11.2 Tangent Lines and Arc Length for Parametric and Polar Curves 
In this section we will derive the formulas required to find slopes, tangent lines, and arc lengths of parametric and polar curves. 
Tangent Lines to Parametric Curves 
We will be concerned in this section with curves that are given by parametric equations
(1)  
Example 1 Find the slope of the tangent line to the unit circle
Solution. From 1, the slope at a general point on the circle is
 
It follows from Formula 1 that the tangent line to a parametric curve will be horizontal at those points where and , since at such points. Two different situations occur when . At points where and , the right side of 1 has a nonzero numerator and a zero denominator; we will agree that the curve has infinite slope and a vertical tangent line at such points. At points where and are both zero, the right side of 1 becomes an indeterminate form; we call such points singular points. No general statement can be made about the behavior of parametric curves at singular points; they must be analyzed case by case.
Example 2 In a disastrous first flight, an experimental paper airplane follows the trajectory
Solution(a). The airplane was flying horizontally at those times when and . From the given trajectory we have
Solution (b). The airplane was flying vertically at those times when and . Setting in 3 yields the equation
 
Example 3 The curve represented by the parametric equations
 
Tangent Lines to Polar Curves 
Our next objective is to find a method for obtaining
slopes of tangent lines to polar curves of the form in which
(6)  
(7)  
Example 5 Find the slope of the tangent line to the circle at the point where . Solution. From 7 with we obtain (verify)
 
Example 6 Find the points on the cardioid at which there is a horizontal tangent line, a vertical tangent line, or a singular point. Solution. A horizontal tangent line will occur where and , a vertical tangent line where and , and a singular point where and . We could find these derivatives from the formulas in 6. However, an alternative approach is to go back to basic principles and express the cardioid parametrically by substituting in the conversion formulas and . This yields
 
Tangent Lines to Polar Curves at the Origin 
Formula 7 reveals some useful information about the behavior of a polar curve that passes through the origin. If we assume that and when , then it follows from Formula 7 that the slope of the tangent line to the curve at is
 
 
11.2.1 Theorem. If the polar curve passes through the origin at , and if at , then the line is tangent to the curve at the origin. 
This theorem tells us that equations of the tangent lines at the origin to the curve can be obtained by solving the equation . It is important to keep in mind, however, that may be zero for more than one value of θ, so there may be more than one tangent line at the origin. This is illustrated in the next example.
Example 7 The threepetal rose in Figure 11.2.9 has three tangent lines at the origin, which can be found by solving the equation
 
ARC Length of a Polar Curve 
A formula for the arc length of a polar curve can be derived by expressing the curve in parametric form and applying Formula (6) of Section 7.4 for the arc length of a parametric curve. We leave it as an exercise to show the following.
11.2.2 ARC Length Formula for Polar
Curves. If no segment of the polar curve is traced more than once as θ increases from α
to β, and if is continuous for , then the arc length

Example 8 Find the arc length of the spiral in Figure 11.2.10 between and .
Solution.  
Example 9 Find the total arc length of the cardioid . Solution. The cardioid is traced out once as θ varies from to . Thus,
 
Quick Check Exercises 11.2  
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Exercise Set 11.2 


510 Find and at the given point without eliminating the parameter. 
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1314 Find all values of

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2126 Find the slope of the tangent line to the polar curve for the given value of θ. 
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2728 Calculate the slopes of the tangent lines indicated in the accompanying figures. 
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2930 Find polar coordinates of all points at which the polar curve has a horizontal or a vertical tangent line. 
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3132 Use a graphing utility to make a conjecture about the number of points on the polar curve at which there is a horizontal tangent line, and confirm your conjecture by finding appropriate derivatives. 
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3338 Sketch the polar curve and find polar equations of the tangent lines to the curve at the pole. 
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3944 Use Formula 8 to calculate the arc length of the polar curve. 
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46. 
Suppose that a bee follows the trajectory

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4954 If and are continuous functions, and if no segment of
the curve

49. 
Find the area of the surface generated by revolving
, about the 
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Find the area of the surface generated by revolving
the curve , about the 
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Find the area of the surface generated by revolving
the curve , about the 
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Find the area of the surface generated by revolving
, about the 
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60. 
Recall from Section 6.9 that the Fresnel sine and cosine functions are defined as

61. 
As illustrated in the accompanying figure, let be a point on the polar curve , let ψ be the smallest counterclockwise angle
from the extended radius to the tangent line at

6263 Use the formula for ψ obtained in Exercise 61. 
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63. 
Show that for a logarithmic spiral , the angle from the radial line to the tangent line is constant along the spiral (see the accompanying figure). [Note: For this reason, logarithmic spirals are sometimes called equiangular spirals.]

Quick Check Answers 11.2  
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Copyright © 2005 John Wiley & Sons, Inc. All rights reserved. 