add notes on mmme 1026 differentiation

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.
 
 </details>
+
+# 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:
+
+$$v = \frac{\mathrm ds}{dt}$$
+
+
+<details>
+<summary>
+
+#### Formal Definition
+
+</summary>
+
+![](./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.
+
+</details>
+
+## 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
+
+$$f(x) = F(u(x))$$
+
+$$ \frac{\mathrm df}{\mathrm dx} = \frac{\mathrm{d}F}{\mathrm du} \frac{\mathrm du}{\mathrm dx} $$
+
+<details>
+<summary>
+
+#### Example 1
+
+Differentiate $f(x) = \cos{x^2}$.
+
+</summary>
+
+Let $u(x) = x^2$, $F(u) = \cos u$
+
+$$ \frac{\mathrm df}{\mathrm dx} = -\sin u \cdot 2x = 2x\sin{x^2} $$
+
+</details>
+
+## 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*}
+
+<details>
+<summary>
+
+#### Example 1
+
+For $x$ near 0, approximate $f(x) = \cos x$ by a quadratic.
+
+</summary>
+
+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
+
+</details>
+
+## 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