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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/9/b/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 9 / b / Solution by

Codex  0 2026-10-18

Real points require

x \geq 0

, and the two branches are

y = \pm x^{3/2}, x \geq 0.

(1)

This is a semicubical parabola, symmetric about the horizontal axis and with a cusp at the origin. The parametrization

x = t^{2}

y = t^{3}

traces both branches. Their limiting slopes at the origin are zero, so the common tangent direction is the positive horizontal axis, even though the curve has a cusp there.

Figure 1.
The two real branches of the semicubical parabola and their cusp at the origin
. **Details**.

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