From b7f931d7cdb8f022126654ccfc2e0781e55957eb Mon Sep 17 00:00:00 2001
From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Thu, 4 Apr 2019 12:31:57 +1100
Subject: [PATCH] [spec] start antidifferentiation revision

---
 spec/calculus.md | 29 +++++++++++++++--------------
 1 file changed, 15 insertions(+), 14 deletions(-)

diff --git a/spec/calculus.md b/spec/calculus.md
index c239084..167a7e7 100644
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -207,13 +207,23 @@ 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 $\int$
@@ -240,6 +250,10 @@ $\int k f(x) dx = k \int f(x) dx$
 | ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
 | $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
 
+### 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 $y=F(x)$
@@ -269,16 +283,3 @@ $$f(x) = {P(x) \over Q(x)} \quad \text{where } P, Q \text{ are polynomial functi
 - when two graphs have the same ordinate, $y$-coordinate is double the ordinate
 - when two graphs have opposite ordinates, $y$-coordinate is 0 i.e. ($x$-intercept)
 - when one of the ordinates is 0, the resulting ordinate is equal to the other ordinate
-
-
-## Implicit Differentiation
-
-On CAS: Action $\rightarrow$ Calculation $\rightarrow$ `impDiff(y^2+ax=5, x, y)`. Returns $y^\prime= \dots$.
-
-Used for differentiating circles etc.
-
-If $p$ and $q$ are expressions in $x$ and $y$ such that $p=q$, for all $x$ nd $y$, then:
-
-$${dp \over dx} = {dq \over dx} \quad \text{and} \quad {dp \over dy} = {dq \over dy}$$
-
-
-- 
2.43.2