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

diff --git a/spec/calculus.md b/spec/calculus.md
index c2390847c889916098938a0431b7fd82a7e53440..ddda40575b93fa94dbef141d5b1d2d8fda3a73ab 100644 (file)
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -1,3 +1,12 @@
+---
+geometry: margin=2cm
+columns: 2
+graphics: yes
+tables: yes
+author: Andrew Lorimer
+classoption: twocolumn
+---
+
 # Differential calculus
 
 ## Limits
@@ -207,13 +216,23 @@ Order of polynomial $n$th derivative decrements each time the derivative is take
 
 ![](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 $\int$
@@ -228,8 +247,10 @@ $\int k f(x) dx = k \int f(x) dx$
 | $f(x)$                          | $\int f(x) \cdot dx$         |
 | ------------------------------- | ---------------------------- |
 | $k$ (constant) | $kx + c$ |
-| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ |
+| $x^n$ | ${x^{n+1} \over {n+1}} + c$ |
 | $a x^{-n}$ | $a \cdot \log_e x + c$ |
+| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
+| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
 | $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
 | $e^k$ | $e^kx + c$ |
 | $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
@@ -237,8 +258,10 @@ $\int k f(x) dx = k \int f(x) dx$
 | ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
 | $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (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
 
@@ -269,16 +292,3 @@ $$f(x) = {P(x) \over Q(x)} \quad \text{where } P, Q \text{ are polynomial functi
 - when two graphs have the same ordinate, $y$-coordinate is double the ordinate
 - when two graphs have opposite ordinates, $y$-coordinate is 0 i.e. ($x$-intercept)
 - when one of the ordinates is 0, the resulting ordinate is equal to the other ordinate
-
-
-## Implicit Differentiation
-
-On CAS: Action $\rightarrow$ Calculation $\rightarrow$ `impDiff(y^2+ax=5, x, y)`. Returns $y^\prime= \dots$.
-
-Used for differentiating circles etc.
-
-If $p$ and $q$ are expressions in $x$ and $y$ such that $p=q$, for all $x$ nd $y$, then:
-
-$${dp \over dx} = {dq \over dx} \quad \text{and} \quad {dp \over dy} = {dq \over dy}$$
-
-