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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 12.1.77
Chapter 12, Problem 12.1.77

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.


x = 4 cos t, y = 4 sin t; slope = 1/2

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Recall that the slope of the tangent line to a parametric curve given by \(x = f(t)\) and \(y = g(t)\) is found by computing \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\).
Differentiate both parametric equations with respect to \(t\): compute \(\frac{dx}{dt}\) for \(x = 4 \cos t\) and \(\frac{dy}{dt}\) for \(y = 4 \sin t\).
Write the expression for the slope \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\) using the derivatives found in the previous step.
Set the slope equal to the given value \(\frac{1}{2}\) and solve the resulting equation for \(t\).
Use the values of \(t\) found to calculate the corresponding points on the curve by substituting back into \(x = 4 \cos t\) and \(y = 4 \sin t\).

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Key Concepts

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

Parametric Equations and Curves

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. Understanding how x and y depend on t allows us to analyze the curve's shape and behavior, which is essential for finding slopes of tangent lines at specific points.
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Derivative of Parametric Equations (dy/dx)

The slope of the tangent line to a parametric curve is found by computing dy/dx = (dy/dt) / (dx/dt). This requires differentiating both x(t) and y(t) with respect to t and then dividing, which gives the instantaneous rate of change of y with respect to x.
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Solving for Parameter Values Given a Slope

After finding the expression for dy/dx in terms of t, we set it equal to the given slope and solve for t. These parameter values correspond to points on the curve where the tangent line has the specified slope, enabling us to find the exact coordinates.
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Understanding Slope Fields
Related Practice
Textbook Question

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola with focus at (3, 0)

Textbook Question

9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.


(-4, 3π/2)

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 a hyperbola centered at the origin is (2b²)/a = 2b√(1 - e²)

Textbook Question

Multiple descriptions Which of the following parametric equations describe the same curve?

a. x = 2t², y = 4 + t; -4 ≤ t ≤ 4

b. x = 2t⁴, y = 4 + t²; -2 ≤ t ≤ 2

c. x = 2t^(2/3), y = 4 + t^(1/3); -64 ≤ t ≤ 64

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Textbook Question

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


r = 3 csc θ

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Textbook Question

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.


y = x²