EBK THOMAS' CALCULUS
14th Edition
ISBN: 9780134654874
Author: WEIR
Publisher: VST
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Question
Chapter 11.4, Problem 26E
a.
To determine
Graph the given limacon expression.
b.
To determine
Graph the given limacon expression.
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Chapter 11 Solutions
EBK THOMAS' CALCULUS
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 - Prob. 9ECh. 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 - Finding Cartesian from Parametric...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 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 - Prob. 51ECh. 11.1 - Prob. 52ECh. 11.1 - Prob. 53ECh. 11.1 - If you have a parametric equation grapher, graph...Ch. 11.1 - Deltoid
x = 2 cos t + cos 2t, y = 2 sin t − sin...Ch. 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 - 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 - Prob. 11ECh. 11.2 - In Exercises 1–14, find an equation for the line...Ch. 11.2 - Prob. 13ECh. 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 - Assuming that the equations in Exercises 15–20...Ch. 11.2 - Assuming that the equations in Exercises 15–20...Ch. 11.2 - Assuming that the equations in Exercises 15–20...Ch. 11.2 - Assuming that the equations in Exercises 15–20...Ch. 11.2 - Assuming that the equations in Exercises 15–20...Ch. 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 - 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 - Find the areas of the surfaces generated by...Ch. 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 - Prob. 42ECh. 11.2 - The curve with parametric equations
is called a...Ch. 11.2 - Prob. 44ECh. 11.2 - Prob. 45ECh. 11.2 - Prob. 46ECh. 11.2 - Prob. 47ECh. 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 - Find the polar coordinates, and , of the...Ch. 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 - Replace the polar equations in Exercises 27–52...Ch. 11.3 - Prob. 38ECh. 11.3 - Replace the polar equations in Exercises 27–52...Ch. 11.3 - Prob. 40ECh. 11.3 - Prob. 41ECh. 11.3 - Prob. 42ECh. 11.3 - Replace the polar equations in Exercises 27–52...Ch. 11.3 - Replace the polar equations in Exercises 27–52...Ch. 11.3 - Prob. 45ECh. 11.3 - Prob. 46ECh. 11.3 - Replace the polar equations in Exercises 27–52...Ch. 11.3 - Replace the polar equations in Exercises 27–52...Ch. 11.3 - Replace the polar equations in Exercises 27–52...Ch. 11.3 - Prob. 50ECh. 11.3 - Prob. 51ECh. 11.3 - Prob. 52ECh. 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 - 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. 19ECh. 11.4 - Prob. 20ECh. 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 - Which of the following has the same graph as r = 1...Ch. 11.4 - Prob. 34ECh. 11.4 - Prob. 35ECh. 11.4 - Prob. 36ECh. 11.4 - Roses Graph the roses r = cos mθ for m = 1/3, 2,...Ch. 11.4 - Spirals Polar coordinates are just the thing for...Ch. 11.4 - Graph the equation for 0 ≤ θ 14 π.
Ch. 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 - 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 - 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 - Find the areas of the regions in Exercises...Ch. 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 - Find the lengths of the curves in Exercises...Ch. 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 - Match the parabolas in Exercises 1–4 with the...Ch. 11.6 - Match the parabolas in Exercises 1–4 with the...Ch. 11.6 - Match each conic section in Exercises 5–8 with one...Ch. 11.6 - Match each conic section in Exercises 5–8 with one...Ch. 11.6 - Match each conic section in Exercises 5–8 with one...Ch. 11.6 - Match each conic section in Exercises 5–8 with one...Ch. 11.6 - Exercises 9–16 give equations of parabolas. Find...Ch. 11.6 - Prob. 10ECh. 11.6 - Exercises 9–16 give equations of parabolas. Find...Ch. 11.6 - Prob. 12ECh. 11.6 - Exercises 9–16 give equations of parabolas. Find...Ch. 11.6 - Prob. 14ECh. 11.6 - Exercises 9–16 give equations of parabolas. Find...Ch. 11.6 - Prob. 16ECh. 11.6 - Prob. 17ECh. 11.6 - Prob. 18ECh. 11.6 - Exercises 17–24 give equations for ellipses. Put...Ch. 11.6 - Prob. 20ECh. 11.6 - Exercises 17–24 give equations for ellipses. Put...Ch. 11.6 - Prob. 22ECh. 11.6 - Exercises 17–24 give equations for ellipses. Put...Ch. 11.6 - Prob. 24ECh. 11.6 - Exercises 25 and 26 give information about the...Ch. 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 - Exercises 35–38 give information about the foci,...Ch. 11.6 - Exercises 35–38 give information about the foci,...Ch. 11.6 - The parabola y2 = 8x is shifted down 2 units and...Ch. 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 - Exercises 39–42 give equations for parabolas and...Ch. 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 - Exercises 9–12 give the foci or vertices and the...Ch. 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 - Sketch the parabolas and ellipses in Exercises...Ch. 11.7 - Prob. 39ECh. 11.7 - Prob. 40ECh. 11.7 - Sketch the parabolas and ellipses in Exercises...Ch. 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 - Give some typical parametrizations for lines,...Ch. 11 - Prob. 3GYRCh. 11 - What is the formula for the slope dy/dx of a...Ch. 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 - Prob. 17GYRCh. 11 - Prob. 18GYRCh. 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 - Prob. 15PECh. 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 - Match each graph in Exercises 39–46 with the...Ch. 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 - Epicycloids When a circle rolls externally along...Ch. 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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- please solve, thank youarrow_forwardEvaluate the definite integral using the given integration limits and the limits obtained by trigonometric substitution. 14 x² dx 249 (a) the given integration limits (b) the limits obtained by trigonometric substitutionarrow_forwardAssignment #1 Q1: Test the following series for convergence. Specify the test you use: 1 n+5 (-1)n a) Σn=o √n²+1 b) Σn=1 n√n+3 c) Σn=1 (2n+1)3 3n 1 d) Σn=1 3n-1 e) Σn=1 4+4narrow_forward
- answer problem 1a, 1b, 1c, 1d, and 1e and show work/ explain how you got the answerarrow_forwardProvethat a) prove that for any irrational numbers there exists? asequence of rational numbers Xn converg to S. b) let S: RR be a sunctions-t. f(x)=(x-1) arc tan (x), xe Q 3(x-1) 1+x² x&Q Show that lim f(x)= 0 14x C) For any set A define the set -A=yarrow_forwardQ2: Find the interval and radius of convergence for the following series: Σ n=1 (-1)η-1 xn narrow_forward8. Evaluate arctan x dx a) xartanx 2 2 In(1 + x²) + C b) xartanx + 1½-3ln(1 + x²) + C c) xartanx + In(1 + x²) + C d) (arctanx)² + C 2 9) Evaluate Inx³ dx 3 a) +C b) ln x² + C c)¾½ (lnx)² d) 3x(lnx − 1) + C - x 10) Determine which integral is obtained when the substitution x = So¹² √1 - x²dx sine is made in the integral πT π π a) √ sin cos e de b) √ cos² de c) c Ꮎ Ꮎ cos² 0 de c) cos e de d) for cos² e de πT 11. Evaluate tan³xdx 1 a) b) c) [1 - In 2] 2 2 c) [1 − In2] d)½½[1+ In 2]arrow_forward12. Evaluate ſ √9-x2 -dx. x2 a) C 9-x2 √9-x2 - x2 b) C - x x arcsin ½-½ c) C + √9 - x² + arcsin x d) C + √9-x2 x2 13. Find the indefinite integral S cos³30 √sin 30 dᎾ . 2√√sin 30 (5+sin²30) √sin 30 (3+sin²30) a) C+ √sin 30(5-sin²30) b) C + c) C + 5 5 5 10 d) C + 2√√sin 30 (3-sin²30) 2√√sin 30 (5-sin²30) e) C + 5 15 14. Find the indefinite integral ( sin³ 4xcos 44xdx. a) C+ (7-5cos24x)cos54x b) C (7-5cos24x)cos54x (7-5cos24x)cos54x - 140 c) C - 120 140 d) C+ (7-5cos24x)cos54x e) C (7-5cos24x)cos54x 4 4 15. Find the indefinite integral S 2x2 dx. ex - a) C+ (x²+2x+2)ex b) C (x² + 2x + 2)e-* d) C2(x²+2x+2)e¯* e) C + 2(x² + 2x + 2)e¯* - c) C2x(x²+2x+2)e¯*arrow_forward4. Which substitution would you use to simplify the following integrand? S a) x = sin b) x = 2 tan 0 c) x = 2 sec 3√√3 3 x3 5. After making the substitution x = = tan 0, the definite integral 2 2 3 a) ៖ ស្លឺ sin s π - dᎾ 16 0 cos20 b) 2/4 10 cos 20 π sin30 6 - dᎾ c) Π 1 cos³0 3 · de 16 0 sin20 1 x²√x²+4 3 (4x²+9)2 π d) cos²8 16 0 sin³0 dx d) x = tan 0 dx simplifies to: de 6. In order to evaluate (tan 5xsec7xdx, which would be the most appropriate strategy? a) Separate a sec²x factor b) Separate a tan²x factor c) Separate a tan xsecx factor 7. Evaluate 3x x+4 - dx 1 a) 3x+41nx + 4 + C b) 31n|x + 4 + C c) 3 ln x + 4+ C d) 3x - 12 In|x + 4| + C x+4arrow_forward1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.arrow_forward2. Observations on the Wronskian. Suppose the functions y₁ and y2 are solutions to the differential equation p(x)y" + q(x)y' + r(x) y = 0 on an open interval I. 1. (a) Prove that if y₁ and y2 both vanish at the same point in I, then y₁ and y2 cannot form a fundamental set of solutions. 2. (b) Prove that if y₁ and y2 both attain a maximum or minimum at the same point in I, then y₁ and Y2 cannot form a fundamental set of solutions. 3. (c) show that the functions & and t² are linearly independent on the interval (−1, 1). Verify that both are solutions to the differential equation t² y″ – 2ty' + 2y = 0. Then justify why this does not contradict Abel's theorem. 4. (d) What can you conclude about the possibility that t and t² are solutions to the differential equation y" + q(x) y′ + r(x)y = 0?arrow_forwardQuestion 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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