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Chapter 8: Problem 58

Use a graphing utility to graph the polar equation and find all points of horizontal tangency. $$ r=3 \cos 2 \theta \sec \theta $$

Short Answer

Expert verified
The complete solution involves the conversion of the polar equation to Cartesian coordinates, the graphing of this function and finally to find where the derivative wrt \( y \) equals zero. This will provide the points of horizontal tangency.

Step by step solution

01

Conversion to Cartesian Coordinates

To graph the function using a graphing utility, we need to convert the polar equation into Cartesian coordinates. Remember that we may express \( r = x^2 + y^2 \) and \( \theta = \arctan(\frac{y}{x}) \). Here, multiply \( 3 \cos 2 \theta \sec \theta = 3 \cos 2 \theta / \cos \theta \) by \( \sqrt{x^2 + y^2} \) to get in terms of 'x' and 'y'. After simplification, you will obtain an equation in terms of 'x' and 'y'.
02

Graphing the function

Use any graphing utility to plot the Cartesian equation you obtained from step 1. For each point, you can then convert the Cartesian coordinates back to polar coordinates.
03

Determining the points of horizontal tangency

A point of horizontal tangency will exist where the derivative of the equation wrt \( y \) equals zero, as the tangent line will be horizontal to the x-axis at these points. Therefore, find the derivative of the equation wrt \( y \), set it equal to zero, and solve to find these points.

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Most popular questions from this chapter

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \text { Circle: } x=h+r \cos \theta, \quad y=k+r \sin \theta $$

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Witch of Agnesi: } x=2 \cot \theta, \quad y=2 \sin ^{2} \theta $$

Use the integration capabilities of a graphing utility to approximate to two decimal places the area of the surface formed by revolving the curve about the polar axis. $$ r=\theta, \quad 0 \leq \theta \leq \pi $$

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \text { Ellipse: } x=h+a \cos \theta, \quad y=k+b \sin \theta $$

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) \(x=e^{t}\) $$ y=2 e^{t}+1 $$
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