Thomas' Calculus: Early Transcendentals (14th Edition)
14th Edition
ISBN: 9780134439020
Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher: PEARSON
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Chapter 11, Problem 10GYR
To determine
Write the consequence due to the lack of uniqueness of polar coordinates that have for graphing.
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Chapter 11 Solutions
Thomas' Calculus: Early Transcendentals (14th Edition)
Ch. 11.1 - Finding Cartesian from Parametric...Ch. 11.1 - Finding Cartesian from Parametric...Ch. 11.1 - Prob. 3ECh. 11.1 - Prob. 4ECh. 11.1 - Prob. 5ECh. 11.1 - Prob. 6ECh. 11.1 - Prob. 7ECh. 11.1 - Prob. 8ECh. 11.1 - Finding Cartesian from Parametric...Ch. 11.1 - Finding Cartesian from Parametric...
Ch. 11.1 - Prob. 11ECh. 11.1 - Finding Cartesian from Parametric...Ch. 11.1 - Prob. 13ECh. 11.1 - Finding Cartesian from Parametric...Ch. 11.1 - Prob. 15ECh. 11.1 - Prob. 16ECh. 11.1 - Finding Cartesian from Parametric...Ch. 11.1 - Prob. 18ECh. 11.1 - In Exercises 19–24, match the parametric equations...Ch. 11.1 - In Exercises 19–24, match the parametric equations...Ch. 11.1 - In Exercises 19–24, match the parametric equations...Ch. 11.1 - In Exercises 19–24, match the parametric equations...Ch. 11.1 - In Exercises 19–24, match the parametric equations...Ch. 11.1 - In Exercises 19–24, match the parametric equations...Ch. 11.1 - In Exercises 25–28, use the given graphs of x =...Ch. 11.1 - In Exercises 25–28, use the given graphs of x =...Ch. 11.1 - In Exercises 25–28, use the given graphs of x =...Ch. 11.1 - In Exercises 25–28, use the given graphs of x =...Ch. 11.1 - Finding Parametric Equations
Find parametric...Ch. 11.1 - Find parametric equations and a parameter interval...Ch. 11.1 - In Exercises 31–36, find a parametrization for the...Ch. 11.1 - In Exercises 31–36, find a parametrization for the...Ch. 11.1 - In Exercises 31–36, find a parametrization for the...Ch. 11.1 - In Exercises 31–36, find a parametrization for the...Ch. 11.1 - In Exercises 31-36, find a parametrization for the...Ch. 11.1 - In Exercises 31-36, find a parametrization for the...Ch. 11.1 - Find parametric equations and a parameter interval...Ch. 11.1 - Find parametric equations and a parameter interval...Ch. 11.1 - Find parametric equations for the...Ch. 11.1 - Find parametric equations tor the circle
using as...Ch. 11.1 - Find a parametrization for the line segment...Ch. 11.1 - Find a parametrization for the curve with...Ch. 11.1 - Find a parametrization for the circle (x − 2)2 +...Ch. 11.1 - Find a parametrization for the circle x2 + y2 = 1...Ch. 11.1 - The witch of Maria Agnesi The bell-shaped witch of...Ch. 11.1 - Hypocycloid When a circle rolls on the inside of a...Ch. 11.1 - Prob. 47ECh. 11.1 - Trochoids A wheel of radius a rolls along a...Ch. 11.1 - Find the point on the parabola x = t, y = t2, −∞ <...Ch. 11.1 - Find the point on the ellipse x = 2 cos t, y = sin...Ch. 11.1 - If you have a parametric equation grapher, graph...Ch. 11.1 - Prob. 52ECh. 11.1 - Prob. 53ECh. 11.1 - Prob. 54ECh. 11.1 - Prob. 55ECh. 11.1 - Prob. 56ECh. 11.1 - a. Epicycloid
x = 9 cos t − cos 9t, y = 9 sin t −...Ch. 11.1 - a. x = 6 cos t + 5 cos 3t, y = 6 sin t − 5 sin...Ch. 11.2 - In Exercises 1–14, find an equation for the line...Ch. 11.2 - Prob. 2ECh. 11.2 - In Exercises 1–14, find an equation for the line...Ch. 11.2 - In Exercises 1–14, find an equation for the line...Ch. 11.2 - Prob. 5ECh. 11.2 - In Exercises 1–14, find an equation for the line...Ch. 11.2 - In Exercises 1–14, find an equation for the line...Ch. 11.2 - In Exercises 1–14, find an equation for the line...Ch. 11.2 - In Exercises 1–14, find an equation for the line...Ch. 11.2 - In Exercises 1–14, find an equation for the line...Ch. 11.2 - In Exercises 1–14, find an equation for the line...Ch. 11.2 - In Exercises 1–14, find an equation for the line...Ch. 11.2 - In Exercises 1–14, find an equation for the line...Ch. 11.2 - In Exercises 1–14, find an equation for the line...Ch. 11.2 - Assuming that the equations in Exercises 15–20...Ch. 11.2 - Prob. 16ECh. 11.2 - Assuming that the equations in Exercises 15–20...Ch. 11.2 - Prob. 18ECh. 11.2 - Prob. 19ECh. 11.2 - Prob. 20ECh. 11.2 - Find the area under one arch of the cycloid
Ch. 11.2 - Find the area enclosed by the y-axis and the...Ch. 11.2 - Find the area enclosed by the ellipse
Ch. 11.2 - Find the area under y = x3 over [0, 1] using the...Ch. 11.2 - Find the lengths of the curves in Exercises...Ch. 11.2 - Find the lengths of the curves in Exercises...Ch. 11.2 - Find the lengths of the curves in Exercises...Ch. 11.2 - Prob. 28ECh. 11.2 - Find the lengths of the curves in Exercises...Ch. 11.2 - Find the lengths of the curves in Exercises...Ch. 11.2 - Prob. 31ECh. 11.2 - Prob. 32ECh. 11.2 - Find the areas of the surfaces generated by...Ch. 11.2 - Prob. 34ECh. 11.2 - Prob. 35ECh. 11.2 - Prob. 36ECh. 11.2 - Prob. 37ECh. 11.2 - Prob. 38ECh. 11.2 - Prob. 39ECh. 11.2 - Prob. 40ECh. 11.2 - Length is independent of parametrization To...Ch. 11.2 - Show that the Cartesian formula
for the length...Ch. 11.2 - The curve with parametric equations
is called a...Ch. 11.2 - The curve with parametric equations
is called a...Ch. 11.2 - Prob. 45ECh. 11.2 - The curves in Exercises 45 and 46 are called...Ch. 11.2 - Cycloid
Find the length of one arch of the...Ch. 11.2 - Prob. 48ECh. 11.2 - Prob. 49ECh. 11.2 - Prob. 50ECh. 11.3 - Prob. 1ECh. 11.3 - Prob. 2ECh. 11.3 - Prob. 3ECh. 11.3 - Prob. 4ECh. 11.3 - Prob. 5ECh. 11.3 - Prob. 6ECh. 11.3 - Prob. 7ECh. 11.3 - Prob. 8ECh. 11.3 - Prob. 9ECh. 11.3 - Prob. 10ECh. 11.3 - Prob. 11ECh. 11.3 - Prob. 12ECh. 11.3 - Prob. 13ECh. 11.3 - Prob. 14ECh. 11.3 - Prob. 15ECh. 11.3 - Prob. 16ECh. 11.3 - Prob. 17ECh. 11.3 - Prob. 18ECh. 11.3 - Prob. 19ECh. 11.3 - Prob. 20ECh. 11.3 - Prob. 21ECh. 11.3 - Prob. 22ECh. 11.3 - Prob. 23ECh. 11.3 - Prob. 24ECh. 11.3 - Prob. 25ECh. 11.3 - Prob. 26ECh. 11.3 - Replace the polar equations in Exercises 27–52...Ch. 11.3 - Prob. 28ECh. 11.3 - Prob. 29ECh. 11.3 - Prob. 30ECh. 11.3 - Replace the polar equations in Exercises 27–52...Ch. 11.3 - Prob. 32ECh. 11.3 - Prob. 33ECh. 11.3 - Replace the polar equations in Exercises 27–52...Ch. 11.3 - Prob. 35ECh. 11.3 - Prob. 36ECh. 11.3 - Prob. 37ECh. 11.3 - Prob. 38ECh. 11.3 - Prob. 39ECh. 11.3 - Prob. 40ECh. 11.3 - Prob. 41ECh. 11.3 - Prob. 42ECh. 11.3 - Prob. 43ECh. 11.3 - Replace the polar equations in Exercises 27–52...Ch. 11.3 - Prob. 45ECh. 11.3 - Prob. 46ECh. 11.3 - Prob. 47ECh. 11.3 - Prob. 48ECh. 11.3 - Prob. 49ECh. 11.3 - Replace the polar equations in Exercises 27–52...Ch. 11.3 - Prob. 51ECh. 11.3 - Replace the polar equations in Exercises 27–52...Ch. 11.3 - Replace the Cartesian equations in Exercises 53–66...Ch. 11.3 - Prob. 54ECh. 11.3 - Prob. 55ECh. 11.3 - Prob. 56ECh. 11.3 - Prob. 57ECh. 11.3 - Prob. 58ECh. 11.3 - Prob. 59ECh. 11.3 - Prob. 60ECh. 11.3 - Prob. 61ECh. 11.3 - Prob. 62ECh. 11.3 - Prob. 63ECh. 11.3 - Prob. 64ECh. 11.3 - Prob. 65ECh. 11.3 - Prob. 66ECh. 11.3 - Prob. 67ECh. 11.3 - Prob. 68ECh. 11.4 - Prob. 1ECh. 11.4 - Prob. 2ECh. 11.4 - Prob. 3ECh. 11.4 - Prob. 4ECh. 11.4 - Prob. 5ECh. 11.4 - Prob. 6ECh. 11.4 - Prob. 7ECh. 11.4 - Prob. 8ECh. 11.4 - Prob. 9ECh. 11.4 - Prob. 10ECh. 11.4 - Prob. 11ECh. 11.4 - Prob. 12ECh. 11.4 - Prob. 13ECh. 11.4 - Prob. 14ECh. 11.4 - Prob. 15ECh. 11.4 - Prob. 16ECh. 11.4 - Prob. 17ECh. 11.4 - Find the slopes of the curves in Exercises 17-20...Ch. 11.4 - Find the slopes of the curves in Exercises 17-20...Ch. 11.4 - Find the slopes of the curves in Exercises 17-20...Ch. 11.4 - Prob. 21ECh. 11.4 - Prob. 22ECh. 11.4 - Prob. 23ECh. 11.4 - Prob. 24ECh. 11.4 - Prob. 25ECh. 11.4 - Prob. 26ECh. 11.4 - Prob. 27ECh. 11.4 - Prob. 28ECh. 11.4 - Prob. 29ECh. 11.4 - Prob. 30ECh. 11.4 - Prob. 31ECh. 11.4 - Prob. 32ECh. 11.4 - Prob. 33ECh. 11.4 - Prob. 34ECh. 11.4 - Prob. 35ECh. 11.4 - Prob. 36ECh. 11.4 - Prob. 37ECh. 11.4 - Prob. 38ECh. 11.4 - Prob. 39ECh. 11.4 - Prob. 40ECh. 11.5 - Finding Polar Areas
Find the areas of the regions...Ch. 11.5 - Finding Polar Areas
Find the areas of the regions...Ch. 11.5 - Prob. 3ECh. 11.5 - Prob. 4ECh. 11.5 - Prob. 5ECh. 11.5 - Prob. 6ECh. 11.5 - Prob. 7ECh. 11.5 - Prob. 8ECh. 11.5 - Prob. 9ECh. 11.5 - Find the areas of the regions in Exercises...Ch. 11.5 - Prob. 11ECh. 11.5 - Find the areas of the regions in Exercises...Ch. 11.5 - Find the areas of the regions in Exercises...Ch. 11.5 - Find the areas of the regions in Exercises...Ch. 11.5 - Find the areas of the regions in Exercises...Ch. 11.5 - Find the areas of the regions in Exercises...Ch. 11.5 - Find the areas of the regions in Exercises...Ch. 11.5 - Find the areas of the regions in Exercises...Ch. 11.5 - Prob. 19ECh. 11.5 - Prob. 20ECh. 11.5 - Find the lengths of the curves in Exercises...Ch. 11.5 - Find the lengths of the curves in Exercises...Ch. 11.5 - Prob. 23ECh. 11.5 - Find the lengths of the curves in Exercises...Ch. 11.5 - Prob. 25ECh. 11.5 - Prob. 26ECh. 11.5 - Find the lengths of the curves in Exercises...Ch. 11.5 - Prob. 28ECh. 11.5 - Prob. 29ECh. 11.5 - Prob. 30ECh. 11.5 - Prob. 31ECh. 11.5 - Prob. 32ECh. 11.6 - Match the parabolas in Exercises 1–4 with the...Ch. 11.6 - Match the parabolas in Exercises 1–4 with the...Ch. 11.6 - Prob. 3ECh. 11.6 - Prob. 4ECh. 11.6 - Prob. 5ECh. 11.6 - Prob. 6ECh. 11.6 - Prob. 7ECh. 11.6 - Prob. 8ECh. 11.6 - Prob. 9ECh. 11.6 - Prob. 10ECh. 11.6 - Prob. 11ECh. 11.6 - Prob. 12ECh. 11.6 - Prob. 13ECh. 11.6 - Prob. 14ECh. 11.6 - Prob. 15ECh. 11.6 - Prob. 16ECh. 11.6 - Prob. 17ECh. 11.6 - Prob. 18ECh. 11.6 - Prob. 19ECh. 11.6 - Prob. 20ECh. 11.6 - Prob. 21ECh. 11.6 - Prob. 22ECh. 11.6 - Prob. 23ECh. 11.6 - Prob. 24ECh. 11.6 - Prob. 25ECh. 11.6 - Prob. 26ECh. 11.6 - Prob. 27ECh. 11.6 - Prob. 28ECh. 11.6 - Prob. 29ECh. 11.6 - Prob. 30ECh. 11.6 - Prob. 31ECh. 11.6 - Prob. 32ECh. 11.6 - Prob. 33ECh. 11.6 - Prob. 34ECh. 11.6 - Prob. 35ECh. 11.6 - Prob. 36ECh. 11.6 - Prob. 37ECh. 11.6 - Prob. 38ECh. 11.6 - Prob. 39ECh. 11.6 - Prob. 40ECh. 11.6 - Prob. 41ECh. 11.6 - Prob. 42ECh. 11.6 - Prob. 43ECh. 11.6 - Prob. 44ECh. 11.6 - Prob. 45ECh. 11.6 - Prob. 46ECh. 11.6 - Prob. 47ECh. 11.6 - Prob. 48ECh. 11.6 - Prob. 49ECh. 11.6 - Prob. 50ECh. 11.6 - Prob. 51ECh. 11.6 - Prob. 52ECh. 11.6 - Prob. 53ECh. 11.6 - Prob. 54ECh. 11.6 - Prob. 55ECh. 11.6 - Prob. 56ECh. 11.6 - Prob. 57ECh. 11.6 - Prob. 58ECh. 11.6 - Prob. 59ECh. 11.6 - Prob. 60ECh. 11.6 - Prob. 61ECh. 11.6 - Prob. 62ECh. 11.6 - Prob. 63ECh. 11.6 - Prob. 64ECh. 11.6 - Prob. 65ECh. 11.6 - Prob. 66ECh. 11.6 - Prob. 67ECh. 11.6 - Prob. 68ECh. 11.6 - Prob. 69ECh. 11.6 - Prob. 70ECh. 11.6 - Prob. 71ECh. 11.6 - Prob. 72ECh. 11.6 - Prob. 73ECh. 11.6 - Prob. 74ECh. 11.6 - Prob. 75ECh. 11.6 - Prob. 76ECh. 11.6 - Prob. 77ECh. 11.6 - Prob. 78ECh. 11.6 - Prob. 79ECh. 11.6 - Prob. 80ECh. 11.6 - Prob. 81ECh. 11.7 - Prob. 1ECh. 11.7 - Prob. 2ECh. 11.7 - Prob. 3ECh. 11.7 - Prob. 4ECh. 11.7 - Prob. 5ECh. 11.7 - Prob. 6ECh. 11.7 - Prob. 7ECh. 11.7 - Prob. 8ECh. 11.7 - Prob. 9ECh. 11.7 - Prob. 10ECh. 11.7 - Prob. 11ECh. 11.7 - Prob. 12ECh. 11.7 - Prob. 13ECh. 11.7 - Prob. 14ECh. 11.7 - Prob. 15ECh. 11.7 - Prob. 16ECh. 11.7 - Prob. 17ECh. 11.7 - Prob. 18ECh. 11.7 - Prob. 19ECh. 11.7 - Prob. 20ECh. 11.7 - Prob. 21ECh. 11.7 - Prob. 22ECh. 11.7 - Prob. 23ECh. 11.7 - Prob. 24ECh. 11.7 - Prob. 25ECh. 11.7 - Prob. 26ECh. 11.7 - Prob. 27ECh. 11.7 - Prob. 28ECh. 11.7 - Prob. 29ECh. 11.7 - Prob. 30ECh. 11.7 - Prob. 31ECh. 11.7 - Prob. 32ECh. 11.7 - Prob. 33ECh. 11.7 - Prob. 34ECh. 11.7 - Prob. 35ECh. 11.7 - Prob. 36ECh. 11.7 - Prob. 37ECh. 11.7 - Prob. 38ECh. 11.7 - Prob. 39ECh. 11.7 - Prob. 40ECh. 11.7 - Prob. 41ECh. 11.7 - Prob. 42ECh. 11.7 - Prob. 43ECh. 11.7 - Prob. 44ECh. 11.7 - Prob. 45ECh. 11.7 - Prob. 46ECh. 11.7 - Prob. 47ECh. 11.7 - Prob. 48ECh. 11.7 - Prob. 49ECh. 11.7 - Prob. 50ECh. 11.7 - Prob. 51ECh. 11.7 - Prob. 52ECh. 11.7 - Prob. 53ECh. 11.7 - Prob. 54ECh. 11.7 - Prob. 55ECh. 11.7 - Prob. 56ECh. 11.7 - Prob. 57ECh. 11.7 - Prob. 58ECh. 11.7 - Prob. 59ECh. 11.7 - Prob. 60ECh. 11.7 - Prob. 61ECh. 11.7 - Prob. 62ECh. 11.7 - Prob. 63ECh. 11.7 - Prob. 64ECh. 11.7 - Prob. 65ECh. 11.7 - Prob. 66ECh. 11.7 - Prob. 67ECh. 11.7 - Prob. 68ECh. 11.7 - Prob. 69ECh. 11.7 - Prob. 70ECh. 11.7 - Prob. 71ECh. 11.7 - Prob. 72ECh. 11.7 - Prob. 73ECh. 11.7 - Prob. 74ECh. 11.7 - Prob. 75ECh. 11.7 - Prob. 76ECh. 11 - Prob. 1GYRCh. 11 - Prob. 2GYRCh. 11 - Prob. 3GYRCh. 11 - Prob. 4GYRCh. 11 - Prob. 5GYRCh. 11 - Prob. 6GYRCh. 11 - Prob. 7GYRCh. 11 - Prob. 8GYRCh. 11 - Prob. 9GYRCh. 11 - Prob. 10GYRCh. 11 - Prob. 11GYRCh. 11 - Prob. 12GYRCh. 11 - Prob. 13GYRCh. 11 - Prob. 14GYRCh. 11 - Prob. 15GYRCh. 11 - Prob. 16GYRCh. 11 - What is the eccentricity of a conic section? How...Ch. 11 - Explain the equation PF = e · PD.
Ch. 11 - Prob. 19GYRCh. 11 - Prob. 1PECh. 11 - Prob. 2PECh. 11 - Prob. 3PECh. 11 - Prob. 4PECh. 11 - Prob. 5PECh. 11 - Prob. 6PECh. 11 - Prob. 7PECh. 11 - Prob. 8PECh. 11 - Prob. 9PECh. 11 - Prob. 10PECh. 11 - Prob. 11PECh. 11 - Prob. 12PECh. 11 - Prob. 13PECh. 11 - Prob. 14PECh. 11 - Find the lengths of the curves in Exercises...Ch. 11 - Prob. 16PECh. 11 - Prob. 17PECh. 11 - Prob. 18PECh. 11 - Prob. 19PECh. 11 - Prob. 20PECh. 11 - Prob. 21PECh. 11 - Prob. 22PECh. 11 - Prob. 23PECh. 11 - Prob. 24PECh. 11 - Prob. 25PECh. 11 - Prob. 26PECh. 11 - Prob. 27PECh. 11 - Prob. 28PECh. 11 - Prob. 29PECh. 11 - Prob. 30PECh. 11 - Prob. 31PECh. 11 - Prob. 32PECh. 11 - Prob. 33PECh. 11 - Prob. 34PECh. 11 - Prob. 35PECh. 11 - Prob. 36PECh. 11 - Prob. 37PECh. 11 - Prob. 38PECh. 11 - Prob. 39PECh. 11 - Prob. 40PECh. 11 - Prob. 41PECh. 11 - Prob. 42PECh. 11 - Prob. 43PECh. 11 - Prob. 44PECh. 11 - Prob. 45PECh. 11 - Prob. 46PECh. 11 - Prob. 47PECh. 11 - Prob. 48PECh. 11 - Prob. 49PECh. 11 - Prob. 50PECh. 11 - Prob. 51PECh. 11 - Prob. 52PECh. 11 - Prob. 53PECh. 11 - Prob. 54PECh. 11 - Prob. 55PECh. 11 - Prob. 56PECh. 11 - Prob. 57PECh. 11 - Prob. 58PECh. 11 - Prob. 59PECh. 11 - Prob. 60PECh. 11 - Prob. 61PECh. 11 - Prob. 62PECh. 11 - Prob. 63PECh. 11 - Prob. 64PECh. 11 - Prob. 65PECh. 11 - Prob. 66PECh. 11 - Prob. 67PECh. 11 - Prob. 68PECh. 11 - Prob. 69PECh. 11 - Prob. 70PECh. 11 - Prob. 71PECh. 11 - Prob. 72PECh. 11 - Prob. 73PECh. 11 - Prob. 74PECh. 11 - Prob. 75PECh. 11 - Prob. 76PECh. 11 - Prob. 77PECh. 11 - Prob. 78PECh. 11 - Prob. 79PECh. 11 - Prob. 80PECh. 11 - Prob. 81PECh. 11 - Prob. 82PECh. 11 - Prob. 83PECh. 11 - Prob. 84PECh. 11 - Prob. 85PECh. 11 - Prob. 86PECh. 11 - Prob. 87PECh. 11 - Prob. 88PECh. 11 - Prob. 1AAECh. 11 - Prob. 2AAECh. 11 - Prob. 3AAECh. 11 - Prob. 4AAECh. 11 - Prob. 5AAECh. 11 - Prob. 6AAECh. 11 - Prob. 7AAECh. 11 - Prob. 8AAECh. 11 - Prob. 9AAECh. 11 - Prob. 10AAECh. 11 - Prob. 11AAECh. 11 - Prob. 12AAECh. 11 - Prob. 13AAECh. 11 - Prob. 14AAECh. 11 - Prob. 15AAECh. 11 - Prob. 16AAECh. 11 - Prob. 17AAECh. 11 - Prob. 18AAECh. 11 - Prob. 19AAECh. 11 - Prob. 20AAECh. 11 - Prob. 21AAECh. 11 - Prob. 22AAECh. 11 - Prob. 23AAECh. 11 - Prob. 24AAECh. 11 - Prob. 25AAECh. 11 - Prob. 26AAECh. 11 - Prob. 27AAECh. 11 - Prob. 28AAECh. 11 - Prob. 29AAECh. 11 - Prob. 30AAE
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