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VICMath MethodsQuick questions
Unit 3
Quick questions on Optimisation and rates of change: VCE Math Methods Unit 3
12short 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 worked example?Show answer
A balloon's volume in cubic centimetres at time $t$ seconds is $V(t) = 100 + 30 t - t^3$ for $t \in [0, \sqrt{30}]$.
What is endpoint check?Show answer
If the domain is a closed interval $[a, b]$, the global max or min might occur at an endpoint, not at a stationary point. Always compare $Q(a)$, $Q(b)$, and $Q$ at each interior stationary point.
What is step 1?Show answer
Optimise surface area $S$.
What is step 2?Show answer
$S = 2 \pi r^2 + 2 \pi r h$ (two circular ends, plus lateral surface).
What is step 3?Show answer
Constraint: $\pi r^2 h = 1000$, so $h = \frac{1000}{\pi r^2}$.
What is step 5?Show answer
$S'(r) = 4 \pi r - \frac{2000}{r^2}$. Set to zero: $4 \pi r = \frac{2000}{r^2}$, so $r^3 = \frac{500}{\pi}$ and $r = \sqrt[3]{\frac{500}{\pi}} \approx 5.42$ cm.
What is step 6?Show answer
$S''(r) = 4 \pi + \frac{4000}{r^3} > 0$, so this is a minimum. $h = \frac{1000}{\pi r^2} \approx 10.83$ cm.
What is forgetting to reduce to one variable?Show answer
You cannot differentiate a function of two variables in Math Methods. Always use the constraint to eliminate one variable before differentiating.
What is ignoring the domain?Show answer
Negative radii, negative populations, or fractional people are not feasible. State the valid domain explicitly and check both interior critical points and endpoints.
What is skipping the classification step?Show answer
A stationary point might be a maximum, minimum or stationary point of inflection. The second derivative test confirms which.
What is wrong units in the final answer?Show answer
Markers expect units (cubic centimetres, dollars per week, metres per second). Missing units is a routine 1-mark penalty.
What is confusing rate of change with the function itself?Show answer
"How fast is $Q$ changing?" asks for $\frac{dQ}{dt}$, not $Q(t)$.