Level 2 Calculus Walkthrough

2025 NCEA Level 2 Calculus Question 2(c)

2025 Paper — Rates of change and justifying when coffee was given

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Question

Hemi is part of a student research project on how caffeine enters the bloodstream. A new device measures caffeine levels in his blood.

He is monitored for 180 minutes (3 hours) and, at some point, he is given a cup of coffee to drink.

\[ C(t)=\frac{t^3}{150}-1.4t^2+80t+240 \qquad \{0 \le t \le 180\} \] \[ \text{where }C\text{ is the concentration }(\mu\text{g/L}),\text{ and }t\text{ is time in minutes.} \] \[ \text{(i) Show that the rate of change of concentration is }-16\text{ after }60\text{ minutes.} \] \[ \text{(ii) Use calculus to justify when Hemi was given the coffee.} \]

Differentiate first, evaluate the derivative at 60 for part (i), then use first and second derivatives together to identify the minimum turning point for part (ii).

First walkthrough idea

Tip to try first

Differentiate \(C(t)\) first, then substitute \(t=60\) for part (i).

Step 1

Differentiate \(C(t)\)

Reveal the explanation and working for this step.

Show the first step’s working

Differentiate \(C(t)\) to obtain the rate-of-change function \(C'(t)\).

Worked result

\[ \frac{t^{2}}{50} - 2.8 t + 80 \]

Differentiate term by term to get \(C'(t)=\frac{t^2}{50}-2.8t+80\).

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91262 — Apply calculus methods in solving problems.

Method: Differentiating, evaluating a rate, and using derivatives to justify the minimum turning point.

This is Question 2(c) from the 2025 NCEA Level 2 Calculus paper for AS91262 — Apply calculus methods in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91262.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise differentiating, evaluating a rate, and using derivatives to justify the minimum turning point. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Finding a stationary value is only part of the argument; justify that it is the required minimum and respect the domain.

Continue practising

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