22M.1.AHL.TZ1.7

pestleMathematicsAIHLPaper 122M· ahl-5-15-slope-fieldssource ↗

A slope field for the differential equation $\frac{d y}{d x} = x^{2} + \frac{y}{2}$ is shown.

Some of the solutions to the differential equation have a local maximum point and a local minimum point.

Write down the equation of the curve on which all these maximum and minimum points lie.

[1]

a.i.

Sketch this curve on the slope field.

[1]

a.ii.

The solution to the differential equation that passes through the point $(0, - 2)$ has both a local maximum point and a local minimum point.

On the slope field, sketch the solution to the differential equation that passes through $(0, - 2)$ .

[2]

Markscheme / solution

$x^{2} + \frac{y}{2} = 0 (y = - 2 x^{2})$ A1

[1 mark]

a.i.

$y = - 2 x^{2}$ drawn on diagram (correct shape with a maximum at $(0, 0)$ ) A1

[1 mark]

a.ii.

correct shape with a local maximum and minimum, passing through $(0, - 2)$ A1

local maximum and minimum on the graph of $y = - 2 x^{2}$ A1

[2 marks]

Examiners’ report

This question was very poorly done and frequently left blank. Few candidates understood the connection between the differential equation and maximum and minimum points. Even when the equation $\frac{d y}{d x} = 0$ was correctly solved, it was rare to see the curve correctly drawn on the slope field. Some were able to draw a solution to the differential equation on the slope field though often not through the given initial condition.

a.i.

[N/A]

a.ii.

[N/A]