Mathematics Extended Module 2 by Angelina Lam

Definition

j8jz2dbnxn — 2022-08-08 05:55:16 UTC

How did we get the word "Surd" ?

Well around 820 AD al-Khwarizmi (the Persian guy who we get the name "Algorithm" from) called irrational numbers "'inaudible" ... this was later translated to the Latin surdus ("deaf" or "mute")

Exercise 1: Rationalize the denominator.

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Exercise 2: Rationalize the denominator.

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Exercise 1: Prove, by mathematical induction, that the proposition is true for all positive integers n.

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Principle of Mathematical Induction

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Principle of Mathematical Induction

Let P(n) be a proposition.

If (i) P(1) is true, and (ii) P(k) is true implies that P(k + 1) is also true for all positive integers k, then P(n) is true for all positive integers n.

Steps for doing the proof

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Step 1: Prove that P(1) is true.
Step 2: Assume that P(k) is true for some positive integer k. Prove that P(k + 1) is also true.

Exericse 2: Prove, by mathematical induction, that the proposition is true for all positive integers n.

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Common Mistakes

j8jz2dbnxn — 2022-08-08 07:58:13 UTC

The following are some common mistakes.

(i) Let n = 1 be true.

(ii) Do not define the proposition as P(n), but employ this notation, and treat P(1) and P(k) as functions.

(iii) State ‘k is real’, ‘k is a constant’ or ‘n is real’, ‘n is a constant’.

(iv) In Step 2, write down ‘Assume that the proposition is true for all positive integers’.

(v) Do not state ‘the proposition is true for n = 1’ and/or ‘the proposition is also true for n = k + 1’ after finishing Step 1 and/or Step 2.

(vi) Misspell the word ‘true’ as ‘ture’.

DSE 2012

j8jz2dbnxn — 2022-08-08 08:12:20 UTC

Prove, by mathematical induction, that the proposition is true for all positive integers n. (5 marks)

Factorial and Combination

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Factorial says to multiply all whole numbers from the chosen number down to 1.

The symbol is "!"

Examples:

4! = 4 × 3 × 2 × 1 = 24

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 7 × 6!
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 7 × 6 × 5!
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 7 × 6 × 5× 4!

Pascal Triangle and Binomial Theorem

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Combinations and Permutations

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When the order doesn't matter, it is a Combination.
When the order does matter it is a Permutation.

https://www.mathsisfun.com/combinatorics/combinations-permutations.html

Example

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Example

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Radian Measure

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(a) If= OA = r, then θ = 1 rad.

(b) π rad = 180°

Arc Length and Area of the Circular Sector

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Completed Example

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Definition of the 6 trigonometric ratios

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Completed Example

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Add Sugar To Coffee

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Completed Example

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Trigonometric identities

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Competed Example 1

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Completed Example 2

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Compound Angles Formula

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Double Angles Formula

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Product-to-Sum, Sum-to-Product

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Other references

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Other exercises

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Limit of a function 函數的極限

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Limits at infinity 於無窮大的極限

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The number e

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Limit about e^x 關於 e^x 的極限

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Limit about sin x 涉及 sin x 的極限

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Derivative (First Principle)

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Example 1

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Basic rules of differentiation

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Example 2

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Addition rule

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Example 3

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Product rule

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Example 4

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Quotient rule

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Example 5

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Chain rule

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Example 6

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Differentiation of Trigonometric Functions (where x is in radians)

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Example 7

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Differentiation of Exponential and Logarithmic Functions

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Example 8

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Example 9

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Implicit Differentiation

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Example 10

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Logarithmic Differentiation

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Example 11

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Example 12

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Second Derivatives

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Example 13

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Example 1

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Example 2

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Example 3

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Example 4

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Example 5

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Example 6

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Example 7

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Example 8

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Example 9

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Example 10

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Example 11

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Example 12

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Example 13

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Example 14

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Tangents

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切線方程

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Example 1

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例題 1

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Example 2

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例題 2

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To find the maximum and minimum points of a curve by the first derivative test.

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Example 1

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Example 2

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To understand the concept of concavity of the graph of a function.

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To find the maxima and minima of a function by the second derivative test.

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Example 1

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Example 2

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To find points of inflextion.

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Example

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Symmetry in the Graphs of Even and Odd Functions

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Example 1

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Horizontal and vertical asymptotes

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Example 2

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Example 3

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Oblique/slant asymptotes

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Graph sketching

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Example

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Rate of Change

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Example 1

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Example 2

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Maximization and Minimization

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Example

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Basic Properties

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Example 1

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Integration by substitution

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Example 1

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Example 2

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Fundamental Theorem of Calculus

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Example 1

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Integration by parts

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Example 1

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Example 2

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Special functions

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Example 1

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Example 2

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Power Rule

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Example 1

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Properties

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Example 1

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Example 2

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Integration of 1/x

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Do remember to write the absolute value.

Example 1

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Integration of e^x

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Example 1

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Integration of trigonometric functions

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Example 1

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Example 2

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Integration by substitution

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Example 1

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Example 2

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Example 3

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Example 4

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Example 5

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Example 6

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Integration by using sum-product formulae

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Example 1

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Integration Involving Powers of Sine and Cosine

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Example 1

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Integration Involving Powers of Trigonometric Functions Other Than Sine and Cosine

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Trigonometric substitution

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Example 1

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Example 2

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Integration by parts

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Example 1

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Geometrical applications

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Example 1

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Physical applications

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Example 1

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Common Mistakes

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Need to show the step how you get the limit based on important theorems.

Common Mistakes

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Forget the integration constant.

Common Mistakes

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Ignore the range of angles given and miss some roots for the equation.

Common Mistakes

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Forget to multiply by Pi when calculating the volume of revolution.

策略

j8jz2dbnxn — 2024-03-20 04:11:08 UTC

按照以下步驟

証明對 n = 1，命題為真。
假設對 n = k，其中 k 為一正數_，命題為真。
利用假設，証明對 n = k + 1，命題為真。

策略

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策略

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Addition of vectors

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Example 1

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Subtraction of vectors

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Example 1

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Example 2

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Scalar multiplication of vectors

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Example 1

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Example 2

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Section formula

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Example 1

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Representation of vectors in 2-dimensional plane

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Example 1

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Example 2

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Representation of vectors in 3-dimensional space

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Example 1

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Example 2

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Parallelism

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Example 1

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Example 2

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Division of a line segment

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Example 1

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To find scalar products of vectors in R2 and R3.

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Example 1

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Example 2

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Example 3

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To perform operations using the properties of scalar products of vectors.

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Example 1

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Example 2

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To find the angle between two vectors and determine whether two vectors are orthogonal.

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Example 1

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Example 2

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To find the angle between two vectors and determine whether two vectors are orthogonal.

j8jz2dbnxn — 2024-05-22 07:20:20 UTC

Example 1

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Properties of vector products

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Example 1

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Example 2

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Vector products of vectors in 3-dimensional space

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Example 1

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Example 2

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Vectors orthogonal to 2 given vectors

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Example 1

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Example 2

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Areas of parallelograms and triangles

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Example 1

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Example 2

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向量的加法

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例題 1

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向量的減法

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例題 1

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例題 2

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進行向量的純量乘法

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例題 1

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例題 2

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利用向量的分點公式求位置向量

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例題 1

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理解向量在平面上的表示法

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例題 1

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例題 2

j8jz2dbnxn — 2024-05-23 06:53:38 UTC

理解向量在空間的表示法

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例題 1

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例題 2

j8jz2dbnxn — 2024-05-23 06:56:27 UTC

利用向量來判斷兩條直線是否平行及解幾何問題

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例題 1

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例題 2

j8jz2dbnxn — 2024-05-23 07:00:27 UTC

解涉及分點的問題

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例題 1

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求在R2和R3中向量的純量積

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例題 1

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例題 2

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例題 3

j8jz2dbnxn — 2024-05-23 07:06:27 UTC

利用向量的純量積的性質進行運算

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例題 1

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例題 2

j8jz2dbnxn — 2024-05-23 07:18:33 UTC

求兩個向量之間的夾角及判斷兩個向量是否正交

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例題 1

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例題 2

j8jz2dbnxn — 2024-05-23 07:21:10 UTC

求某向量在另一向量上的向量投影及投影

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例題 1

j8jz2dbnxn — 2024-05-23 07:23:38 UTC

向量積的性質

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例題 1

j8jz2dbnxn — 2024-05-23 07:26:16 UTC

例題 2

j8jz2dbnxn — 2024-05-23 07:26:50 UTC

求在R3中兩個向量的向量積

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例題 1

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例題 2

j8jz2dbnxn — 2024-05-23 07:31:06 UTC

求和兩個已知向量皆正交的向量

j8jz2dbnxn — 2024-05-23 07:32:21 UTC

例題 1

j8jz2dbnxn — 2024-05-23 07:33:06 UTC

例題 2

j8jz2dbnxn — 2024-05-23 07:33:49 UTC

利用向量積求平行四邊形及三角形的面積

j8jz2dbnxn — 2024-05-23 07:35:17 UTC

例題 1

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例題 2

j8jz2dbnxn — 2024-05-23 07:36:49 UTC

使用定積分法求曲線與x軸之間的面積

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例題 1

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例題 2

j8jz2dbnxn — 2024-05-23 07:39:53 UTC

使用定積分法求曲線與y軸之間的面積

j8jz2dbnxn — 2024-05-23 07:40:55 UTC

例題 1

j8jz2dbnxn — 2024-05-23 07:41:33 UTC

例題 2

j8jz2dbnxn — 2024-05-23 07:42:05 UTC

使用定積分法求兩條曲線之間的面積

j8jz2dbnxn — 2024-05-23 07:43:10 UTC

例題 1

j8jz2dbnxn — 2024-05-23 07:44:02 UTC

例題 2

j8jz2dbnxn — 2024-05-23 07:44:31 UTC

使用定積分法求繞坐標軸旋轉所成 (空心) 旋轉體的體積

j8jz2dbnxn — 2024-05-23 07:45:37 UTC

例題 1

j8jz2dbnxn — 2024-05-23 07:46:14 UTC

使用定積分法求繞平行於坐標軸的直線旋轉所成 (空心) 旋轉體的體積

j8jz2dbnxn — 2024-05-23 07:47:39 UTC

例題 1

j8jz2dbnxn — 2024-05-23 07:48:18 UTC

Concept of matrices

j8jz2dbnxn — 2024-09-02 05:24:14 UTC

Example 1

j8jz2dbnxn — 2024-09-02 05:26:16 UTC

Addition and Subtraction of Matrices

j8jz2dbnxn — 2024-09-02 05:27:27 UTC

Example 1

j8jz2dbnxn — 2024-09-02 05:28:04 UTC

Example 2

j8jz2dbnxn — 2024-09-02 05:28:46 UTC

Scalar Multiplication

j8jz2dbnxn — 2024-09-02 05:29:48 UTC

Example 1

j8jz2dbnxn — 2024-09-02 05:31:09 UTC

Properties of Basic Operations

j8jz2dbnxn — 2024-09-02 05:32:15 UTC

Example 1

j8jz2dbnxn — 2024-09-02 05:35:26 UTC

Matrix multiplication

j8jz2dbnxn — 2024-09-02 05:36:43 UTC

Example 1

j8jz2dbnxn — 2024-09-02 05:37:15 UTC

Example 2 (This one is super important)

j8jz2dbnxn — 2024-09-02 05:38:25 UTC

Example 3

j8jz2dbnxn — 2024-09-02 05:41:07 UTC

Properties of Matrix Multiplication

j8jz2dbnxn — 2024-09-02 05:43:10 UTC

Example 1

j8jz2dbnxn — 2024-09-02 05:43:48 UTC

Determinant

j8jz2dbnxn — 2024-09-02 05:49:24 UTC

Example 1

j8jz2dbnxn — 2024-09-02 05:49:59 UTC

Expanding a Determinant

j8jz2dbnxn — 2024-09-02 05:50:36 UTC

Example 2

j8jz2dbnxn — 2024-09-02 05:51:42 UTC

Another useful property

j8jz2dbnxn — 2024-09-02 05:53:19 UTC

|AB| = |A| |B|

The proof is out of the scope.

Properties of Determinants (For your reference)

j8jz2dbnxn — 2024-09-02 05:55:22 UTC

Extra Example 1

j8jz2dbnxn — 2024-09-02 05:56:12 UTC

Extra Example 2

j8jz2dbnxn — 2024-09-02 05:57:20 UTC

The Adjoint Method to Find the Inverse

j8jz2dbnxn — 2024-09-02 05:59:43 UTC

Example 1

j8jz2dbnxn — 2024-09-02 06:00:57 UTC

MUST remember the term "invertible".

Example 2

j8jz2dbnxn — 2024-09-02 06:01:34 UTC

Properties of Inverses

j8jz2dbnxn — 2024-09-02 06:02:54 UTC

Example 1

j8jz2dbnxn — 2024-09-02 06:03:23 UTC

Example 2

j8jz2dbnxn — 2024-09-02 06:03:52 UTC

Exercise 1

j8jz2dbnxn — 2024-09-02 06:04:41 UTC

Solving the system (order 2) by Inverse Matrix Method

j8jz2dbnxn — 2024-09-02 06:06:50 UTC

Example 1

j8jz2dbnxn — 2024-09-02 06:07:38 UTC

Cramer's Rule (order 2)

j8jz2dbnxn — 2024-09-02 06:09:00 UTC

Example 1

j8jz2dbnxn — 2024-09-02 06:09:36 UTC

Solving the system (order 3) by Inverse Matrix Method

j8jz2dbnxn — 2024-09-02 06:11:44 UTC

Example 2

j8jz2dbnxn — 2024-09-02 06:13:10 UTC

Cramer's Rule (order 3)

j8jz2dbnxn — 2024-09-02 06:14:09 UTC

Example 2

j8jz2dbnxn — 2024-09-02 06:15:04 UTC

Exercise 1

j8jz2dbnxn — 2024-09-02 06:17:24 UTC

Pay attention to the followings.

The second equation has no terms of y.
k is a constant.

Gaussian Eliminiation (order 2)

j8jz2dbnxn — 2024-09-02 06:18:33 UTC

Gaussian Elimination (order 3)

j8jz2dbnxn — 2024-09-02 06:19:19 UTC

Example 1

j8jz2dbnxn — 2024-09-02 06:20:18 UTC

Example 2

j8jz2dbnxn — 2024-09-02 06:20:57 UTC

Exercise 2

j8jz2dbnxn — 2024-09-02 06:21:43 UTC

Part (a) is pretty easy.

The focus is part (b)

Number of Solutions

j8jz2dbnxn — 2024-09-02 06:23:12 UTC

MUST familarise this part.

Example 1

j8jz2dbnxn — 2024-09-02 06:24:10 UTC

Example 2

j8jz2dbnxn — 2024-09-02 06:24:38 UTC

Example 3

j8jz2dbnxn — 2024-09-02 06:25:31 UTC

Example 4

j8jz2dbnxn — 2024-09-02 06:25:43 UTC

Exercise 3

j8jz2dbnxn — 2024-09-02 06:26:34 UTC

Make sure you know the conditions for different numbers of solutions.

Graphical representations of order 2 system (Review)

j8jz2dbnxn — 2024-09-02 06:29:15 UTC

You have learnt these in Compulsory Part

Example (when there are Infinitely Many solutions)

j8jz2dbnxn — 2024-09-02 06:30:30 UTC

Solutions for order 3 system

j8jz2dbnxn — 2024-09-02 06:32:10 UTC

Example

j8jz2dbnxn — 2024-09-02 06:33:12 UTC

Exercise 4

j8jz2dbnxn — 2024-09-02 06:33:55 UTC

System of Homogeneous Linear Equations

j8jz2dbnxn — 2024-09-02 06:37:10 UTC

The prefix "homo" means "the same / identical".

The prefix "hetero" means "different".

Example 1

j8jz2dbnxn — 2024-09-02 06:38:15 UTC

Make sure you know the meaning of the word "trivial".

Example 2

j8jz2dbnxn — 2024-09-02 06:40:32 UTC

Example 5

j8jz2dbnxn — 2024-09-02 06:40:47 UTC