A curve is given by the parametric equations and

MATHEMATICS · Multi-part · 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^{2t}

and

y=3e^{t}-t

. The parameter

t

satisfies

\sin t=\dfrac{1}{2}

with

0\le t\le \pi

. (i) Find all values of

t

in the interval

[0,\pi]

that satisfy

\sin t=\dfrac{1}{2}

. (ii) Taking the smaller of the values obtained in part (i), calculate the numerical value of

\dfrac{dy}{dx}

at that value of

t

, correct to three decimal places. (iii) Indicate whether the tangent at this point is steep or gentle on the basis of the value found in part (ii).