Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problema 12.3.8

Capitolo 12, Problema 12.3.8

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

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Recognize that the curve is given in polar form as \(r = 8 \cos \theta - 4\). To find the slopes of tangent lines at the origin, we need to find the values of \(\theta\) where the curve passes through the origin, i.e., where \(r = 0\).

Set \(r = 0\) and solve for \(\theta\): \(0 = 8 \cos \theta - 4\). Rearranging gives \(8 \cos \theta = 4\), so \(\cos \theta = \frac{1}{2}\).

Find the values of \(\theta\) that satisfy \(\cos \theta = \frac{1}{2}\). These are \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{5\pi}{3}\) (within one full rotation \(0 \leq \theta < 2\pi\)).

Recall that the slope of the tangent line to a polar curve at a point where \(r \neq 0\) is given by \(\frac{dy}{dx} = \frac{r' \sin \theta + r \cos \theta}{r' \cos \theta - r \sin \theta}\), where \(r' = \frac{dr}{d\theta}\). However, since the problem asks to avoid calculating derivatives, use the geometric fact that at the origin, the tangent lines correspond to the lines passing through the origin at angles \(\theta\) where \(r=0\).

Therefore, the slopes of the tangent lines at the origin are the slopes of lines making angles \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{5\pi}{3}\) with the positive x-axis. Use \(\text{slope} = \tan \theta\) to find the slopes of these tangent lines.

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Polar Coordinates and the Curve Equation

Polar coordinates represent points using radius r and angle θ. The given curve r = 8 cos θ − 4 defines the distance from the origin as a function of θ. Understanding how the curve behaves near the origin (r = 0) is essential to find tangent lines there.

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05:32

Intro to Polar Coordinates

Tangent Lines in Polar Coordinates

Tangent lines to a polar curve at the origin correspond to directions where the curve passes through r = 0. At these points, the slope of the tangent line can be found by analyzing the angle θ where r = 0, since the tangent line direction is related to θ at the origin.

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05:32

Intro to Polar Coordinates

Slope of Tangent Lines Without Derivatives

Without using derivatives, slopes of tangent lines at the origin can be found by converting polar coordinates to Cartesian form and using geometric reasoning. The slope is given by tan(θ) for the tangent line direction at r = 0, where the curve intersects the origin.

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Percorso guidato

05:13

Slopes of Tangent Lines

Pratica correlata

Domanda del libro di testo

15–22. Sets in polar coordinates Sketch the following sets of points.

2 ≤ r ≤ 8

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Domanda del libro di testo

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.

x=2 sin 8t, y=2 cos 8t

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Domanda del libro di testo

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

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Domanda del libro di testo

53–56. Simple curves Tabulate and plot enough points to sketch a graph of the following equations.

r = 1 - cos θ

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Domanda del libro di testo

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.

r = 6 cos θ + 8 sin θ

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Domanda del libro di testo

63–74. Arc length of polar curves Find the length of the following polar curves.

The complete cardioid r = 4 + 4 sin θ

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