Differentiation

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Question 5

A precision-engineered micro-shuttle follows a path $C$ in a magnetic field. The position of the shuttle at time $t$ , where $t$ is a parameter in radians, is given by the parametric equations

\cos 2t, \quad y = 8 \sin^3 t, \quad -\frac{\pi}{2} < t < \frac{\pi}{2}

The shuttle passes through point $P$ when $\frac{\pi}{6}$ .

The line $l$ represents the tangent to the shuttle's path at point $P$ .

Use parametric differentiation to show that (i) $dydx=ksin⁡t\frac{\mathrm{d}y}{\mathrm{d}x} = k \sin t$ where $k$ is a constant to be found. (ii) an equation for the tangent line $l$ is $3 x + 4 y - 10 = 0$ .

[7]

The path $C$ is intersected again by the line $l$ at the point $Q$ .

Using algebra and showing detailed reasoning, find the exact coordinates of $Q$ .

[6]

Differentiation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Question 5

A precision-engineered micro-shuttle follows a path $C$ in a magnetic field. The position of the shuttle at time $t$ , where $t$ is a parameter in radians, is given by the parametric equations

\cos 2t, \quad y = 8 \sin^3 t, \quad -\frac{\pi}{2} < t < \frac{\pi}{2}

The shuttle passes through point $P$ when $\frac{\pi}{6}$ .

The line $l$ represents the tangent to the shuttle's path at point $P$ .

[7]

The path $C$ is intersected again by the line $l$ at the point $Q$ .

Using algebra and showing detailed reasoning, find the exact coordinates of $Q$ .

[6]

Differentiation

Question 5

Differentiation Questions

Differentiation

Question 5

Differentiation Questions