For , a curve is defined parametrically by x(t)=e^ 2t , y(t)= (3t). The point is…

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

For

t\ge 0

, a curve is defined parametrically by

x(t)=e^{2t},\qquad y(t)=\sin(3t).

The point

P(0,1)

is fixed in the plane. (1) Derive an expression for

\dfrac{dy}{dx}

in terms of

t

. (2) Determine all values of

t

in the interval

0\le t<2\pi

for which the tangent line to the curve passes through

P

. Justify that your values of

t

satisfy this condition. (3) For the smallest positive value of

t

found in part (b), write the equation of the corresponding tangent line. (4) For the same value of

t

, write the equation of the normal line to the curve and verify that it is perpendicular to the tangent line.