chem precipitation, diff / antidiff

diff --git a/chem/water.md b/chem/water.md
index 34beb005a14d29d2e51b190dd353314d1628ac96..15e95f6410b3f46ac683b143c779843291aa4e26 100644 (file)
--- a/chem/water.md
+++ b/chem/water.md
@@ -79,6 +79,8 @@
 - special concentration unit in mol / L (abbbreviation M)
 $$c={n \over v}$$
 
+$$c_1v_2=c_2v_2=n\quad\leftarrow(\operatorname{constant})$$
+
 | Quantity               | Symbol  | Unit          |
 | ---------------------- | ------- | ------------- |
 | molar mass             | $M$     | g / mol       |
@@ -87,6 +89,7 @@ $$c={n \over v}$$
 
 ## Precipitation reactions
 Ions in solution combine to form a new compound.
+Occurs when concentration exceeds solubility.
 Precipitation reactions happen when water cannot dissociate ionic lattice structure.
 Precipitate (product):
 - generally insoluble in water

diff --git a/methods/calculus.md b/methods/calculus.md
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+++ b/methods/calculus.md
@@ -0,0 +1,48 @@
+# Calculus
+
+## Planner
+
+1. 16A Recognising relationships and 16B Constant rate of change
+2. 16C Average rate of change and 16D Instantaneous rate of change
+3. 17F Limits and continuity
+4. 17A First principles
+5. 17B Rules for differentiation and 17C Negative integers
+6. 17D Graphs of derivatives
+7. 18A Tangents and normals
+8. 18B Rates of change
+9. 18C and 18D Stationary point
+10. 18E Applications of Max and Min
+11. Revision
+12. Test
+
+
+## Average rate of change
+
+Average rate of change between $x=[a,b]$ given two points $P(a, f(a))$ and $Q(b, f(b))$ is the gradient $m$ of line $\overleftrightarrow{PQ}$
+
+## Instantaneous rate of change
+Tangent to a curve at a point - has same slope as graph at this point.
+Values for $\Delta$ are always approximations.
+
+Secant - line passing through two points on a curve
+Chord - line segment joining two points on a curve
+
+Instantaneous rate of change is estimated by using two given points on each side of the concerned point. Evaluate as in average rate of change.
+
+Each point $Q_n<P$ becomes closer to $Q_P$.
+
+## Position and velocity
+
+Position - location relative to a reference point
+
+Average velocity - average rate of change in position over time
+
+Instantaneous velocity - calculated the same way as averge $\Delta$
+
+## Derivatives
+
+**Derivative** denoted by $f^\prime(x)$:
+
+$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
+
+**Tangent line** of function $f$ at point $M(a, f(a))$ is the line through $M$ with gradient $f^\prime(a)$.

diff --git a/spec/calculus.md b/spec/calculus.md
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+# Differential calculus
+
+## Limits
+
+$$\lim_{x \rightarrow a}f(x)$$
+
+$L^-$ - limit from below
+
+$L^+$ - limit from above
+
+$\lim_{x \to a} f(x)$ - limit of a point
+
+- Limit exists if $L^-=L^+$
+- If limit exists, point does not.
+
+Limits can be solved using normal techniques (if div 0, factorise)
+
+## Limit theorems
+
+1. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
+2. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
+3. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
+4. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
+
+Corollary: $\lim_{x \rightarrow a} c \times f(x)=cF$ where $c=$ constant
+
+## Solving limits for $x\rightarrow\infty$
+
+Factorise so that all values of $x$ are in denominators.
+
+e.g.
+
+$$\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2$$
+
+
+
+## Continuous functions
+
+A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.