Chapter 11: Problem 23
Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola symmetric about the \(y\) -axis that passes through the point (2,-6)
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Sketch the graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$\frac{x^{2}}{4}+\frac{y^{2}}{16}=1$$
Find an equation of the line tangent to the following curves at the given point. $$y^{2}=8 x ;(8,-8)$$
Sketch the graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$$
Give the property that defines all parabolas.
Consider the curve \(r=f(\theta)=\cos \left(a^{\theta}\right)-1.5\) where \(a=(1+12 \pi)^{1 / 2 \pi} \approx 1.78933\) (see figure). a. Show that \(f(0)=f(2 \pi)\) and find the point on the curve that corresponds to \(\theta=0\) and \(\theta=2 \pi\) b. Is the same curve produced over the intervals \([-\pi, \pi]\) and \([0,2 \pi] ?\) c. Let \(f(\theta)=\cos \left(a^{\theta}\right)-b,\) where \(a=(1+2 k \pi)^{1 / 2 \pi}, k\) is an integer, and \(b\) is a real number. Show that \(f(0)=f(2 \pi)\) and that the curve closes on itself. d. Plot the curve with various values of \(k\). How many fingers can you produce?
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