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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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Parameterizing Equations

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

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.


x=cos t+t sin t,y=sin t−t cos t; t=π/4

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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. 

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Textbook Question

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


r = 3

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Textbook Question

85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)


A circular corral of unit radius is enclosed by a fence. A goat inside the corral is tied to the fence with a rope of length 0≤a≤2 (see figure). What is the area of the region (inside the corral) that the goat can graze? Check your answer with the special cases a=0 and a=2.


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Textbook Question

69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.

x² = -6y; (-6, -6)

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Textbook Question

Plot the points with polar coordinates (2, π/6) and (−3, −π/2). Give two alternative sets of coordinate pairs for both points.

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