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Applications of Maximum and Minimum
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Solving Quadratic Equations by Factors
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When No Need to Apply Quotient Rule for Differentiating a Fraction
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Perfect Numbers
Some of the first patterns of natural numbers spotted and studied by Pythagoreans were $\textit{perfect numbers}$. These are natural numbers[...]
Method of Exhaustion
Eudoxus, Greek mathematician 408-355 BC, developed the idea of seeking mathematical solutions using he $\textit{Method on Exhaustion}$: at this stage,[...]
The Golden Ratio
Since the days of old, artists as well as mathematicians have known that there is a special, aesthetically pleasing rectangle[...]
Digital Cubes
The number $153$ is equal to the sum of the cubes of its digits: $$1^3 + 5^3 + 3^3 =[...]
Laws of Exponents (Index Laws)
$\textbf{Laws of Exponents (Index Laws)}$ $a^x \times a^y = a^{x+y}$ To $\textit{multiply}$ numbers with the $\textit{same base}$, keep the base[...]
Proof by Contradiction
$\textbf{Introduction to Proof by Contradiction}$ The basic idea of $\textit{Proof by Contradiction}$ is to assume that the statement that we[...]
The Sign of One
Make the following numbers using FOUR 1s using any mathematics operators and/or symbols, such as $\dfrac{x}{y}$, $\sqrt{x}$, decimal dots, $+$,[...]
Number Curiosity
Are these numbers beautiful? \begin{align} \displaystyle 1 \times 8 + 1 &= 9 \\ 12 \times 8 + 2 &=[...]
Nicomachus Theorem
$\textbf{Nicomachus}$ discovered "Nicomachus Theorem" interesting number patterns involving cubes and sums of odd numbers. Nicomachus was born in Roman Syria[...]
How to get top marks in VCE Mathematical Methods Units 3 & 4
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Logarithmic Differentiation
Basic Rule of Logarithmic Differentiation $$ \displaystyle \dfrac{d}{dx}\log_e{x} = \dfrac{1}{x} \\ \dfrac{d}{dx}\log_e{f(x)} = \dfrac{f'(x)}{f(x)} $$ Practice Questions Question 1 Differentiate[...]
Binomial Expansion | Binomial Theorem
Binomial Expansion is based on two terms, that is binomial. Any expression of the form \( (a+b)^n \) is called[...]
Compound Interest | Series and Sequences
Compound Interest is being used to calculate the total investment across over time. Suppose John invests $1000 in the bank.[...]
Geometric Sequence | Math Help
A geometric sequence is also referred as a geometric progression. Each term of a geometric sequence can be obtained from[...]
Arithmetic Sequence | Maths Help
An Arithmetic Sequence is a sequence in which each term differs from the previous one by the same fixed number.[...]
General Term of a Number Sequence
The general term of a number sequence is one of many ways of defining sequences. Consider the tower of bricks.[...]
Understanding Number Sequence
Number Sequence or progression is ordered list of numbers defined by a pattern or rule. The numbers in the sequence[...]
Implicit Differentiation | Calculus Help
Implicit Differentiation for Calculus Problems This very powerful differentiation process follows from the chain rule. $$u = g(f(x)) \\ \frac{du}{dx}[...]
Surd Equations Reducible to Quadratic | Math Algebra
Surd Equations Reducible to Quadratic | Math Algebra Surd Equations Reducible to Quadratic for Math Algebra is done squaring both[...]
Trigonometric Equations Reducible to Quadratic | Math Skills
Trigonometric Equations Reducible to Quadratic for Math Skills Trigonometric Equations Reducible to Quadratic for Math Skills are based on trigonometric[...]
Exponential Equations Reducible to Quadratic | Math Help
Exponential Equations Reducible to Quadratic for Math Help Exponential Equations Reducible to Quadratic for Math Help is based on various[...]
Fraction Equations Reducible to Quadratic | Free Math Help
Fraction Equations Reducible to Quadratic | Free Math Help iitutor provides full explains of Fraction Equations Reducible to Quadratic for[...]
Algebra Equations Reducible to Quadratic | Math Help
Algebra Equations Reducible to Quadratic Form | Math Help Algebra Equations Reducible to Quadratic Form for Math Help is done[...]
Best Examples of Mathematical Induction Inequality
Mathematical Induction Inequality Proofs Mathematical Induction Inequality is being used for proving inequalities. It is quite often applied for the[...]
Best Examples of Mathematical Induction Divisibility
Mathematical Induction Divisibility Proofs Mathematical Induction Divisibility can be used to prove divisibility, such as divisible by 3, 5 etc.[...]
5 Important Patterns of First Principles
First Principles The First Principles defines how basic rule of derivative of a function. $$\displaystyle f'(x) = \lim_{h\to\infty} \frac{f(x+h)-f(x)}{h}$$ Patterns[...]
10 Deadly Common Algebra Mistakes
Common Patterns of Algebra Mistakes Students often make Common Algebra Mistakes due to confusions such as expand and simplify rules,[...]
Mathematical Induction Fundamentals
Mathematical Induction Fundamentals The Mathematical Induction Fundamentals are defined for applying 3 steps, such as step 1 for showing its[...]
Integrating Trigonometric Functions by Double Angle Formula
Integrating Trigonometric Functions by Double Angle Formula Integrating Trigonometric Functions can be done by Double Angle Formula reducing the power[...]
Trigonometric Integration by Substitution
Trigonometric Integration by Substitution Trigonometric Integration by Substitution can be handled using the basic rules such as \( \displaystyle \frac{d}{dx}[...]
Definite Integration by Substitution
Definite Integration by Substitution Definite Integration by Substitution requires to convert upper and lower limits of definite integration. $$ \displaystyle[...]
Integration by Substitution
Integration by Substitution Integration by Substitution is performed by replacing the pronumeral (variable) to other pronumeral to simplify the expression[...]
Definite Integrals
Background of Definite Integrals Definite Integrals are calculated by subtracting the function values of upper and lower limits. $$\displaystyle \int_{a}^{b}{f(x)}dx[...]
Indefinite Integral of Rational Functions
Understanding Indefinite Integral of Rational Functions Using Indefinite Integral of Rational Functions requires that the format of the expression must[...]
Radical Indefinite Integrals
Radical Indefinite Integrals should be performed after converting its radical or surd notations into index form. $$\displaystyle \sqrt[n]{x^m} = x^{\frac{m}{n}}$$[...]
Indefinite Integral Formula
Basics of Indefinite Integral Formula Indefinite Integral Formula ahs been made from the reverse operations of differentiation or anti-differentiation. \([...]
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4 important types of absolute value equation
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Logarithmic Inequalities
Solving logarithmic inequalities, it is important to understand the direction of the inequality changes if the base of the logarithms[...]
Quartic Graph Sketching
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Cubic Graph Sketching
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Algebra - Harder Factorise or Factoring can be done by collecting common factors and using sums and products of factors.[...]
Sketching Quadratic Graphs
Sketching Quadratic Graphs using Transfomration Sketching Quadratic Graphs are drawn based on \( y=x^2 \) graph for transforming and translating.[...]
Internal Division
Internal Division of Line Segments Internal Division an interval in a given ratio is that if the interval \( (x_1,y_1)[...]
Angle Between Lines
Linear Functions - Angle Between Lines If the acute angle \( \theta \) between two straight lines \( y =[...]
Solving Radical Equations
Solving radical equations are required to isolate the radicals or surds to one side of the equations. Then square both[...]
Integration Reverse Chain Rule
By recalling the chain rule, Integration Reverse Chain Rule comes from the usual chain rule of differentiation. This skill is[...]
Inequality with Variables in the Denominator
Solving Inequality with Variables in the Denominator requires special cares due to the direction of the inequalities. Let's have a[...]
12 Patterns of Logarithmic Equations
Solving logarithmic equations is done many ways using properties of logarithmic functions, such as multiply of logs, change the base[...]
Geometric Series for Time Payments
Time payments is are calculated based on Geometric Series for reducible compound interests. Basically geometric series formula is used for[...]
Complex Numbers with Vector Addition
Complex Numbers with Vector Addition is obtained using the sides of triangles, the sum of sides of two sides of[...]
Rates of Change involving Integration
Rates of Change for finding height and time can be solved by integration. Once integrated, find the integral constant using[...]
Three Dimensional Trigonometry
One of the most effective ways to solve Three Dimensional Trigonometry questions is to list all of the trigonometric ratios[...]
Quadratic Equations in Square Roots
For solving Quadratic Equations in Square Roots, it is required to square both sides entirely, but not individually. For instance,[...]
Area of a Triangle using Radius and Perimeter
Area of a triangle can be calculated using its perimeter and the radius of the circle which is inscribed in[...]
Collision of Projectile Motion
Collision of Projectile Motion is handled by setting the same vertical height and horizontal distance at a given time. Worked[...]
Collinear Proof in Circle Geometry
Collinear Proof can be done in Circle Geometry by showing Three or more points lie on a single straight line.[...]
Integration by Parts
Integration by Parts is made of product rule of differentiation. The derivative of \(uv\) is \(u'v + uv'\) and integrate[...]
Volumes by Cross Sections
Volumes by Cross Sections can be setting a unit volume, \( \delta V \) of a cross section. Then establish[...]
Mathematical Induction Inequality Proof with Factorials
Mathematical Induction Inequality Proof with Factorials uses one of the properties of factorials, \( n! = n(n-1)! = n(n-1)(n-2)! \).[...]
Cyclic Quadrilateral in Circle Geometry
Cyclic Quadrilateral is inscribed into a circle, whose vertices all lie on a circle. The properties of Cyclic Quadrilateral in[...]
Locus of Complex Numbers
Locus of Complex Numbers is obtained by letting \( z = x+yi \) and simplifying the expressions. Operations of modulus,[...]
Mechanics Circular Motions
Mechanics Circular Motions are handled by resolving forces horizontally and vertically in conjunction with the tension of the string, normal[...]
Ellipse Geometry
Ellipse Geometry is used for proving ellipse questions relating to geometry such as similar triangles and circle geometry. Worked Examples[...]
Inequality Proofs
Inequality Proofs can be done many ways. For proving \(A \ge B \), one of the easiest ways is to[...]
Complex Number Regions
Regions defined by complex numbers \( \displaystyle z = x + yi \) where \(x\) and \(y\) are real numbers,[...]
Ellipse Discriminant Eccentricity
Discriminant can be used to ellipses for identifying the status of their intersections in conjunction with eccentricity. The eccentricity of[...]
Rationalising Denominators
Rationalising Denominators simplifies the fractions by multiplying the conjugates of the denominators. Worked Example of Rationalising Denominators Simplify \( \displaystyle\frac{1}{\sqrt{1}[...]
Probability with Replacement
Probability with Replacement is used for questions where the outcomes is returned back to the sample space again. Which means[...]
Integration Recurrence Formula
For Integration Recurrence Formula or reduction formula, it is important set a relationship between two consecutive terms by using mostly[...]
Polynomial Equation and Maximum Coefficient
If \( \alpha \) is a root of a polynomial equation \( P(x) = 0 \), then \( P(\alpha) =[...]
Absolute Value Inequalities
Absolute Value Inequalities are usually proved by the absolute value of a certain value is greater than or equal to[...]
Integration using Trigonometric Properties
Trigonometric properties such as the sum of squares of sine and cosine with the same angle is one, $$ \displaystyle[...]
Circle Geometry with Semicircles
There are many properties of circle geometry with semi circles, such as equal arcs on circles of equal radii subtend[...]
Volumes by Cylindrical Shells Method
Let’s consider the problem of finding the volume of the solid obtained by rotating about the \(x\)-axis or parallel to[...]
Integrating Binomial Expansion
Integrating Binomial Expansion is being used for evaluating certain series or expansions by substituting particular values after integrating binomial expansion.[...]
Probability Ratio using Combination
Counting Techniques for Probability Ratio using Combination The probability ratio of an event is the likelihood of the chance that[...]
Inequalities using Arithmetic Mean Geometric Mean
Arithmetic Mean of \(a\) and \(b\) is always greater than or equal to the Geometric Mean of \(a\) and \(b\),[...]
Mathematical Induction Inequality Proof with Two Initials
Usually, mathematical induction inequality proof requires one initial value, but in some cases, two initials are to be required, such[...]
Trigonometric Proof using Compound Angle Formula
There are many areas to apply the compound angle formulas, and trigonometric proof using compound angle formula is one of[...]
Finding a Function from Differential Equation
The solution of a differential equation is to find an expression without \(\displaystyle \frac{d}{dx} \) notations using given conditions. Note[...]
3 Ways of Evaluating Nested Square Roots
Nested square roots or nested radical problems are quite interesting to solve. The key skill for this question is to[...]