A twice differentiable polar curve is given by r=r(θ), with Cartesian coordinates…

A twice differentiable polar curve is given by \(r=r(\theta)\), with Cartesian coordinates \(x=r\cos\theta\) and \(y=r\sin\theta\). (a) Derive the formula \[ \frac{dy}{dx}=\frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta} \] at every point where the denominator is nonzero. (b) For \(r=2+\sin\theta\), evaluate \(\dfrac{dy}{dx}\) at \(\theta=0\). Give the corresponding Cartesian point. (c) For the same curve, evaluate \(\dfrac{dy}{dx}\) at \(\theta=\pi/2\). Give the corresponding Cartesian point and write an equation of the tangent line. (d) A particle traces the curve so that \(\dfrac{d\theta}{dt}=3\) radians per second. At the instant \(\theta=0\), find \(\dfrac{dr}{dt}\) and the speed \[ \sqrt{\Bigl(\frac{dr}{dt}\Bigr)^2+\Bigl(r\frac{d\theta}{dt}\Bigr)^2}. \]

Cartesian coordinates of a polar graph are already parametric in \(\theta\); the chain rule produces the displayed slope formula. Polar speed uses the given combination of \(dr/dt\) and \(r\,d\theta/dt\).