x) = 0$ alone however does not guarantee a minimum or maximum. +It only means that point is a *stationary point*. + +There are three main types of stationary points: + +- maximum +- minimum +- point of inflection + +### Local Maximum + +The point $x = a$ is a local maximum if: + +$$f'(a) = 0 \text{ and } f''(a) < 0$$ + +This is because $f'(x)$ is a decreasing function of $x$ near $x=a$. + +### Local Minimum + +The point $x = a$ is a local minimum if: + +$$f'(a) = 0 \text{ and } f''(a) > 0$$ + +This is because $f'(x)$ is a increasing function of $x$ near $x=a$. + +### Point of Inflection + +$$f'(a) = 0 \text{ and } f''(a) = 0 \text { and } f'''(a) \ne 0$$ + +#### $f'''(a) > 0$ + +![](./images/vimscrot-2021-12-27T15:38:11,125781274+00:00.png) + +#### $f'''(a) < 0$ + +![](./images/vimscrot-2021-12-27T15:38:29,395666506+00:00.png) + +# Taylor series + +The expansion + +$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$ + +is an example of a *Taylor series*. +These enable us to approximate a given function f(x) using a series which is often easier to +calculate. +Among other uses, they help us: + +- calculate complicated function using simple arithmetic operations +- find useful analytical approximations which work for $x$ near a given value + (e.g. $e^x \approx 1 + x$ for $x$ near 0) +- Understand the behaviour of a function near a stationary point + +## Strategy + +Suppose we know information about $f(x)$ only at the point $x=0$. +How can we find out about $f$ for other values of $x$? + +We could approximate the function by successive polynomials, +each time matching more derivatives at $x=0$. + +\begin{align*} + g(x) = a_0 &\text{ using } f(0) \\ + g(x) = a_0 + a_1x &\text{ using } f(0), f'(0) \\ + g(x) = a_0 + a_1x + a_2x^2 &\text{ using } f(0), f'(0), f''(0) \\ + &\text{and so on...} +\end{align*} + +
+ + +#### Example 1 + +For $x$ near 0, approximate $f(x) = \cos x$ by a quadratic. + + + +1. Set $f(0) = g(0$: + + $$f(0) = 1 \rightarrow g(0) = a_0 = 1$$ + +2. Set $f'(0) = g'(0$: + + $$f'(0) = -\sin0 = 0 \rightarrow g'(0) = a_1 = 0$$ + +3. Set $f''(0) = g''(0$: + + $$f''(0) = -\cos = -1 \rightarrow g''(0) = 2a_2 = -1 \rightarrow a_2 = -0.5$$ + +So for $x$ near 0, + +$$\cos x \approx 1 - \frac 1 2 x^2$$ + +Check: + +$x$ | $\cos x$ | $1 - 0.5x^2$ +--- | -------- | ------------ +0.4 | 0.921061 | 0.920 +0.2 | 0.960066 | 0.980 +0.1 | 0.995004 | 0.995 + +
+ +## General Case + +### Maclaurin Series + +A Maclaurin series is a Taylor series expansion of a function about 0. + +Any function $f(x)$ can be written as an infinite *Maclaurin Series* + +$$f(x) = a_0 + a_1x + a_2x^2 + a_3x^2 + \cdots$$ + +where + +$$a_0 = f(0) \qquad a_n = \frac 1 {n!} \frac{\mathrm d^nf}{\mathrm dx^n} \bigg|_{x=0}$$ + +($|_{x=0}$ means evaluated at $x=0$) + +### Taylor Series + +We may alternatively expand about any point $x=a$ to give a Taylor series: + +\begin{align*} +f(x) = &f(a) + (x-a)f'(a) \\ + & + \frac 1 {2!}(x-a)^2f''(a) \\ + & + \frac 1 {3!}(x-a)^3f'''(a) \\ + & + \cdots + \frac 1 {n!}(x-a)^nf^{(n)}(a) +\end{align*} + +a generalisation of a Maclaurin series. + +An alternative form of Taylor series is given by setting $x = a+h$ where $h$ is small: + +$$f(a+h) = f(a) + hf'(a) + \cdots + \frac 1 {n!}h^nf^{(n)}(a) + \cdots$$ + + +## Taylor Series at a Stationary Point + +If f(x) has a stationary point at $x=a$, then $f'(a) = 0$ and the Taylor series begins + +$$f(x) = f(a) + \frac 1 2 f''(a)(x-a)^2 + \cdots$$ + +- If $f''(a) > 0$ then the quadratic part makes the function increase going away from $x=a$ and we + have a minimum +- If $f''(a) < 0$ then the quadratic part makes the function decrease going away from $x=a$ and we + have a maximum +- If $f''(a) = 0$ then we must include a higer order terms to determine what happens + have a minimum diff --git a/uni/mmme/1026_maths_for_engineering/images/vimscrot-2021-12-27T14:33:20,836330991+00:00.png b/uni/mmme/1026_maths_for_engineering/images/vimscrot-2021-12-27T14:33:20,836330991+00:00.png new file mode 100644 index 0000000000000000000000000000000000000000..ae2448db643a88cfc9870324d7da4b7b44b4cb05 GIT binary patch literal 54058 zcmeFZ^;eYL7dJeppdxXjAV_(Oq)13NDBax>(hbrb1_%g9Nq2V+DJ9YkLzjreFoXyL z0s`;3hWq=h=lu)bU*1`>U>56|>)L1U&;IN>hfozIX?$FATo??7FDoOV27}$0fx)i9 zZ{dKykmIR;1;1{%ypYwn1zx_l%s#+i|H5P?o@;ofZZCQznwqB_?Vp(`7pvJ$k1sx5 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