Level 3 Integration Walkthrough
2020 NCEA Level 3 Integration Question 3(c)
2020 Paper
Question
An object originally moving at a constant velocity suddenly starts to accelerate. From the start of the object's acceleration, its motion can be modelled by \(\frac{dv}{dt}=t+e^{0.2t}\) for \(0\le t\le15\), where \(v\) is the velocity in metres per second and \(t\) is the time in seconds after the object starts to accelerate.
When \(t=0\), the velocity of the object was \(8\) metres per second. Find the velocity of the object when \(t=10\).
You must use calculus and give the results of any integration needed to solve this problem.
First walkthrough idea
Focus to try first
Acceleration is \(\frac{dv}{dt}\). Integrate it, use \(v(0)=8\), then evaluate \(v(10)\).
Step 1
Integrate the acceleration
Integrate the polynomial and exponential terms separately, including the reverse-chain-rule factor for \(e^{0.2t}\).
Show the first step’s working
Walkthrough overview
What this question practises
This 2020 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Integrating acceleration to find velocity.
This is Question 3(c) from the 2020 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 integrating acceleration to find velocity. Use the hints to plan the method, then repeat the question without hints and check each step.
Common mistake to avoid
Check the antiderivative by differentiating it, and handle constants and bounds explicitly.
Continue practising
- All 2020 Integration walkthroughs
- All AS91579 Integration years
- Practise more questions using this skill: Antidifferentiation