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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 12.3.17
Chapter 12, Problem 12.3.17

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


r = 4 cos 2θ; at the tips of the leaves

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1
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 formula: \(\frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)}\)
Identify the given function: \(r = 4 \cos 2\theta\). Next, compute its derivative with respect to \(\theta\): \(\frac{dr}{d\theta} = \frac{d}{d\theta} (4 \cos 2\theta) = -8 \sin 2\theta\)
Determine the values of \(\theta\) at the tips of the leaves. The tips occur where \(r\) is at a maximum or minimum, which happens when \(\cos 2\theta = \pm 1\). Solve for \(\theta\) such that \(\cos 2\theta = \pm 1\) to find these points.
For each \(\theta\) found in the previous step, substitute \(r\), \(\frac{dr}{d\theta}\), \(\sin \theta\), and \(\cos \theta\) into the slope formula: \(\frac{dy}{dx} = \frac{(-8 \sin 2\theta) \sin \theta + (4 \cos 2\theta) \cos \theta}{(-8 \sin 2\theta) \cos \theta - (4 \cos 2\theta) \sin \theta}\)
Simplify the expression for each \(\theta\) to find the slope of the tangent line at the tips of the leaves. This will give you the slope values of the tangent lines at those points.

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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 (x, y). Polar curves are equations expressed as r = f(θ), describing shapes based on angle θ. Understanding how to interpret and plot these curves is essential for analyzing their properties, such as tangent lines.
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Intro to Polar Coordinates

Slope of Tangent Lines in Polar Coordinates

The slope of a tangent line to a polar curve at a point is found by converting the curve to Cartesian coordinates and using the derivative dy/dx = (dy/dθ) / (dx/dθ). This requires differentiating x = r cos θ and y = r sin θ with respect to θ, then evaluating at the given point.
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Slopes of Tangent Lines

Finding Specific Points on Polar Curves (Tips of the Leaves)

The 'tips of the leaves' on a polar curve like r = 4 cos 2θ correspond to points where r reaches local maxima or minima, often where the derivative dr/dθ = 0. Identifying these points is crucial to evaluate the slope of the tangent line at those specific locations.
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Slope of Polar Curves
Related Practice
Textbook Question

57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.


r = 3/(2 + cos θ)

Textbook Question

Find the area of the region bounded by the astroid x = cos³ t, y = sin³ t, for 0 ≤ t ≤ 2π

Textbook Question

Without calculating derivatives, determine the slopes of each of the lines tangent to the curve r=8 cos θ−4 at the origin.

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.


x²/9 + y²/4 = 1

Textbook Question

{Use of Tech} Implicit function graph Explain and carry out a method for graphing the curve x = 1 + cos² y − sin² y using parametric equations and a graphing utility.

Textbook Question

57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.


r = 1/(2 - 2 sin θ)