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Chapter 8: Problem 30

Find all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results. $$ x=\cos \theta, \quad y=2 \sin 2 \theta $$

Short Answer

Expert verified
The horizontal points of tangency are at \( \theta = \pm \pi/2, \pm 3\pi/2 \) and the vertical points of tangency are at \( \theta = 0, \pi, 2\pi \). A graphing utility can be used to visually confirm these points on the curve.

Step by step solution

01

Derive parametric equations

Derive the given parametric equations with respect to \( \theta \). \nDerivative of x with respect to \( \theta \) is \( dx/d\theta = -\sin(\theta) \)\nDerivative of y with respect to \( \theta \) is \( dy/d\theta = 4\cos(2\theta) \)
02

Find the derivative of y with respect to x

Calculate dy/dx which is the derivative of y with respect to x. Using the Chain rule, it can be computed as \( dy/d\theta \) divided by \( dx/d\theta \), i.e.\n \( dy/dx = (4\cos(2\theta))/(-\sin(\theta)) \)
03

Find points of horizontal tangency

Set dy/dx to zero to find points of horizontal tangency i.e. when \( \theta = \pm \pi/2, \pm 3\pi/2 \), since cosines of these values are zero.
04

Find points of vertical tangency

Vertical tangency will occur when dy/dx is undefined, i.e. \( \theta = 0, \pi, 2\pi \), since sin(\theta) of these values will result in denominator becoming zero. This results in values of x and y as x = 1, x = -1 and x = 1 with corresponding y values as 0, 0 and 0 respectively.

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Most popular questions from this chapter

Conjecture (a) Use a graphing utility to graph the curves represented by the two sets of parametric equations. \(x=4 \cos t \quad x=4 \cos (-t)\) \(y=3 \sin t \quad y=3 \sin (-t)\) (b) Describe the change in the graph when the sign of the parameter is changed. (c) Make a conjecture about the change in the graph of parametric equations when the sign of the parameter is changed. (d) Test your conjecture with another set of parametric equations.

Determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? $$ \text { (a) } \begin{aligned} x &=t \\ y &=2 t+1 \end{aligned} $$ $$ \text { (b) } \begin{aligned} x &=\cos \theta \\ y &=2 \cos \theta+1 \end{aligned} $$ $$ \text { (c) } \begin{aligned} x &=e^{-t} \\ y &=2 e^{-t}+1 \end{aligned} $$ (d) \(x=e^{t}\) $$ y=2 e^{t}+1 $$

In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{-1}{1-\cos \theta}\)

Show that the polar equation for \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(r^{2}=\frac{b^{2}}{1-e^{2} \cos ^{2} \theta} \cdot \quad\) Ellipse

Graphical Reasoning In Exercises 1-4, use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. \(r=\frac{2 e}{1+e \cos \theta}\)
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