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NSWMaths AdvancedQuick questions
Year 12: Calculus
Quick questions on Applications of differentiation: stationary points, inflection, optimisation and related rates
11short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is stationary points?Show answer
A stationary point of $f(x)$ occurs where $f'(x) = 0$. To classify it, use one of two tests.
What is concavity and points of inflection?Show answer
A point of inflection is where the concavity changes. Find candidates by solving $f''(x) = 0$, then confirm the concavity actually changes by checking the sign of $f''$ either side.
What is optimisation word problems?Show answer
1. Draw a diagram and label variables. 2. Write the quantity to optimise (area, volume, cost) as a function.
What is related rates?Show answer
When two quantities $x$ and $y$ both change with time and are linked by an equation, differentiate the equation with respect to $t$ (implicit differentiation). Substitute the given rate and the instantaneous value to solve for the unknown rate.
What is first derivative test?Show answer
Check the sign of $f'(x)$ just before and just after the stationary point.
What is second derivative test?Show answer
Evaluate $f''(x)$ at the stationary point.
What is forgetting to classify the stationary point?Show answer
Setting $f'(x) = 0$ only finds candidates. You must justify whether each is a maximum, minimum or horizontal inflection.
What is assuming $f'' = 0$ guarantees an inflection?Show answer
The concavity must actually change. For example, $f(x) = x^4$ has $f''(0) = 0$ but no inflection there.
What is not eliminating one variable in optimisation?Show answer
You cannot differentiate a function of two variables in Maths Advanced. Use the constraint to reduce to one.
What is ignoring units and domain?Show answer
A negative radius is meaningless. Always check the feasible domain and state units in the final answer.
What is confusing $\frac{dy}{dx}$ with $\frac{dy}{dt}$ in related rates?Show answer
When time is the underlying variable, every quantity must be differentiated with respect to $t$.