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Chapter 8: Problem 31

Find all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results. $$ x=4+2 \cos \theta, \quad y=-1+\sin \theta $$

Short Answer

Expert verified
The points of vertical tangency to the curve are (6, -1) and (2, -1). The points of horizontal tangency to the curve are (4, 0) and (4, -2).

Step by step solution

01

Define the derivatives

Begin by defining the derivatives dx/d胃 and dy/d胃 from the given parametric equations, so\(dx/d胃 = -2 \sin \theta\) and \(dy/d胃 = \cos \theta\)
02

Find the points of vertical tangency

Vertical tangency occurs when the derivative in x is zero or undefined. Set dx/d胃 to 0 and solve for 胃:\(-2 \sin \theta = 0\)Therefore \(\theta = 0, \pi\)Substitute these 胃 values in the original equations for x and y to get the points of vertical tangency: For \(\theta = 0\), \(x = 4 + 2(1)\), \(y = -1 + 0\) which gives (6, -1)For \(\theta = \pi\), \(x = 4 + 2(-1)\), \(y = -1 + 0\) which gives (2, -1)
03

Find the points of horizontal tangency

Horizontal tangency occurs when the derivative in y is zero or undefined. Set dy/d胃 to 0 and solve for 胃:\(\cos \theta = 0\)Therefore \(\theta = \pi/2, (3\pi)/2\)Substitute these 胃 values in the original equations for x and y to get the points of horizontal tangency: For \(\theta = \pi/2\), \(x = 4 + 2(0)\), \(y = -1 + 1\) which gives (4,0)For \(\theta = (3\pi)/2\), \(x = 4 - 2(0)\), \(y = -1 - 1\) which gives (4, -2)
04

Confirm your results

The final step is to confirm these results using a graphing utility. Plot the curve and see if it crosses the y-axis at the points found for vertical tangency and likewise for the x-axis and horizontal tangency. The curve should be tangent to these points.

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Most popular questions from this chapter

$$ \text { State the definition of a smooth curve } $$

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

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