Level 3 Integration Walkthrough
2019 NCEA Level 3 Integration Question 1(e)
2019 Paper
Question
The rate of change of quantity \(N\) at any instant is given by the differential equation \(\frac{dN}{dt}=kN\).
If \(N\) has positive values \(N_1\) and \(N_2\) at times \(t_1\) and \(2t_1\), respectively, prove that \(k=\frac{1}{t_1}\ln\!\left(\frac{N_2}{N_1}\right)\).
You must use calculus and show the results of any integration needed to solve the problem.
First walkthrough idea
Focus to try first
Separate the differential equation, integrate logarithmically, then compare the two given positive values of \(N\).
Step 1
Separate and integrate
Move \(N\) to the left and \(dt\) to the right.
Show the first step’s working
Walkthrough overview
What this question practises
This 2019 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Proving an exponential-growth relation from a differential equation.
This is Question 1(e) from the 2019 NCEA Level 3 Integration paper for AS91579 — Apply integration 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 AS91579.
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Learning summary
Review the method, not only the answer
This walkthrough helps you practise proving an exponential-growth relation from a differential equation. Use the hints to plan the method, then repeat the question without hints and check each step.
Common mistake to avoid
Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.
Continue practising
- All 2019 Integration walkthroughs
- All AS91579 Integration years
- Practise more questions using this skill: Differential equations