Practice section 4 of 6

A curve has $x=3t+2$, $y=t+3/t$, where $t\ne0$. Find the tangent at $P$ where $x=11$, and determine whether that tangent cuts the curve again. [8]

Curve $C$ has $x=\theta-\sin\theta$, $y=1-\cos\theta$, $0\le\theta\le2\pi$.

(i) Show that $dy/dx=\cot(\theta/2)$ and find the gradient at $\theta=\pi$. Describe the tangents as $\theta\to0$ and $\theta\to2\pi$.

(ii) Sketch $C$, clearly showing its features at $\theta=0,\pi,2\pi$.

(iii) For $P$ with parameter $p$, $0<p<\pi/2$, show that the normal at $P$ crosses the x-axis at $(p,0)$.

Find the cartesian equation of the curve with parametric equations: $x=-4t-\dfrac1t$, $y=3t+\dfrac1{2t}$. [4]

The curve $C$ is defined by $x=t^2$, $y=\ln t$, $t>0$.

(i) Sketch $C$ for $2\le x\le5$, indicating the coordinates of the endpoints. [2]

(ii) Find the cartesian equation of $C$. [2]

(a) Find the cartesian equation for $x=\cos\theta$, $y=2\sin^2\theta$. [2]

(b) A curve has $x=t^3+t$, $y=2t-1$, $-1\le t\le1$.

(i) Sketch the curve, indicating its axial intercepts. [3]

(ii) Without using a calculator, find its intersection with $y=\tfrac12$. [2]

A curve has $x=t/(1+t)$, $y=\ln(\cos t)$, where the printed exclusions are $t\ne-1,\pi/2$. Find $dy/dx$ in terms of $t$. [3]