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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.3.11
Chapter 12, Problem 12.3.11

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.


r = 1 - sin θ; (1/2, π/6)

Verified step by step guidance
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Recall that for a polar curve given by \(r = f(\theta)\), the slope of the tangent line \(\frac{dy}{dx}\) can be found using the formulas for \(x\) and \(y\) in terms of \(r\) and \(\theta\): \(x = r \cos \theta\) and \(y = r \sin \theta\).
Express \(x\) and \(y\) as functions of \(\theta\): \(x(\theta) = r(\theta) \cos \theta\) and \(y(\theta) = r(\theta) \sin \theta\). For the given curve, \(r = 1 - \sin \theta\), so substitute this into \(x(\theta)\) and \(y(\theta)\).
Find the derivatives \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\) by applying the product rule: \(\frac{d}{d\theta}[r(\theta) \cos \theta]\) and \(\frac{d}{d\theta}[r(\theta) \sin \theta]\). Remember to differentiate both \(r(\theta)\) and the trigonometric functions.
Calculate the slope of the tangent line using the formula \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\). This gives the slope in terms of \(\theta\).
Evaluate \(\frac{dy}{dx}\) at the given point \(\theta = \frac{\pi}{6}\) to find the slope of the tangent line at that point.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar Coordinates and Polar Curves

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian coordinates (x, y). Polar curves are equations expressed as r = f(θ), describing how the radius changes with the angle. Understanding this system is essential for interpreting the given curve and point.
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Intro to Polar Coordinates

Slope of Tangent Line in Polar Coordinates

The slope of the tangent line to a polar curve at a point is found by converting the curve to parametric form (x = r cos θ, y = r sin θ) and then computing dy/dx = (dy/dθ) / (dx/dθ). This method links polar derivatives to Cartesian slopes.
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Slopes of Tangent Lines

Differentiation of Parametric Functions

To find dy/dx for parametric equations, differentiate x(θ) and y(θ) with respect to θ separately, then divide dy/dθ by dx/dθ. This technique is crucial for determining the slope of the tangent line to curves defined parametrically, such as polar curves.
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Related Practice
Textbook Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


x² = 12y

Textbook Question

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

An ellipse with vertices (±5, 0), passing through the point (4, 3/5)

Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


A circle centered at the origin with radius 4, generated counterclockwise

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Textbook Question

Circles in general Show that the polar equation

r² - 2r r₀ cos(θ - θ₀) = R² - r₀²

describes a circle of radius R whose center has polar coordinates (r₀, θ₀)

Textbook Question

45–60. Areas of regions Find the area of the following regions.


The region inside the outer loop but outside the inner loop of the limaçon r = 3 - 6 sin θ

Textbook Question

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.


x = cos t, y = sin² t; 0 ≤ t ≤ π

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