HSC Tutor: Mathematics Advanced 2U - Dr Amir tutoring

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13.5 hours: (8:30-10am)

Date: Dec 2024 - Jan 2025

Dr Amir presents online,  in a lecture format, to a group of approved registered participants. Dr Amir designs the questions presented in this session based on Dr Amir's understanding of HSC and Mathematics Advanced  and Dr Amir's Pedagogy. The materials presented are subject to Copy Right and can not be saved or copied or distributed in any format. Dr Amir presents from a selection of the  following topics:

• Algebra 
• Functions, graphs and transformations 
• Horizontal/Vertical Dilation 
• Horizontal/Vertical Translation
• Combined transformations
• Trigonometric Functions
• Trigonometric Identities, Equations
• Exponential functions
• Logarithmic functions/Equations
• Differentiation 
• Logarithmic functions
• Trigonometric functions
• Exponential functions
• Anti-derivatives 
• Increasing/decreasing curves
• Stationary/Turning Points
• Optimisation problems using differentiation
• Curve-sketching using the derivative 
• Extending calculus
• Integration
• Trapezoidal rule
• Definite and indefinite integrals
• Logarithmic functions
• Trigonometric functions
• Exponential functions
• Areas enclosed by x or y axis
• Motion and rates 
• Sequences and series 
• Arithmetic sequences and series
• Geometric sequences and series
• Limiting sum
• Series and finance
• Arithmetic growth/decay
• Geometric growth/decay
• Simple/compound interest
• Annuities
• Loans and geometric series
• Probability/Statistics
• Displaying and interpreting data
• Bi-variate data
• Correlation and regression
• Line of best fit
• Least squares regression line
• Pearson correlation 
• Standard deviation
• Variance
• Discrete/Continuous probability distributions 
• Probability density functions (PDF)
• Cumulative distribution functions (CDF)
• Quantile/Percentile
• Z-scores
• Normal distribution

Let’s embark on a mathematical odyssey through the realms of high  school mathematics, guided by the sagacious Dr. Amir. Brace yourself for  a comprehensive exploration of these captivating topics:

Algebra, the magical art of balancing equations and deciphering  unknowns, awaits! Dr. Amir will introduce you to algebraic expressions,  equations, and inequalities. You’ll dance with linear functions,  quadratic polynomials, and systems of equations. Algebra is like a  symphony of symbols, where variables harmonize to reveal hidden truths.

Functions are mathematical machines that transform inputs into  outputs. Dr. Amir will guide you through function notation, domain,  range, and composition. Graphs will come alive—lines, parabolas, and  exponential curves. Transformations will twist and stretch these graphs,  revealing their secrets.

Imagine stretching or compressing a graph horizontally or vertically.  Dr. Amir will show you how dilation affects functions. You’ll encounter  scaling factors, amplitude, and phase shifts. It’s like adjusting a  cosmic telescope to explore mathematical constellations.

Functions love to slide! Dr. Amir will teach you the art of  translation. Shifting left or right (horizontal) or up and down  (vertical) changes the game. You’ll explore intercepts, asymptotes, and  intercepting aliens in the Cartesian plane.

Why settle for one transformation when you can have a party? Dr. Amir  will combine dilation, translation, and reflection. Functions will  twist, flip, and pirouette. It’s like choreographing a mathematical  ballet.

Triangles and angles unite! Dr. Amir will introduce sine, cosine, and  tangent. You’ll sail the unit circle, decode radians, and unravel  periodicity. Trig functions oscillate like cosmic waves—prepare for a  celestial voyage.

Dr. Amir’s treasure trove includes Pythagorean identities, sum and  difference formulas, and double-angle magic. You’ll solve trig  equations, prove identities, and navigate the mysterious world of  trigonometry.

Exponential growth and decay await! Dr. Amir will reveal the secrets  of base e. You’ll explore compound interest, radioactive decay, and  population explosions. Exponential functions are like wildfire—spreading  knowledge across mathematical landscapes.

Logarithms—the inverse of exponentials—hold mystical powers. Dr. Amir  will guide you through logarithmic properties, change of base, and  solving logarithmic equations. It’s like decoding ancient scrolls with a  mathematical Rosetta Stone.

Calculus time! Dr. Amir will introduce derivatives—the slopes of  curves. You’ll find rates of change, tangents, and critical points.  Anti-derivatives (indefinite integrals) are like unraveling time-travel  equations. Hold on tight!

Dr. Amir loves hills and valleys! You’ll explore concavity,  inflection points, and stationary curves. Increasing or  decreasing—functions reveal their secrets. It’s like hiking through  mathematical landscapes.

Real-world puzzles await! Dr. Amir will tackle  optimization—maximizing areas, minimizing costs, and optimizing rocket  trajectories. Calculus becomes a compass for navigating life’s  challenges.

Dr. Amir’s canvas is the Cartesian plane. You’ll sketch curves using  derivatives—finding intercepts, asymptotes, and turning points. It’s  like painting with mathematical brushes.

Beyond the basics lies a universe of calculus. Dr. Amir will delve  into parametric equations, polar coordinates, and vectors. Calculus  becomes a cosmic adventure.

Anti-derivatives meet definite integrals. Dr. Amir will calculate  areas under curves, volumes of revolution, and centroids. Imagine  finding the area enclosed by a dragon-shaped curve—it’s like  mathematical origami.

Numerical integration enters the scene. Dr. Amir will use trapezoids  to estimate areas. Definite integrals become your mathematical  GPS—navigating through mathematical landscapes.

Uncertainty meets data. Dr. Amir will display and interpret data.  You’ll explore correlation, regression, and the line of best fit.  Standard deviation and variance become your statistical companions.

Probability density functions (PDFs) and cumulative distribution  functions (CDFs) await. Dr. Amir will introduce quartiles, z-scores, and  the normal distribution. Probability becomes a cosmic dice game.

Probability Density Functions (PDFs):
 

• A PDF is a mathematical function that describes the likelihood of a continuous random variable taking on a specific value.
• For continuous random variables, we can’t assign probabilities to  individual points, but we can talk about the probability of a value  falling within a certain range.
• The PDF, denoted as f(x), satisfies the following conditions: 
  • (f(x) \geq 0) for all (x) in the real numbers.
  • (f) is piecewise continuous.
  • (\int_{-\infty}^{\infty} f(x) , dx = 1).
• It provides the relative likelihood of observing different values of the random variable. For example, in the context of waiting times for an elevator, we can use a  PDF to calculate the probability that a person waits less than a certain  time.

Cumulative Distribution Functions (CDFs):
 

• The CDF gives the cumulative probability that a random variable takes on a value less than or equal to a specific point.
• It’s the integral of the PDF from negative infinity up to that point.
• The CDF, denoted as F(x), starts at 0 and gradually approaches 1 as we move to the right.
• For any value (x), (F(x)) represents the probability that the random variable is less than or equal to (x).
• The CDF is useful for calculating percentiles and understanding the overall distribution of the random variable.

Quantile/Percentile:
 

• Quantiles divide the data into equal portions.
• The p-th percentile is the value below which (p%) of the data falls.
• For example, the median (50th percentile) is the value that splits the data into two equal halves.
• Percentiles are essential for understanding data distribution and making comparisons.

Z-scores:
 

• A Z-score (also called a standard score) measures how many standard deviations a data point is from the mean.
• It standardizes data, allowing us to compare values from different distributions.
• A positive Z-score indicates a value above the mean, while a negative Z-score is below the mean.
• Z-scores are particularly useful in assessing outliers and understanding relative positions within a dataset.

Normal Distribution:
 

• The normal distribution, also known as the Gaussian distribution, is a bell-shaped curve.
• Many natural phenomena follow this distribution (e.g., heights, IQ scores, errors in measurements).
• It’s characterized by its mean ((\mu)) and standard deviation ((\sigma)).
• The standard normal distribution has a mean of 0 and a standard deviation of 1.
• Z-scores are directly related to the normal distribution, making it a fundamental concept in statistics.

In summary, these concepts provide the foundation for understanding  continuous random variables, their distributions, and how to analyze and  interpret data. Dr. Amir’s tutoring has equipped you with essential  tools for tackling real-world problems involving uncertainty and  variability.

So, dear student, sharpen your pencils, ignite your curiosity, and  let Dr. Amir’s mathematical lantern guide you through this enchanting  labyrinth

Registration is required. 

13.5 hours: (10-11:30am)

Date: Dec 2024 - Jan 2025

Dr Amir presents online,  in a lecture format, to a group of approved registered participants. Dr Amir designs the questions presented in this session based on Dr Amir's understanding of HSC and Mathematics Standard and Dr Amir's Pedagogy. The materials presented are subject to Copy Right and can not be saved or copied or distributed in any format. Dr Amir presents from a selection of the  following topics:

• Financial Mathematics
• Tax
• Investments
• Depreciation
• Loans
• Simple/Compound Interest
• Inflation/Appreciation/Depreciation
• Shares
• Credit cards
• Declining-balance method
• Annuities
• Future/Present Value of an Annuity
• Loan Repayments
• Budgeting
• Simultaneous equations
• Break-even analysis
• Rates
• Ratios
• Fuel consumption rate
• Energy consumption rate
• Right angled trigonometry
• Angle of Elevation/Depression
• Sine Rule
• Cosine Rule
• Compass radial surveys
• Bearings
• Bi-variate data analysis
• Pearson Correlation
• Lines of best fit
• Z-scores
• Interpolation
• Extrapolation
• Probability/Statistical analysis
• Relative frequency
• Expected frequency
• Normal distribution
• Standardised score
• Pareto charts
• Box and whisker plots
• Outliers
• The critical path
• The Maximum flow minimum cut theorem
• Graphs and functions
• Parabola
• Exponential
• Hyperbola
• Reciprocal functions
• Networks
• Paths
• Cycles
• Trees
• Minimum Spanning Trees
• The shortest path
• The critical path
• Formulas 
• Equations
• Linear relations
• Measurement
• Perimeter
• Area
• Volume
• Latitude/Longitude
• Time zones

Let’s explore these topics in depth:

Financial mathematics deals with managing money, investments, and  financial transactions. Key concepts include interest rates, present  value, and future value. Here are some essential components:

• Simple Interest: Calculated as (I = P \cdot r \cdot  t), where (I) is the interest, (P) is the principal amount, (r) is the  interest rate, and (t) is the time (in years).
• Compound Interest: Incorporates interest on both  the principal and accumulated interest. The formula is (A = P \left(1 +  \frac{r}{n}\right)^{nt}), where (A) is the final amount, (n) is the  number of compounding periods per year, and (t) is the time.
• Future Value (FV): Represents the value of an investment at a future date, considering interest.
• Present Value (PV): The current worth of a future sum of money, discounted at a specific interest rate.

Understanding taxation is crucial for financial planning. Topics include income tax, deductions, credits, and tax brackets.

Investments involve allocating funds to assets (stocks, bonds, real  estate) to generate returns. Concepts include risk, diversification, and  portfolio management.

Depreciation accounts for the decrease in value of assets over time.  Methods include straight-line depreciation and declining balance.

Loans involve borrowing money with interest. Types include personal loans, mortgages, and student loans.

Inflation reduces the purchasing power of money. Appreciation refers  to an asset’s increase in value, while depreciation is its decrease.

Shares (stocks) represent ownership in a company. Understanding stock markets, dividends, and stock valuation is essential.

Credit cards allow borrowing against a credit limit. Managing credit card debt and interest rates is crucial.

Used for calculating depreciation, this method allocates more depreciation in the early years of an asset’s life.

Annuities involve regular payments over time. Examples include retirement pensions and insurance policies.

Calculating the value of annuity payments at a future or present date.

Understanding loan amortization and repayment schedules.

Creating and managing a budget to allocate income effectively.

Solving systems of equations with multiple variables.

Determining the point at which costs equal revenue.

Understanding rates of change and comparing quantities.

• Right-Angled Trigonometry: Involves sine, cosine, and tangent ratios.
• Angle of Elevation/Depression: Used in surveying and navigation.

• Sine Rule: Relates side lengths and angles in non-right triangles.
• Cosine Rule: Calculates side lengths or angles in non-right triangles.

Used in navigation and land surveying.

Examining relationships between two variables.

Measures the strength and direction of linear association between two variables.

Used in regression analysis to model data trends.

Standardizing data for comparison.

Estimating values within or beyond a given data range.

Understanding probability distributions, sampling, and hypothesis testing.

Used in probability calculations.

The bell-shaped curve and Z-scores.

Visualizing data distributions and outliers.

In project management, identifies the longest path for completing tasks.

Graph theory concept related to network flow.

Understanding functions, domain, range, and graphing.

• Parabola: U-shaped curve.
• Exponential Function: Rapid growth or decay.
• Hyperbola: Two intersecting curves.
• Reciprocal Functions: Inverse of linear functions.

Registration is required. 

13.5 hours: (12-1:30pm)

Date: Dec 2024 - Jan 2025

Dr Amir presents online,  in a lecture format, to a group of approved registered participants. Dr Amir designs the questions presented in this session based on Dr Amir's understanding of HSC and Mathematics Extension 1 and Dr Amir's Pedagogy. The materials presented are subject to Copy Right and can not be saved or copied or distributed in any format. Dr Amir presents from a selection of the  following topics:

• Graphs sketching
• Graphing Sums and products
• Absolute functions
• Square root functions
• Inverse functions
• Polynomials
• The remainder theorem
• The factor theorem
• Multiple zeros
• Probability/Statistics
• Combinatorics
• Factorial
• Ordered/Unordered selections
• Circle arrangements
• Pigeonhole principle
• Pascal's Triangle
• Binomial expansion/Theorem
• Binomial probability
• Binomial distributions
• Bernoulli distribution
• Normal approximations
• Sample proportions
• Exponential Growth/Decay
• Trigonometry
• Compound angles
• The sum of sine and cosine
• Inverse Trigonometric Functions
• Differentiating
• Integrating
• Integration by substitution
• Integration of trig functions
• Graphing inverse trig functions
• Double angle
• The t-formula
• Products to sums in trigonometry
• Mathematical induction: Series/Divisibility
• Vectors
• Vector components
• The dot product
• Projection of vectors
• Projectile motion
• The time equations of movement
• The equation of path
• Differential equations and its applications

Dr. Amir’s Mathematics Tutoring: Unlocking Potential for High School Students in Extension 1 Mathematics

Dr. Amir, an experienced online mathematics tutor, is committed to  helping high school students excel in their mathematical journey. His  personalised approach, deep subject knowledge, and effective teaching  methods make him a sought-after mentor. Let’s delve into the quality  tutoring Dr. Amir provides across various topics:

• Graph Sketching:
   
  • Dr. Amir emphasizes understanding the behavior of functions through  graph sketching. Students learn to identify critical points, asymptotes,  and concavity.
  • He encourages hands-on practice, ensuring students can confidently  sketch graphs of polynomial, rational, and trigonometric functions.
• Graphing Sums and Products:
   
  • Dr. Amir simplifies complex expressions involving sums and products.  Students gain clarity on how to graph these composite functions.
  • His step-by-step guidance helps students visualize the interplay of different functions.
• Absolute Functions and Square Root Functions:
   
  • Dr. Amir demystifies absolute value and square root functions. He covers domain, range, and transformations.
  • Students learn to graph these functions accurately and understand their significance.
• Inverse Functions:
   
  • Dr. Amir ensures students grasp the concept of inverse functions. He  explores one-to-one functions, finding inverses, and their graphical  representation.
  • His patient explanations help students navigate this fundamental topic.
• Polynomials and The Remainder Theorem:
   
  • Dr. Amir dives into polynomial functions, including long division and synthetic division.
  • He connects the remainder theorem to polynomial evaluation, reinforcing algebraic skills.
• The Factor Theorem and Multiple Zeros:
   
  • Dr. Amir simplifies the factor theorem, helping students find zeros and factors of polynomials.
  • His focus on multiple zeros enhances problem-solving abilities.
• Probability and Statistics:
   
  • Dr. Amir covers probability distributions, expected values, and variance.
  • Students gain insights into real-world applications and statistical reasoning.
• Combinatorics and Factorial:
   
  • Dr. Amir introduces combinatorics, permutations, and combinations.
  • His engaging examples make counting principles accessible.
• Binomial Expansion and Binomial Probability:
   
  • Dr. Amir demystifies binomial expansion using Pascal’s triangle.
  • Students learn to calculate probabilities in binomial experiments.
• Trigonometry and Compound Angles:
   
  • Dr. Amir navigates trigonometric functions, identities, and compound angle formulas.
  • Students master concepts like sum and difference of angles.
• Integration and Differentiation:
   
  • Dr. Amir simplifies integration techniques, including substitution and trigonometric integrals.
  • His differentiation lessons cover rules, applications, and optimization.
• Vectors and Projectile Motion:
   
  • Dr. Amir introduces vectors, dot products, and vector components.
  • Students explore projectile motion, understanding parabolic paths.
• Differential Equations and Applications:
   
  • Dr. Amir delves into differential equations, emphasizing practical applications.
  • Students appreciate their relevance in modeling real-world phenomena.

In summary, Dr. Amir’s tutoring transcends rote learning. He fosters  critical thinking, problem-solving, and mathematical intuition. With Dr.  Amir, students not only conquer topics but also develop a lifelong love  for mathematics.

Registration is required. 

13.5 hours:  (2-3:30pm)

Date: Dec 2024 - Jan 2025

Dr Amir presents online,  in a lecture format, to a group of approved registered participants. Dr Amir designs the questions presented in this session based on Dr Amir's understanding of HSC and Mathematics Extension 2 and Dr Amir's Pedagogy. The materials presented are subject to Copy Right and can not be saved or copied or distributed in any format. Dr Amir presents from a selection of the  following topics:

• Complex numbers
• Square root of a complex number
• Argand Diagram
• Modulus and Argument
• Euler formula
• De Moivre's Theorem
• Complex numbers in quadratic equations
• Polynomials in complex numbers
• Roots of unity
• Roots of complex numbers
• Curves on complex plane
• Mathematical Proof
• Proof by contradiction
• Proof by counter-example
• Proofs by numbers
• Proofs by inequalities
• Proofs in inequalities
• 3D Vectors
• Angle between vectors
• Using vectors in geometry proofs
• Mathematical induction
• Series and sigma notation
• Recursive formula proofs
• Integration by substitution
• Integration: Partial fractions
• Integration by parts
• Integration of trigonometric functions
• Integration of inverse trig functions
• Integration of logarithmic functions
• Integration using t-formula
• Recurrence relations
• Velocity in terms of x
• Acceleration in terms of x
• Simple harmonic motion
• Projectile motion
• Forces and motion
• Equations of motion
• Resisted vertical motion
• Resisted projectile motion

Dr. Amir’s Mathematics Tutoring Services: Unlocking Potential for High School Students

Are you a high school student seeking expert guidance in mathematics?  Look no further! Dr. Amir’s Mathematics Tutoring offers exceptional  services designed to empower students, enhance their understanding, and  boost their academic performance. Let’s delve into the comprehensive  topics covered by Dr. Amir’s tutoring:

• Complex Numbers:
   
  • Dr. Amir introduces complex numbers, emphasizing their real and  imaginary components. Students learn to perform arithmetic operations,  represent complex numbers geometrically, and solve equations involving  them.
• Square Root of a Complex Number:
   
  • Dr. Amir demystifies the square root of complex numbers, exploring  both principal and secondary roots. Students gain insights into the  Argand plane representation.
• Argand Diagram:
   
  • Dr. Amir guides students through the Argand diagram—a powerful tool  for visualizing complex numbers. Understanding quadrants, polar form,  and plotting complex points becomes second nature.
• Modulus and Argument:
   
  • Dr. Amir delves into the modulus (magnitude) and argument (angle) of  complex numbers. Students grasp their significance in trigonometric  form and polar coordinates.
• Euler’s Formula:
   
  • Dr. Amir unveils Euler’s remarkable formula: e^(iθ) = cos(θ) + i sin(θ). Students explore its applications in complex analysis and exponential growth.
• De Moivre’s Theorem:
   
  • Dr. Amir equips students with De Moivre’s theorem, enabling them to  find powers and roots of complex numbers. The polar form shines as they  tackle trigonometric identities.
• Complex Numbers in Quadratic Equations:
   
  • Dr. Amir bridges theory and practice, solving quadratic equations  with complex roots. Students conquer the quadratic formula and explore  discriminants.
• Polynomials in Complex Numbers:
   
  • Dr. Amir navigates polynomial functions involving complex  coefficients. Students learn to factor, find roots, and analyze  behavior.
• Roots of Unity:
   
  • Dr. Amir introduces the concept of roots of unity, connecting them  to complex numbers. Students explore their geometric arrangement on the  unit circle.
• Curves on the Complex Plane:
   
  • Dr. Amir unravels the beauty of curves traced by complex functions.  Students visualize transformations, branch cuts, and singularities.
• Mathematical Proof Techniques:
   
  • Dr. Amir emphasizes rigorous proof methods: 
    • Proof by Contradiction: Students learn to assume the opposite and derive contradictions.
    • Proof by Counter-Example: Dr. Amir illustrates how counter-examples disprove conjectures.
    • Proofs by Numbers and Inequalities: Students wield mathematical tools to validate statements.
• 3D Vectors and Geometry:
   
  • Dr. Amir extends tutoring to 3D vectors. Students explore cross  products, dot products, and vector equations of lines and planes.
• Mathematical Induction:
   
  • Dr. Amir demystifies induction, empowering students to prove statements for all natural numbers.
• Series and Sigma Notation:
   
  • Dr. Amir dives into series, convergence, and summation notation. Students master arithmetic, geometric, and telescoping series.
• Integration Techniques:
   
  • Dr. Amir covers integration strategies: 
    • Integration by Substitution
    • Partial Fractions
    • Integration by Parts
    • Trigonometric and Inverse Trig Integrals
    • Logarithmic Integrals
    • Using t-Formulas
• Physics Applications:
   
  • Dr. Amir relates mathematics to physics: 
    • Projectile Motion
    • Forces and Motion
    • Equations of Motion
    • Simple Harmonic Motion
    • Resisted Vertical and Projectile Motion

Dr. Amir’s mission is clear: unleash every student’s potential.  With personalized lessons, expert guidance, and a passion for teaching,  Dr. Amir transforms mathematical challenges into triumphs. Whether  you’re preparing for exams or seeking deeper understanding, Dr. Amir’s  Mathematics Tutoring is your compass to success! 

Registration is required.