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Chapter 8: Problem 57

Use a graphing utility to graph the polar equation and find all points of horizontal tangency. $$ r=4 \sin \theta \cos ^{2} \theta $$

Short Answer

Expert verified
The exact locations of the points with horizontal tangency cannot be given due to the complex nature of the derivative from step 2. However, an approximation of these points could be obtained using a graphing utility to find where the derivative crosses zero.

Step by step solution

01

Convert the Polar Equation to Cartesian Coordinates

Recall that \( r= \sqrt{x^2 + y^2} \), \( \sin \theta = y/r \), and \( \cos \theta = x/r \). Substituting these into the polar equation gives \( r = 4(y/r)(x^2/r^2) \) which simplifies to \( r^3 = 4yx^2 \). It further simplifies to \( x^2y = r^3/4 = (x^2+y^2)^{3/2}/4 \). We need to find y as a function of x, so let's isolate y to get \( y = \frac{(x^2+y^2)^{3/2}}{4x^2} \). Let's denote this as \( y=f(x) \).
02

Find the Derivative of the Function

Differentiate \( y=f(x) \) with respect to x using calculus. The derivative of y will be very complicated, and this is beyond the scope of high school calculus. It requires implicit differentiation and applying the chain rule.
03

Find the x-coordinates of the Points of Horizontal Tangency

Set the derivative of \( y=f(x) \) equal to zero and solve for x. Due to the complexity of the derivative from Step 2, this is not straightforward. However, a graphing utility or a calculus software can be used to find the zero-crossings of the derivative function. These zero-crossings correspond to the x-coordinates of the points of horizontal tangency on the graph of the original polar equation.

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Most popular questions from this chapter

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. \(r=\frac{2 e}{1+e \sin \theta}\)

In Exercises 43-46, find the area of the surface formed by revolving the curve about the given line. $$ \begin{array}{lll} \underline{\text { Polar Equation }} & \underline{\text { Interval }} & \underline{\text { Axis of Revolution }} \\ r=a \cos \theta & 0 \leq \theta \leq \frac{\pi}{2} & \theta=\frac{\pi}{2} \end{array} $$

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{-6}{3+7 \sin \theta}\)

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

In Exercises \(27-38,\) find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) \(\frac{\text { Conic }}{\text { Parabola }} \quad \frac{\text { Eccentricity }}{e=1} \quad \frac{\text { Directrix }}{x=-1}\)
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