A curve is given by the parametric equations x = e^ t t

MATHEMATICS · Subjective · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A curve is given by the parametric equations

x = e^{t}\cos t, \quad y = e^{t}\sin t, \quad t \in \mathbb R.

Answer the following questions. (i) Obtain

\dfrac{dy}{dx}

in terms of

t

. (ii) Show that

\dfrac{dy}{dx} = \tan\left(t + \dfrac{\pi}{4}\right)

and hence express the slope solely in terms of

x

and

y

. (iii) Determine all real values of

t

for which the tangent to the curve is perpendicular to the straight line

y = x.