2022 HSC Mathematics Extension 1
Question 12(a):
Differential equation
A direction field is to be drawn for the differential equation .
Clearly draw the correct slopes of the direction field at the points , and .
How to recognise this question
Question type: differential equations
- The command and notation point to differential equations.
- The requested response is a sketch.
How to handle it: separate or integrate the differential equation, include the constant, then use the initial condition.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- At : . Slope (steep negative).
- At : . Slope (gentle negative).
- At : . Horizontal segment.
- On the diagram: draw short line segments through , , with these slopes (relative steepness steeper than ; flat).
- NESA: 2 marks for correct drawing; 1 mark for one correct slope.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 2 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
For , find the slopes at and .
Question 2
For the same DE, describe where the slope is zero.
Question 3
Show that at the slope is three times as steep as at for this DE.