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QLDMath MethodsQuick questions
Unit 3: Further calculus and statistics
Quick questions on Optimisation and rates of change (QCE Mathematical Methods Unit 3)
13short 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 the first derivative?Show answer
The first derivative $f'(x)$ measures the instantaneous rate of change. Its sign reveals function behaviour.
What is the second derivative?Show answer
The second derivative $f''(x)$ measures the rate of change of $f'$, equivalent to the concavity of $f$.
What is the optimisation method?Show answer
Every optimisation problem follows the same five steps.
What is related rates of change?Show answer
When two or more related quantities change with time, the chain rule links their rates.
What is curve sketching?Show answer
For $f(x) = x^3 - 3 x$, find and classify all stationary points.
What is optimisation in context?Show answer
A rectangle is inscribed under the curve $y = 9 - x^2$ in the first quadrant, with two sides on the axes. Find the dimensions of the rectangle of maximum area.
What is related rates (sliding ladder)?Show answer
A 5 m ladder rests against a wall. The base slides away from the wall at 0.2 m/s. How fast is the top sliding down when the base is 3 m from the wall?
What is skipping the domain restriction?Show answer
In the open-box example, $x$ must satisfy $0 < x < 10$, otherwise the base is non-physical. The other stationary point ($x \approx 12.74$) is mathematically valid but contextually impossible.
What is forgetting to classify?Show answer
A stationary point is not automatically a maximum. Use the second-derivative test or a sign chart.
What is reporting the optimising variable instead of the optimal quantity?Show answer
If the question asks "find the maximum volume", report the volume, not the value of $x$. If it asks for both, give both.
What is mixing up the chain rule in related rates?Show answer
Always write the chain rule statement explicitly. Substituting numerical values before differentiating loses the relationship between rates.
What is omitting units?Show answer
PSMT and EA markers strip marks for missing or wrong units. Always state cm, cm$^2$, cm$^3$, m/s as appropriate.
What is treating endpoints as stationary points?Show answer
On a closed interval, the maximum may occur at an endpoint where the derivative is non-zero. Always evaluate the function at the endpoints as well.