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Chapter 8: Problem 32

Find all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results. $$ x=4 \cos ^{2} \theta, \quad y=2 \sin \theta $$

Short Answer

Expert verified
The points of horizontal tangency are at \((4,0)\), \((-4,0)\) every integer multiple of \(\pi\) and points of vertical tangency are at \((0,\pm 2)\) for \(\theta = \frac{(2n+1)\pi}{2}\) where \(n\) is an integer.

Step by step solution

01

Define Expression for Derivative

A tangent line to a curve defined by parametric equations has a slope given by \(dy/dx\), which can be obtained using the chain rule as \((dy/d\theta) / (dx/d\theta)\). Here, by differentiating the given equations with respect to \(\theta\), the derivative \(dy/dx\) has the form \(dy/dx = (dy/d\theta) / (dx/d\theta)\).
02

Calculate Derivative

After differentiating, compute \((dy/d\theta) = 2 \cos(\theta)\) and \((dx/d\theta) = -8\cos(\theta)\sin(\theta)\). Then, divide \(dy/d\theta\) by \(dx/d\theta\) to obtain the derivative \(dy/dx = -\frac{1}{4\sin(\theta)}\).
03

Points of Horizontal Tangency

Horizontal tangency occurs when \(dy/dx = 0\), which yields \(\sin(\theta) = 0\). Solve for \(\theta\) to get \(\theta= n\pi\), where \(n\) is an integer. By substituting these `\(\theta\)` values into the original equations will yield the points with horizontal tangency which are \((4,0)\), \((-4,0)\) for \(n = 0, \pm 1, \pm 2, \pm 3, . . .\).
04

Points of Vertical Tangency

Vertical tangency occurs when \(dx/d\theta = 0\), which yields \(-8\cos(\theta)\sin(\theta)=0\), or \(\cos(\theta) = 0\) and \(\sin(\theta) = 0\). The first condition gives \(\theta = \frac{(2n+1)\pi}{2}\), where \(n\) is an integer. By substituting these `\(\theta\)` values into the original equations, we find that the vertical tangents lie on the \(y\)-axis, at points \((0,\pm 2)\) for \(n = 0, \pm 1, \pm 2, \pm 3, . . .\).
05

Confirm with Graphing Utility

Use a graphing utility to graph the curve in the x-y plane using the parametric equations. The points of horizontal and vertical tangency should match the coordinates calculated earlier.

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Most popular questions from this chapter

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 t^{2}, \quad y=t^{4}+1 $$

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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.
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