Differentiation

The trajectory of a particle in a high-energy magnetic field is described by the implicit curve $C\,$ defined by the equation

x^2 \sin y + y^2 \cos x = K

where $K\,$ is a constant. The particle is observed to pass through the point $P(π,π2)\displaystyle P\left(\pi, \frac{\pi}{2}\right)$ .

Show that $K=3π24\displaystyle K = \frac{3\pi^2}{4}$ .

[2]

bi.

Show that $dydx=y2sin⁡x−2xsin⁡yx2cos⁡y+2ycos⁡x\displaystyle \frac{dy}{dx} = \frac{y^2 \sin x - 2x \sin y}{x^2 \cos y + 2y \cos x}$ .

[4]

bii.

Hence, determine the numerical gradient of the trajectory at point $P$ .

[2]

biii.

The tangent to the trajectory at $P\,$ intersects the $x$ -axis at the point $Q$ . Find the exact $x$ -coordinate of $Q$ .

[3]

Differentiation Questions