Chapter 10: Problem 69
Find an equation of the line tangent to the following curves at the given point. $$x^{2}=-6 y ;(-6,-6)$$
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Circles in general $$r^{2}-2 r(a \cos \theta+b \sin \theta)=R^{2}-a^{2}-b^{2}$$ describes a circle of radius \(R\) centered at \((a, b)\).
Find parametric equations (not unique) of the following ellipses (see Exercises \(75-76\) ). Graph the ellipse and find a description in terms of \(x\) and \(y.\) An ellipse centered at the origin with major axis of length 6 on the \(x\) -axis and minor axis of length 3 on the \(y\) -axis, generated counterclockwise
Consider the polar curve \(r=2 \sec \theta\). a. Graph the curve on the intervals \((\pi / 2,3 \pi / 2),(3 \pi / 2,5 \pi / 2)\) and \((5 \pi / 2,7 \pi / 2) .\) In each case, state the direction in which the curve is generated as \(\theta\) increases. b. Show that on any interval \((n \pi / 2,(n+2) \pi / 2),\) where \(n\) is an odd integer, the graph is the vertical line \(x=2\).
Water flows in a shallow semicircular channel with inner and outer radii of \(1 \mathrm{m}\) and \(2 \mathrm{m}\) (see figure). At a point \(P(r, \theta)\) in the channel, the flow is in the tangential direction (counterclockwise along circles), and it depends only on \(r,\) the distance from the center of the semicircles. a. Express the region formed by the channel as a set in polar coordinates. b. Express the inflow and outflow regions of the channel as sets in polar coordinates. c. Suppose the tangential velocity of the water in \(\mathrm{m} / \mathrm{s}\) is given by \(v(r)=10 r,\) for \(1 \leq r \leq 2 .\) Is the velocity greater at \(\left(1.5, \frac{\pi}{4}\right)\) or \(\left(1.2, \frac{3 \pi}{4}\right) ?\) Explain. d. Suppose the tangential velocity of the water is given by \(v(r)=\frac{20}{r},\) for \(1 \leq r \leq 2 .\) Is the velocity greater at \(\left(1.8, \frac{\pi}{6}\right)\) or \(\left(1.3, \frac{2 \pi}{3}\right) ?\) Explain. e. The total amount of water that flows through the channel (across a cross section of the channel \(\theta=\theta_{0}\) ) is proportional to \(\int_{1}^{2} v(r) d r .\) Is the total flow through the channel greater for the flow in part (c) or (d)?
Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work. A hyperbola with vertices (±2,0) and asymptotes \(y=\pm 3 x / 2\)
What do you think about this solution?
We value your feedback to improve our textbook solutions.
