Mathematical Methods: should I choose it?

What will I learn?

How will I be assessed?

Progression and companion subjects

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Will I find it hard?

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• How do you feel about explaining a method and working through unfamiliar multi-step problems?
• Try the sample without looking at a worked answer, then check the steps you need to practise.

Does it scale up or down?

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Try a little of the subject

Rates of change, first principles and derivatives of power functions

1. Average rates and the slope of a secant A rate of change compares a change in one quantity with the corresponding change in another. If a quantity Q is measured in litres and time t in minutes, the rate has units litres per minute. For a function f over an interval [a,b], where a≠b, the average rate is [f(b)−f(a)]/(b−a). Both differences must refer to the same ordered pair of inputs; reversing only one difference changes the sign incorrectly. For f(x)=x²+2x, the values at x=1 and x=4 are 3 and 24. The average rate over [1,4] is (24−3)/(4−1)=7. Graphically this is the gradient of the secant through (1,3) and (4,24). It describes the whole interval, not the slope of the curve at every point in it. The secant equation is y−3=7(x−1). A linear function f(x)=mx+c has the same average rate m over every nonzero interval. In a table, equal input increments therefore produce equal output increments. Unequal input increments require comparing ratios, not raw output differences. A table with outputs increasing by 6 while the input steps change from 1 to 3 does not show a constant rate: the corresponding rates are 6 and 2. For a nonlinear function, the chosen interval matters. With f(x)=x²+2x, the average rate over [1,2] is 5, whereas over [2,4] it is 8. An average rate of zero means the endpoint outputs agree; the function may still rise and fall between them. In a motion problem, average velocity uses signed displacement divided by elapsed time. Average speed instead uses total distance divided by elapsed time, so it cannot always be read from the same endpoint calculation. A useful diagnostic reverses the calculation. If f(2)=5 and the average rate over [2,6] is −3, then f(6)−5=−3(6−2), giving f(6)=−7. The rate determines the endpoint change, but it does not determine all inte

Original practice question: For f(x)=(x²+1)e^x, calculate the equation of the tangent at x=0. Show your derivative and the coordinates of the point of tangency.

This is ATARMAxxing’s original sample learning material. It is not an official assessment task or a prediction of your school’s workload.

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• Mathematics — SACE Stage 1 (opens in a new tab)

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