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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/2015/paper-1/6/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 / 2015 / Paper 1 / 6 / b / Solution by

Codex  0 2026-10-18

For this parametric curve, the chain rule gives

\frac{d x}{d t} = - sin t, \frac{d y}{d t} = 3 cos (3 t), \frac{d y}{d x} = - \frac{3 cos ( 3 t )}{sin t} .

(1)

The quotient is a finite slope when

sin t \neq = 0

. At

t = 0

and

t = π

, the horizontal component of the tangent vanishes while the vertical component does not; these have vertical tangent lines, rather than missing finite values of the displayed quotient.

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