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Mastering Continuity in Calculus: Key Concepts & Applications

NZ Year 13 Math: Master Advanced Concepts & Skills

Functions

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One to one functions

Function notation

Composite functions

Adding functions

Subtracting functions

Multiplying functions

Dividing functions

Operations of Functions – In a Nutshell

Inverse functions

Operations with functions

Inequalities of combined functions

Inequalities of Combined Functions

Difference quotient: applications of functions

Transformations of Functions

Combining transformations of functions

Transformations of functions: Horizontal translations

Transformations of functions: Vertical translations

Reflection across the y-axis: y = f(-x)

Reflection across the x-axis: y = -f(x)

Transformations of functions: Horizontal stretches

Transformations of functions: Vertical stretches

Even and odd functions

Quadratic Functions

Characteristics of quadratic functions

Transformations of quadratic functions

Quadratic function in general form: y = ax^2 + bx + c

Converting from general to vertex form by completing the square

Shortcut: Vertex formula

Converting general form into vertex form by applying the vertex formula

Vertex Formula Application

Graphing parabolas for given quadratic functions

Finding the quadratic functions for given parabolas

Applications of quadratic functions

Quadratic function in vertex form: y = a(x-p)^2 + q

Completing the square

Completing the Square

Polynomial Functions

Descartes&apos; rule of signs

Descartes' Rule of Signs

Descartes' rule of signs

Descartes" rule of signs

Polynomial long division

Polynomial synthetic division 

What is a polynomial function?

Polynomial functions

Recognizing a Polynomial Function

Remainder theorem

Finding the Remainder Using Synthetic Division and the Remainder Theorem

Factor theorem

Using the Factor Theorem to Test For Factors of a Polynomial

Rational zero theorem

Rational Zero Theorem

Characteristics of polynomial graphs

Determining the equation of a polynomial function

Applications of polynomial functions

Multiplicities of polynomials 

Imaginary zeros of polynomials

Solving polynomial inequalities

Fundamental theorem of algebra

Rational Functions

What is a rational function?

Point of discontinuity

Slant asymptote

Determining the Equation of a Slant Asymptote Using Long Division

Determining the Equation of a Slant Asymptote Using Synthetic Division

Solving equations with algebraic fractions

Solving rational equations

Solving rational inequalities

Solving Rational Inequalities With Two Fractions

Vertical asymptote

Horizontal asymptote

Horizontal Asymptote

Graphs of rational functions

Exponential Functions

Exponents: Product rule (a^x)(a^y) = a^(x+y)

Exponents: Product rule (a^x)(a^y) = a^(x+y)

Exponents: Division rule (a^x / a^y) = a^(x-y)

Exponents: Division rule: a^x / a^y = a^(x-y)

Exponents: Power rule (a^x)^y = a^(x * y)

Exponents: Power rule: (a^x)^y = a^(xy)

Exponents: Power rule

Exponents: Negative exponents

Exponents: Zero exponent: a^0 = 1

Exponents: Zero exponent: a^0 = 1

Exponents: Rational exponents

Graphing exponential functions

Graphing transformations of exponential functions

Finding an exponential function given its graph

Logarithmic Functions

Solving logarithmic equations

What is a logarithm?

Logarithms and Exponential Growth

Understanding Logarithms through Exponential Growth

Understanding Logarithms

Converting from logarithmic form to exponential form

Evaluating logarithms without a calculator

Common logarithms

Natural log: ln

Evaluating logarithms using change-of-base formula

Converting from exponential form to logarithmic form

Solving exponential equations with logarithms

Product rule of logarithms

Quotient rule of logarithms

Combining product rule and quotient rule in logarithms

Evaluating logarithms using logarithm rules

Graphing logarithmic functions

Finding a logarithmic function given its graph

Applications of Exponential and Logarithmic Functions

Exponential growth and decay by a factor

Exponential decay: Half-life

Exponential growth and decay by percentage

Finance: Compound interest

Finance: Compound Interest

Continuous growth and decay

Logarithmic scale: Richter scale (earthquake)

Logarithmic scale: pH scale

Logarithmic scale: dB scale

Finance: Future value and present value

Quadratic Equations and Inequalities

Solving quadratic equations by factorising

Solving quadratic equations by factoring

Solving quadratic equations by completing the square

Using quadratic formula to solve quadratic equations

Solving a quadratic equation with TWO REAL SOLUTIONS

Solving a quadratic equation with TWO COMPLEX SOLUTIONS

Nature of roots of quadratic equations: The discriminant

Applications of quadratic equations

Solving quadratic inequalities

Solving Simultaneous Equations

Money related questions in linear equations

Unknown number related questions in linear equations

Distance and time related questions in linear equations

Rectangular shape related questions in linear equations

Determining number of solutions to linear equations

Solving simultaneous equations by graphing

Solving systems of linear equations by graphing

Solving simultaneous equations by elimination

Solving systems of linear equations by elimination

Solving simultaneous equations by substitution

Solving systems of linear equations by substitution

Simultaneous Equations (Advanced)

Solving 3 variable simultaneous equations by substitution

Solving 3 variable systems of equations by substitution

Solving 3 variable simultaneous equations by elimination

Solving 3 variable systems of equations by elimination

Word problems relating 3 variable simultaneous equations

Word problems relating 3 variable systems of equations

Solving 3 variable simultaneous equations with no or infinite solutions

Solving 3 variable systems of equations (no solution, infinite solutions)

System of Equations With No Solution

System of Equations With Infinite Solutions

System of Equations With Infinite Solutions - Extended

Simultaneous linear equations

System of linear equations

Simultaneous linear-quadratic equations

System of linear-quadratic equations

Simultaneous quadratic-quadratic equations

Introduction to Trigonometry

Use sine ratio to calculate angles and sides (Sin = \(  \frac{o}{h}\) )

Use cosine ratio to calculate angles and sides (Cos =  \(  \frac{a}{h}\)  )

Combination of SohCahToa questions

Word problems relating ladder in trigonometry

Word problems relating guy wire in trigonometry

Year 13 Maths

math

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