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Math Calculus Calculus: Early Transcendentals (2nd Edition)

Enhanced butterfly curve The butterfly curve of Example 8 is enhanced by adding a term: r = e sin θ − 2 cos 4 θ + sin 5 ( θ / 12 ) , for 0 ≤ θ ≤ 24 π . a. Graph the curve. b. Explain why the new term produces the observed effect. ( Source: S. Wagon and E. Packel, Animating Calculus. Freeman, 1994)

Enhanced butterfly curve The butterfly curve of Example 8 is enhanced by adding a term: r = e sin θ − 2 cos 4 θ + sin 5 ( θ / 12 ) , for 0 ≤ θ ≤ 24 π . a. Graph the curve. b. Explain why the new term produces the observed effect. ( Source: S. Wagon and E. Packel, Animating Calculus. Freeman, 1994)

Solution Summary: The author explains that the graph of r=emathrmsintheta -2 math

Calculus: Early Transcendentals (2nd Edition)

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Textbook Question

Chapter 10.2, Problem 107E

Enhanced butterfly curve The butterfly curve of Example 8 is enhanced by adding a term:

r = e sin θ − 2 cos 4 θ + sin 5 ( θ / 12 ) , for 0 ≤ θ ≤ 24 π .

a. Graph the curve.
b. Explain why the new term produces the observed effect.

(Source: S. Wagon and E. Packel, Animating Calculus. Freeman, 1994)

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Chapter 10.2, Problem 106E

Chapter 10.2, Problem 108E

Chapter 10 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 10.1 - Explain how a pair of parametric equations...Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E Ch. 10.1 - Give parametric equations that generate the line...Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Explain how to find points on the curve x = f(t),...

Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - Prob. 15E Ch. 10.1 - Prob. 16E Ch. 10.1 - Prob. 17E Ch. 10.1 - Prob. 18E Ch. 10.1 - Prob. 19E Ch. 10.1 - Prob. 20E Ch. 10.1 - Prob. 21E Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - Prob. 24E Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Parametric equations of circles Find parametric...Ch. 10.1 - Parametric equations of circles Find parametric...Ch. 10.1 - Parametric equations of circles Find parametric...Ch. 10.1 - Prob. 30E Ch. 10.1 - Parametric equations of circles Find parametric...Ch. 10.1 - Prob. 32E Ch. 10.1 - Prob. 33E Ch. 10.1 - Prob. 34E Ch. 10.1 - Prob. 35E Ch. 10.1 - Prob. 36E Ch. 10.1 - Parametric lines Find the slope of each line and a...Ch. 10.1 - Parametric lines Find the slope of each line and a...Ch. 10.1 - Parametric lines Find the slope of each line and a...Ch. 10.1 - Prob. 40E Ch. 10.1 - Prob. 41E Ch. 10.1 - Prob. 42E Ch. 10.1 - Prob. 43E Ch. 10.1 - Prob. 44E Ch. 10.1 - Curves to parametric equations Give a set of...Ch. 10.1 - Curves to parametric equations Give a set of...Ch. 10.1 - Prob. 47E Ch. 10.1 - Prob. 48E Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - Prob. 55E Ch. 10.1 - Beautiful curves Consider the family of curves...Ch. 10.1 - Prob. 57E Ch. 10.1 - Prob. 58E Ch. 10.1 - Prob. 59E Ch. 10.1 - Derivatives Consider the following parametric...Ch. 10.1 - Derivatives Consider the following parametric...Ch. 10.1 - Prob. 62E Ch. 10.1 - Derivatives Consider the following parametric...Ch. 10.1 - Prob. 64E Ch. 10.1 - Explain why or why not Determine whether the...Ch. 10.1 - Tangent lines Find an equation of the line tangent...Ch. 10.1 - Tangent lines Find an equation of the line tangent...Ch. 10.1 - Tangent lines Find an equation of the line tangent...Ch. 10.1 - Tangent lines Find an equation of the line tangent...Ch. 10.1 - Prob. 70E Ch. 10.1 - Prob. 71E Ch. 10.1 - Prob. 72E Ch. 10.1 - Prob. 73E Ch. 10.1 - Prob. 74E Ch. 10.1 - Prob. 75E Ch. 10.1 - Prob. 76E Ch. 10.1 - Prob. 77E Ch. 10.1 - Prob. 78E Ch. 10.1 - Prob. 79E Ch. 10.1 - Prob. 80E Ch. 10.1 - Prob. 81E Ch. 10.1 - Prob. 82E Ch. 10.1 - Eliminating the parameter Eliminate the parameter...Ch. 10.1 - Eliminating the parameter Eliminate the parameter...Ch. 10.1 - Prob. 85E Ch. 10.1 - Prob. 86E Ch. 10.1 - Prob. 87E Ch. 10.1 - Prob. 88E Ch. 10.1 - Slopes of tangent lines Find all the points at...Ch. 10.1 - Slopes of tangent lines Find all the points at...Ch. 10.1 - Slopes of tangent lines Find all the points at...Ch. 10.1 - Slopes of tangent lines Find all the points at...Ch. 10.1 - Prob. 93E Ch. 10.1 - Prob. 94E Ch. 10.1 - Prob. 95E Ch. 10.1 - Lissajous curves Consider the following Lissajous...Ch. 10.1 - Lam curves The Lam curve described by...Ch. 10.1 - Prob. 98E Ch. 10.1 - Prob. 99E Ch. 10.1 - Prob. 100E Ch. 10.1 - Prob. 101E Ch. 10.1 - Prob. 102E Ch. 10.1 - Prob. 103E Ch. 10.1 - Air drop A plane traveling horizontally at 80 m/s...Ch. 10.1 - Air dropinverse problem A plane traveling...Ch. 10.1 - Prob. 106E Ch. 10.1 - Implicit function graph Explain and carry out a...Ch. 10.1 - Prob. 108E Ch. 10.1 - Prob. 109E Ch. 10.1 - Prob. 110E Ch. 10.2 - Plot the points with polar coordinates (2,6) and...Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - What is the polar equation of the vertical line x...Ch. 10.2 - What is the polar equation of the horizontal line...Ch. 10.2 - Prob. 7E Ch. 10.2 - Prob. 8E Ch. 10.2 - Graph the points with the following polar...Ch. 10.2 - Graph the points with the following polar...Ch. 10.2 - Prob. 11E Ch. 10.2 - Prob. 12E Ch. 10.2 - Prob. 13E Ch. 10.2 - Points in polar coordinates Give two sets of polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Prob. 27E Ch. 10.2 - Prob. 28E Ch. 10.2 - Prob. 29E Ch. 10.2 - Prob. 30E Ch. 10.2 - Prob. 31E Ch. 10.2 - Prob. 32E Ch. 10.2 - Prob. 33E Ch. 10.2 - Prob. 34E Ch. 10.2 - Prob. 35E Ch. 10.2 - Prob. 36E Ch. 10.2 - Prob. 37E Ch. 10.2 - Prob. 38E Ch. 10.2 - Prob. 39E Ch. 10.2 - Prob. 40E Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Prob. 43E Ch. 10.2 - Prob. 44E Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Prob. 49E Ch. 10.2 - Prob. 50E Ch. 10.2 - Prob. 51E Ch. 10.2 - Prob. 52E Ch. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Prob. 55E Ch. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Prob. 59E Ch. 10.2 - Prob. 60E Ch. 10.2 - Prob. 61E Ch. 10.2 - Cartesian-to-polar coordinates Convert the...Ch. 10.2 - Cartesian-to-polar coordinates Convert the...Ch. 10.2 - Cartesian-to-polar coordinates Convert the...Ch. 10.2 - Cartesian-to-polar coordinates Convert the...Ch. 10.2 - Prob. 66E Ch. 10.2 - Prob. 67E Ch. 10.2 - Prob. 68E Ch. 10.2 - Prob. 69E Ch. 10.2 - Prob. 70E Ch. 10.2 - Prob. 71E Ch. 10.2 - Prob. 72E Ch. 10.2 - Prob. 73E Ch. 10.2 - Prob. 74E Ch. 10.2 - Circles in general Show that the polar equation...Ch. 10.2 - Prob. 76E Ch. 10.2 - Prob. 77E Ch. 10.2 - Prob. 78E Ch. 10.2 - Prob. 79E Ch. 10.2 - Prob. 80E Ch. 10.2 - Prob. 81E Ch. 10.2 - Equations of circles Find equations of the circles...Ch. 10.2 - Prob. 83E Ch. 10.2 - Prob. 84E Ch. 10.2 - Prob. 85E Ch. 10.2 - Prob. 86E Ch. 10.2 - Prob. 87E Ch. 10.2 - Prob. 88E Ch. 10.2 - Prob. 89E Ch. 10.2 - Limiting limaon Consider the family of limaons r =...Ch. 10.2 - Prob. 91E Ch. 10.2 - Prob. 92E Ch. 10.2 - Prob. 93E Ch. 10.2 - The lemniscate family Equations of the form r2 = a...Ch. 10.2 - The rose family Equations of the form r = a sin m...Ch. 10.2 - Prob. 96E Ch. 10.2 - Prob. 97E Ch. 10.2 - The rose family Equations of the form r = a sin m...Ch. 10.2 - Prob. 99E Ch. 10.2 - Prob. 100E Ch. 10.2 - Prob. 101E Ch. 10.2 - Spirals Graph the following spirals. Indicate the...Ch. 10.2 - Prob. 103E Ch. 10.2 - Prob. 104E Ch. 10.2 - Prob. 105E Ch. 10.2 - Prob. 106E Ch. 10.2 - Enhanced butterfly curve The butterfly curve of...Ch. 10.2 - Prob. 108E Ch. 10.2 - Prob. 109E Ch. 10.2 - Prob. 110E Ch. 10.2 - Prob. 111E Ch. 10.2 - Cartesian lemniscate Find the equation in...Ch. 10.2 - Prob. 113E Ch. 10.2 - Prob. 114E Ch. 10.3 - Prob. 1E Ch. 10.3 - Prob. 2E Ch. 10.3 - Explain why the slope of the line tangent to the...Ch. 10.3 - What integral must be evaluated to find the area...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Horizontal and vertical tangents Find the points...Ch. 10.3 - Horizontal and vertical tangents Find the points...Ch. 10.3 - Horizontal and vertical tangents Find the points...Ch. 10.3 - Prob. 18E Ch. 10.3 - Prob. 19E Ch. 10.3 - Prob. 20E Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Prob. 37E Ch. 10.3 - Prob. 38E Ch. 10.3 - Prob. 39E Ch. 10.3 - Prob. 40E Ch. 10.3 - Prob. 41E Ch. 10.3 - Prob. 42E Ch. 10.3 - Prob. 43E Ch. 10.3 - Prob. 44E Ch. 10.3 - Prob. 45E Ch. 10.3 - Multiple identities Explain why the point (1, 3/2)...Ch. 10.3 - Area of plane regions Find the areas of the...Ch. 10.3 - Area of plane regions Find the areas of the...Ch. 10.3 - Area of plane regions Find the areas of the...Ch. 10.3 - Area of plane regions Find the areas of the...Ch. 10.3 - Prob. 51E Ch. 10.3 - Prob. 52E Ch. 10.3 - Regions bounded by a spiral Let Rn be the region...Ch. 10.3 - Area of polar regions Find the area of the regions...Ch. 10.3 - Area of polar regions Find the area of the regions...Ch. 10.3 - Area of polar regions Find the area of the regions...Ch. 10.3 - Prob. 57E Ch. 10.3 - Prob. 58E Ch. 10.3 - Grazing goat problems Consider the following...Ch. 10.3 - Grazing goat problems Consider the following...Ch. 10.3 - Prob. 61E Ch. 10.3 - Tangents and normals Let a polar curve be...Ch. 10.3 - Prob. 63E Ch. 10.4 - Give the property that defines all parabolas.Ch. 10.4 - Prob. 2E Ch. 10.4 - Give the property that defines all hyperbolas.Ch. 10.4 - Prob. 4E Ch. 10.4 - Prob. 5E Ch. 10.4 - What is the equation of the standard parabola with...Ch. 10.4 - Prob. 7E Ch. 10.4 - Prob. 8E Ch. 10.4 - Given vertices (a, 0) and eccentricity e, what are...Ch. 10.4 - Prob. 10E Ch. 10.4 - What are the equations of the asymptotes of a...Ch. 10.4 - Prob. 12E Ch. 10.4 - Graphing parabolas Sketch a graph of the following...Ch. 10.4 - Prob. 14E Ch. 10.4 - Prob. 15E Ch. 10.4 - Prob. 16E Ch. 10.4 - Prob. 17E Ch. 10.4 - Graphing parabolas Sketch a graph of the following...Ch. 10.4 - Prob. 19E Ch. 10.4 - Equations of parabolas Find an equation of the...Ch. 10.4 - Equations of parabolas Find an equation of the...Ch. 10.4 - Prob. 22E Ch. 10.4 - Prob. 23E Ch. 10.4 - Equations of parabolas Find an equation of the...Ch. 10.4 - From graphs to equations Write an equation of the...Ch. 10.4 - From graphs to equations Write an equation of the...Ch. 10.4 - Prob. 27E Ch. 10.4 - Prob. 28E Ch. 10.4 - Prob. 29E Ch. 10.4 - Prob. 30E Ch. 10.4 - Prob. 31E Ch. 10.4 - Prob. 32E Ch. 10.4 - Equations of ellipses Find an equation of the...Ch. 10.4 - Equations of ellipses Find an equation of the...Ch. 10.4 - Equations of ellipses Find an equation of the...Ch. 10.4 - Prob. 36E Ch. 10.4 - Prob. 37E Ch. 10.4 - Prob. 38E Ch. 10.4 - Prob. 39E Ch. 10.4 - Prob. 40E Ch. 10.4 - Prob. 41E Ch. 10.4 - Prob. 42E Ch. 10.4 - Prob. 43E Ch. 10.4 - Prob. 44E Ch. 10.4 - Equations of hyperbolas Find an equation of the...Ch. 10.4 - Equations of hyperbolas Find an equation of the...Ch. 10.4 - Equations of hyperbolas Find an equation of the...Ch. 10.4 - Prob. 48E Ch. 10.4 - From graphs to equations Write an equation of the...Ch. 10.4 - From graphs to equations Write an equation of the...Ch. 10.4 - Eccentricity-directrix approach Find an equation...Ch. 10.4 - Eccentricity-directrix approach Find an equation...Ch. 10.4 - Eccentricity-directrix approach Find an equation...Ch. 10.4 - Eccentricity-directrix approach Find an equation...Ch. 10.4 - Prob. 55E Ch. 10.4 - Prob. 56E Ch. 10.4 - Prob. 57E Ch. 10.4 - Prob. 58E Ch. 10.4 - Prob. 59E Ch. 10.4 - Prob. 60E Ch. 10.4 - Tracing hyperbolas and parabolas Graph the...Ch. 10.4 - Tracing hyperbolas and parabolas Graph the...Ch. 10.4 - Tracing hyperbolas and parabolas Graph the...Ch. 10.4 - Tracing hyperbolas and parabolas Graph the...Ch. 10.4 - Prob. 65E Ch. 10.4 - Hyperbolas with a graphing utility Use a graphing...Ch. 10.4 - Prob. 67E Ch. 10.4 - Prob. 68E Ch. 10.4 - Tangent lines Find an equation of the tine tangent...Ch. 10.4 - Tangent lines Find an equation of the tine tangent...Ch. 10.4 - Tangent lines Find an equation of the tine tangent...Ch. 10.4 - Prob. 72E Ch. 10.4 - Prob. 73E Ch. 10.4 - Prob. 74E Ch. 10.4 - Prob. 75E Ch. 10.4 - The ellipse and the parabola Let R be the region...Ch. 10.4 - Tangent lines for an ellipse Show that an equation...Ch. 10.4 - Prob. 78E Ch. 10.4 - Volume of an ellipsoid Suppose that the ellipse...Ch. 10.4 - Area of a sector of a hyperbola Consider the...Ch. 10.4 - Volume of a hyperbolic cap Consider the region R...Ch. 10.4 - Prob. 82E Ch. 10.4 - Prob. 83E Ch. 10.4 - Golden Gate Bridge Completed in 1937, San...Ch. 10.4 - Prob. 85E Ch. 10.4 - Prob. 86E Ch. 10.4 - Prob. 87E Ch. 10.4 - Prob. 88E Ch. 10.4 - Shared asymptotes Suppose that two hyperbolas with...Ch. 10.4 - Focal chords A focal chord of a conic section is a...Ch. 10.4 - Focal chords A focal chord of a conic section is a...Ch. 10.4 - Focal chords A focal chord of a conic section is a...Ch. 10.4 - Prob. 93E Ch. 10.4 - Prob. 94E Ch. 10.4 - Confocal ellipse and hyperbola Show that an...Ch. 10.4 - Approach to asymptotes Show that the vertical...Ch. 10.4 - Prob. 97E Ch. 10.4 - Prob. 98E Ch. 10.4 - Prob. 99E Ch. 10 - Explain why or why not Determine whether the...Ch. 10 - Prob. 2RE Ch. 10 - Prob. 3RE Ch. 10 - Prob. 4RE Ch. 10 - Prob. 5RE Ch. 10 - Prob. 6RE Ch. 10 - Prob. 7RE Ch. 10 - Prob. 8RE Ch. 10 - Eliminating the parameter Eliminate the parameter...Ch. 10 - Prob. 10RE Ch. 10 - Parametric description Write parametric equations...Ch. 10 - Parametric description Write parametric equations...Ch. 10 - Prob. 13RE Ch. 10 - Prob. 14RE Ch. 10 - Parametric description Write parametric equations...Ch. 10 - Parametric description Write parametric equations...Ch. 10 - Tangent lines Find an equation of the line tangent...Ch. 10 - Prob. 18RE Ch. 10 - Prob. 19RE Ch. 10 - Sets in polar coordinates Sketch the following...Ch. 10 - Prob. 21RE Ch. 10 - Prob. 22RE Ch. 10 - Polar conversion Write the equation...Ch. 10 - Polar conversion Consider the equation r = 4/(sin ...Ch. 10 - Prob. 25RE Ch. 10 - Prob. 26RE Ch. 10 - Prob. 27RE Ch. 10 - Slopes of tangent lines a. Find all points where...Ch. 10 - Slopes of tangent lines a. Find all points where...Ch. 10 - Slopes of tangent lines a. Find all points where...Ch. 10 - Prob. 31RE Ch. 10 - The region enclosed by all the leaves of the rose...Ch. 10 - The region enclosed by the limaon r = 3 cos Ch. 10 - The region inside the limaon r = 2 + cos and...Ch. 10 - Prob. 35RE Ch. 10 - Prob. 36RE Ch. 10 - The area that is inside the cardioid r = 1 + cos ...Ch. 10 - Prob. 38RE Ch. 10 - Prob. 39RE Ch. 10 - Prob. 40RE Ch. 10 - Conic sections a. Determine whether the following...Ch. 10 - Prob. 42RE Ch. 10 - Prob. 43RE Ch. 10 - Prob. 44RE Ch. 10 - Tangent lines Find an equation of the line tangent...Ch. 10 - Prob. 46RE Ch. 10 - Tangent lines Find an equation of the line tangent...Ch. 10 - Tangent lines Find an equation of the line tangent...Ch. 10 - Prob. 49RE Ch. 10 - Prob. 50RE Ch. 10 - Prob. 51RE Ch. 10 - Prob. 52RE Ch. 10 - Prob. 53RE Ch. 10 - Prob. 54RE Ch. 10 - Eccentricity-directrix approach Find an equation...Ch. 10 - Prob. 56RE Ch. 10 - Prob. 57RE Ch. 10 - Prob. 58RE Ch. 10 - Prob. 59RE Ch. 10 - Prob. 60RE Ch. 10 - Prob. 61RE Ch. 10 - Prob. 62RE Ch. 10 - Prob. 63RE Ch. 10 - Prob. 64RE Ch. 10 - Prob. 65RE Ch. 10 - Prob. 66RE Ch. 10 - Prob. 67RE Ch. 10 - Prob. 68RE Ch. 10 - Prob. 69RE Ch. 10 - Prob. 70RE Ch. 10 - Prob. 71RE

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