From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Tue, 9 Apr 2019 06:08:16 +0000 (+1000)
Subject: Merge branch 'master' of ssh://charles/tank/andrew/school/notes
X-Git-Tag: yr12~168
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/9b4ce53d414fedba7ccafc9534dee37f57d74372?ds=sidebyside;hp=-c

Merge branch 'master' of ssh://charles/tank/andrew/school/notes
---

9b4ce53d414fedba7ccafc9534dee37f57d74372
diff --combined spec/calculus.md
index 5be3218,167a7e7..ddda405
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@@ -1,12 -1,3 +1,12 @@@
 +---
 +geometry: margin=2cm
 +columns: 2
 +graphics: yes
 +tables: yes
 +author: Andrew Lorimer
 +classoption: twocolumn
 +---
 +
  # Differential calculus
  
  ## Limits
@@@ -216,13 -207,23 +216,23 @@@ Order of polynomial $n$th derivative de
  
  ![](graphics/second-derivatives.png)
  
+ ## Implicit Differentiation
+ 
+ On CAS: Action $\rightarrow$ Calculation $\rightarrow$ `impDiff(y^2+ax=5, x, y)`. Returns $y^\prime= \dots$.
+ 
+ Used for differentiating circles etc.
+ 
+ If $p$ and $q$ are expressions in $x$ and $y$ such that $p=q$, for all $x$ nd $y$, then:
+ 
+ $${dp \over dx} = {dq \over dx} \quad \text{and} \quad {dp \over dy} = {dq \over dy}$$
+ 
  ## Antidifferentiation
  
  $$y={x^{n+1} \over n+1} + c$$
  
  ## Integration
  
- $$\int f(x) dx = F(x) + c$$
+ $$\int f(x) dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)$$
  
  - area enclosed by curves
  - $+c$ should be shown on each step without $\int$
@@@ -237,10 -238,8 +247,10 @@@ $\int k f(x) dx = k \int f(x) dx
  | $f(x)$                          | $\int f(x) \cdot dx$         |
  | ------------------------------- | ---------------------------- |
  | $k$ (constant) | $kx + c$ |
 -| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ |
 +| $x^n$ | ${x^{n+1} \over {n+1}} + c$ |
  | $a x^{-n}$ | $a \cdot \log_e x + c$ |
 +| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
 +| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
  | $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
  | $e^k$ | $e^kx + c$ |
  | $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
@@@ -248,7 -247,13 +258,11 @@@
  | ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
  | $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
  | $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
 -| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
 -| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
  
+ ### Definite integrals
+ 
+ $$\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)_{}$$
+ 
  ## Applications of antidifferentiation
  
  - $x$-intercepts of $y=f(x)$ identify $x$-coordinates of stationary points on $y=F(x)$
@@@ -269,13 -274,12 +283,12 @@@ $${dy \over dx} = {{dy \over dt} \over 
  
  $${d^2 \over dx^2} = {d(y^\prime) \over dx} = {{dy^\prime \over dt} \over {dx \over dt}} \> \vert \> y^\prime = {dy \over dx}$$
  
- # Rational functions
+ ## Rational functions
  
  $$f(x) = {P(x) \over Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}$$
  
- ## Addition of ordinates
+ ### Addition of ordinates
  
  - when two graphs have the same ordinate, $y$-coordinate is double the ordinate
  - when two graphs have opposite ordinates, $y$-coordinate is 0 i.e. ($x$-intercept)
  - when one of the ordinates is 0, the resulting ordinate is equal to the other ordinate
-