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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.
Recommended video:
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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

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 = 3 + t, y = 1 − t; 0 ≤ t ≤ 1

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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.


4x = -y²

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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)

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Textbook Question

90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

The length of the latus rectum of the parabola y ² =4px or x ² =4py is 4|p|.

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Textbook Question

53–56. Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 53–55, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle.


The tip of the 15-inch second hand of a clock completes one revolution in 60 seconds.

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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.


The line that passes through the points P(1, 1) and Q(3, 5), oriented in the direction of increasing x

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