Cheatsheets for CM1015 Computational Mathematics

level-4/computational-mathematics/student-notes/fabio-lama/README.md

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+# About
+
+Listed here is a collection of cheatsheet by topic. Those cheatsheets do not
+explain the topics in depth, but rather serve as quick lookup documents.
+Therefore, the course material provided by the lecturer should still be studied
+and understood. Not everything that is tested at the mid-terms or final exams is
+covered and the Author does not guarantee that the cheatsheets are free of
+errors.
+
+* [Number bases](./cheatsheet_number_bases.pdf)
+* [Sequences and Series](./cheatsheet_sequence_series.pdf)
+* [Modular Arithmetic](./cheatsheet_modular_arithmetic.pdf)
+* [Trigonometry](./cheatsheet_trigonometry.pdf)
+* [Graphs of Functions & Kinematics](./cheatsheet_graphs_functions_kinematics.pdf)
+* [Trigonometrical Functions & Identities](./cheatsheet_trigonometrical_functions_identities.pdf)
+* [Exponential & Logarithm Functions](./cheatsheet_exponential_logarithm_funcions.pdf)
+* [Gradients of Curves & Differentiation](./cheatsheet_gradients_curves_differentiation.pdf)
+* [Vectors](./cheatsheet_vectors.pdf)
+* [Matrices](./cheatsheet_matrices.pdf)
+* [Statistics](./cheatsheet_statistics.pdf)
+* [Probability & Combinatorics](./cheatsheet_probability_combinatorics.pdf)
+
+# Building
+
+_NOTE_: This step is only necessary if you chose to modify the base docments.
+
+The base documents are written in [AsciiDoc](https://asciidoc.org/) and can be
+found in the `src/` directory.
+
+The following dependencies must be installed (Ubuntu):
+
+```console
+$ apt install -y ruby-dev wkhtmltopdf
+$ gem install asciidoctor
+$ chmod +x build.sh
+```
+
+To build the documents (PDF version):
+
+```console
+$ ./build.sh pdf
+```
+
+Optionally, for the HTML version:
+
+```console
+$ ./build.sh html
+```
+
+and for the PNG version:
+
+```console
+$ ./build.sh png
+```
+
+The generated output can be deleted with `./build.sh clean`.
+
+# Disclaimer
+
+The Presented Documents ("cheatsheets") by the Author ("Fabio Lama") are
+summaries of specific topics. The term "cheatsheet" implies that the Presented
+Documents are intended to be used as learning aids or as references for
+practicing and does not imply that the Presented Documents should be used for
+inappropriate practices during exams such as cheating or other offences.
+
+The Presented Documents are heavily based on the learning material provided by
+the University of London, respectively the VLeBooks Collection database in the
+Online Library, especially on the books:
+
+* _Foundation maths (Harlow: Pearson, 2016) 6th edition by Croft, A. and R. Davison_
+* _Precalculus with Limits (January 1, 2017) 4th edition by Ron Larson_
+
+The Presented Documents may incorporate direct or indirect definitions,
+examples, descriptions, graphs, sentences and/or other content used in those
+provided materials. **At no point does the Author present the work or ideas
+incorporated in the Presented Documents as their own.**

level-4/computational-mathematics/student-notes/fabio-lama/build.sh

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+#!/bin/bash
+
+# Because `make` sucks.
+
+gen_html() {
+    # Remove suffix and prefix
+    FILE=$1
+    OUT=${FILE%.adoc}
+    HTML_OUT="cheatsheet_${OUT}.html"
+
+    asciidoctor $FILE -o ${HTML_OUT}
+}
+
+# Change directory to src/ in order to have images included correctly.
+cd "$(dirname "$0")/src/"
+
+case $1 in
+    html)
+        for FILE in *.adoc
+        do
+            # Generate HTML file.
+            gen_html ${FILE}
+        done
+
+        # Move up from src/
+        mv *.html ../
+        ;;
+    pdf)
+        for FILE in *.adoc
+        do
+            # Generate HTML file.
+            gen_html ${FILE}
+
+            # Convert HTML to PNG.
+            PDF_OUT="cheatsheet_${OUT}.pdf"
+            wkhtmltopdf \
+                --enable-local-file-access \
+                --javascript-delay 2000\
+                $HTML_OUT $PDF_OUT
+        done
+
+        # Move up from src/
+        mv *.pdf ../
+
+        # Cleanup temporarily generated HTML files.
+        rm *.html > /dev/null 2>&1
+        ;;
+    png | img)
+        for FILE in *.adoc
+        do
+            # Generate HTML file.
+            gen_html ${FILE}
+
+            # Convert HTML to PNG.
+            IMG_OUT="cheatsheet_${OUT}.png"
+            wkhtmltopdf \
+                --enable-local-file-access \
+                --javascript-delay 2000\
+                $HTML_OUT $IMG_OUT
+        done
+
+        # Move up from src/
+        mv *.png ../
+
+        # Cleanup temporarily generated HTML files.
+        rm *.html > /dev/null 2>&1
+        ;;
+    clean)
+        rm *.html > /dev/null 2>&1
+        rm *.png > /dev/null 2>&1
+        rm ../*.html > /dev/null 2>&1
+        rm ../*.png > /dev/null 2>&1
+        ;;
+    *)
+        echo "Unrecognized command"
+        ;;
+esac

level-4/computational-mathematics/student-notes/fabio-lama/src/exponential_logarithm_funcions.adoc

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+= Cheatsheet - Exponential & Logarithm Functions
+Fabio Lama <[email protected]>
+:description: Module: CM1015 Computational Mathematics, started 04. April 2022
+:doctype: article
+:sectnums: 4
+:toclevels: 4
+:stem:
+
+== Exponential Expressions
+
+An expression of the form stem:[a^x] is called an **exponential expression**,
+where _a_ is the **base** and _x_ is the **exponent**,**power** or **index**.
+
+NOTE: The most common exponential expression is stem:[e^x], where _e_ is the
+constant _2.71828..._. Also known as "Euler's number". This number is found in
+many natural phenomena.
+
+== Exponential Functions
+
+An exponential function has the form of stem:[y = e^x]. This function has important properties:
+
+* _y_ is never negative.
+* When stem:[x=0], then stem:[y = 1]
+* As _x_ increases, then _y_ increases (**exponential growth**).
+
+image::assets/exponential_logarithm_functions/exponential_function.png[align="center"]
+
+== Logarithms
+
+The logarithm is the inverse function to the exponentiation. The following two
+equations are equivalent.
+
+[stem]
+++++
+y = a^x hArr log_a (y) = x
+++++
+
+For example:
+
+[stem]
+++++
+5^3 = 125 hArr log_5 (125) = 3
+++++
+
+Also, conventionally we define **natural logarithms** as:
+
+[stem]
+++++
+log = log_10\
+ln = log_e
+++++
+
+=== Calculating Logarithms to any Base
+
+[stem]
+++++
+log_a X = (log_10 X)/(log_10 a)\
+log_a X = (ln X)/(ln a)
+++++
+
+This also means that - for example:
+
+[stem]
+++++
+log_5 (125) = (log (125))/(log (5))
+++++
+
+Additionally:
+
+[stem]
+++++
+log_a a = 1
+++++
+
+=== Laws
+
+**First Law**:
+
+[stem]
+++++
+log A + log B = log AB
+++++
+
+**Second Law**:
+
+[stem]
+++++
+log A - log B = log (A/B)
+++++
+
+Also note that:
+
+[stem]
+++++
+log 1 = 0
+++++
+
+**Third Law**
+
+[stem]
+++++
+n log A = log A^n
+++++
+
+such as stem:[3 log 2 = log 2^3 = log 8]. This applies for all stem:[n in RR].
+