 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/4/b/solution

At the stated point, the radial distance is

7 π /2

, so

4 π - t = 7 π /2

and

t = π /2

. Differentiate the parametric curve:

\overset{x}{˙} = - sin t + (4 π - t) cos t, \overset{y}{˙} = - cos t - (4 π - t) sin t .

t = π /2

, these are

\overset{x}{˙} = - 1

and

\overset{y}{˙} = - 7 π /2

. Since

\overset{x}{˙} \neq = 0

, the tangent line has finite slope

\frac{d y}{d x} = \frac{y ˙}{x ˙} = \frac{7 π}{2} .

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