150 years of potato production
The humble potato has been a staple crop for many years. This note provides a brief overview of potato production in Australia for the period 1861 - 2011.
The humble potato has been a staple crop for many years. This note provides a brief overview of potato production in Australia for the period 1861 - 2011.
The last question in this year's HSC Extension 1 Mathematics exam asked students to locate the point on a given parabola that minimised the distance to a related point on another parabola. Students were told that the lines perpendicular to the tangents at the two points coincided. The plot on the left shows lines that do not coincide and therefore the distance is not minimised. However the plot on the right shows that the correct point has been located and the distance minimised. The full analysis is on my website.
Proof of a cubic inequality that is often set in maths competitions.
The solution to the last question of the 2020 Australian Mathematics Challenge for Years 7 and 8.
A number puzzle can require you to find the numbers in circles that add to numbers on connecting lines. This note shows how to do this easily.
Schedule of 20 sessions covering all the algebra required to enter Year 11. The page numbers are shown for the references used.
An application of a differential equation to a mixing problem, showing the solution that cannot be obtained by separation of variables.
An example of an integral used to evaluate arc length. The integral cannot be evaluated numerically so the inverse function is used. An alternative substitution is also adopted.
The Poisson probability function models discrete (count) data. This note looks at a linear transformation of the standard probability function and also derives the conditional probability of the number of outcomes of one Poisson random variable given the total number of outcomes of two independent Poisson random variables.
This note looks at how some mathematical relationships that can be simply stated produce vastly different results when a parameter value is altered.
Here is an article in linear algebra - how to calculate the transition matrix and use it to obtain coordinates with non-standard bases.
Some comments on the formula for Blood Alcohol Concentration (BAC) used in General Maths.
This note looks at the locus of the modulus of a function of complex numbers. The locus has some interesting properties. Increasing a parameter draws two identical entities together; after they touch they join and also give up a part of themselves to produce an offspring.
Plots of some interesting families of curves not normally encountered.
My solutions to the two problems given in December 2020
Ever wondered why the multiples (1,4,1) are used in Simpson's Rule? This article shows one method for obtaining these multiples.
This notes derives the formula for the term given the sum and the sum given the term for a specific family of series.
Formulae for calculating functions of distances between points and complex roots of unity. One formula for a point on the unit circle and another for a point on the real axis.
The standard method of finding the inflection point for a rational function with quadratic numerator and denominator can involve a fair bit of algebra and solving a difficult cubic equation. An alternative method using partial fractions makes the job a lot easier. This note works through such an example.
This note shows how to calculate the force required to lift a wheel up and over a step.
Derivation of the Fourier Series for a piecewise function.
More Extension 1 questions for revision
Revision questions for Mathematics
This note look at the location of inflection points relative to stationary points.
This note shows which integers can be expressed as the difference of two squares.
Even relatively simple functions are not simple to integrate as shown in these examples. The results are commonly available in a Table of Integrals.
An app to illustrate the limacon curve. The parametric coordinates are given by x = 2cos(t) - h cos(2t), y = 2sin(t) - h sin(2t). This app uses two arms with variable lengths a and b to control the shape of the curve, taking the place of the parameter h, and t to control the angle of rotation. See what happens when these variables are altered, either using the arrow buttons or inputting a value yourself.
A diverse family of curves are the rose curves. The parametric coordinates are x = 2b cos(t) cos(nt), y = 2b sin(t) cos(nt). Here b controls the radius of the figure and n controls the frequency. Changing n by a small amount (0.1 for example) gives a vastly different image. Increment n by 0.01 manually and watch what happens.
A trochoid is a curve formed when one circle rolls around another circle, either inside (hypo-) or outside (epi-). When the point describing the curve lies on the circumference of the moving circle we have hypocycloids and epicycloids. The parametric coordinates are ((a+1)cos(t) - cos(a+1)t, (a+1)sin(t) - sin(a+1)t). The interactive app shows the resulting curves for (-5
This note shows how the matrix exponential is used to solve a simple system of differential equations.
This note looks at three types of Means - Arithmetic, Geometric and Harmonic - and the situations where they are appropriate.
Modular arithmetic looks at relationships between numbers through the remainders when divided by the same integer. Powers of integers produce patterns in the remainders and these patterns can be used to solve problems.
This note looks at the distribution of letters in the English language as well as how the distribution can change significantly when a cipher is used.
Google used a novel method to find potential employees
Given a system of differential equations with two variables we can plot the gradient in 2D. A critical point, where the gradient is zero, can be classified as a centre, a spiral, a node or a saddlepoint, depending upon the values of the eigenvalues of the coefficient matrix. This note illustrates all possible types of points for a linear system. Also shown is an example of the predator-prey model.
A logical way to express the Product and Quotient Rules for differentiation.
Six questions to extend your mathematical knowledge and understanding
A set of 10 revision questions for Term 1 of Year 11. Other sets of questions are available only to students attending Kenderdine Maths Tutoring.
A common statement made by students is 'Standard Mathe will give you a higher ATAR than Advanced Maths' This note makes some observations regarding the difference between the HSC and Scaled marks for Standard Mathematics compared to Mathematics in the 2019 HSC.
This note gives one example of the construction and interpretation of a slope field for a first-order differential equation. The complete document with 14 examples is available to students of Kenderdine Maths Tutoring.
How to calculate the sum of the sequence 1,2,1,2,2,1,2,2,2,1,2,2,2,2,......
How are formula derived for the sums of powers of the natural numbers? Archimedes was able to sum the square numbers but it wasn't until 1713 that Bernoulli published a method for summing other powers.
Solution to the tangential circles problem
This note gives the possible patterns when the numbers 1 to 6 are arranged in a triangular pattern with constant sum along each edge.
Video showing how angles of any magnitude are related to acute angles and how to determine the appropriate sign of the trigonometric functions.
A note on two similar standard integrals that appear to be unrelated but more advanced maths than HSC level shows them to be related.
Links to software and apps to view swf videos on your computer or smart phone
School students are told that 0! equal 1 without knowing why. This note explains why
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