Problem 64
For the following exercises, sketch the graph of each conic. $$ 4 x^{2}+9 y^{2}=36 $$
Problem 64
Sketch a graph of the polar equation and identify any symmetry. $$ r=2 \theta $$
Problem 64
For the following exercises, find the arc length of the curve on the indicated interval of the parameter.\(x=e^{t} \cos t, \quad y=e^{t} \sin t, \quad 0 \leq t \leq \frac{\pi}{2}\) (express answer as a decimal rounded to three places)
Problem 64
For the following exercises, find the slope of a tangent line to a polar curve \(r=f(\theta) .\) Let \(x=r \cos \theta=f(\theta) \cos \theta\) and \(y=r \sin \theta=f(\theta) \sin \theta\), so the polar equation \(r=f(\theta)\) is now written in parametric form.Find the points on the interval \(-\pi \leq \theta \leq \pi\) at which the cardioid \(r=1-\cos \theta\) has a vertical or horizontal tangent line.
Problem 65
For the following exercises, sketch the graph of each conic. $$ 25 x^{2}-4 y^{2}=100 $$
Problem 65
The graph of \(r=2 \cos (2 \theta) \sec (\theta) .\) is called a strophoid. Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.
Problem 65
For the following exercises, find the arc length of the curve on the indicated interval of the parameter.\(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\) on the interval \([0,2 \pi)\) (the hypocycloid)
Problem 65
For the following exercises, find the slope of a tangent line to a polar curve \(r=f(\theta) .\) Let \(x=r \cos \theta=f(\theta) \cos \theta\) and \(y=r \sin \theta=f(\theta) \sin \theta\), so the polar equation \(r=f(\theta)\) is now written in parametric form.For the cardioid \(r=1+\sin \theta\), find the slope of the tangent line when \(\theta=\frac{\pi}{3}\).
Problem 66
Find the slope of the tangent line to the given polar curve at the point given by the value of \(\theta\).\(r=3 \cos \theta, \theta=\frac{\pi}{3}\)
Problem 66
For the following exercises, sketch the graph of each conic. $$ \frac{x^{2}}{16}-\frac{y^{2}}{9}=1 $$
