← Integration methods: past papers
AS 91579 · 2024 paper · exponentials, a coffee, equal areas
Question 3(b) Achieved
Practises Differential equations
Two exponential terms to integrate, a log-form differential equation, an upper limit that appears on both sides, a bisected cosine area, and Newton’s law of cooling built from scratch.
Solve the differential equation dy/dx = 5/(4x − 3), given that y = 10 when x = 1.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 A number over a linear bracket: what does it integrate to?
A log, divided by the number in front of x:
y = int (5/(4x − 3)) dx = (5/4) ln |4x − 3| + c -
2 Use y = 10 when x = 1.
10 = (5/4) ln 1 + c = 0 + c, so c = 10.
What the marker wanted
- Achieved The whole solution, with c = 10 and the integration shown. Any sensible rounding is accepted.
In our words, from NZQA’s 2024 marking schedule.
✕Common mistake
Leaving out the 1/4 and writing 5 ln |4x − 3|. NZQA’s 2024 report lists treating every fraction as a plain log of its bottom among the Not Achieved answers, and this integral, with its missing factor of 4, is its example.How did it go?
Got it moves it further out. Review brings it back tomorrow.