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Math Calculus Calculus: Early Transcendentals (2nd Edition)

Slopes of tangent lines a. Find all points where the following curves have vertical and horizontal tangent lines. b. Find the slope of the lines tangent to the curve at the origin (when relevant). c. Sketch the curve and all the tangent lines identified in parts (a) and (b) . 28. r = 2 cos 2 θ

Slopes of tangent lines a. Find all points where the following curves have vertical and horizontal tangent lines. b. Find the slope of the lines tangent to the curve at the origin (when relevant). c. Sketch the curve and all the tangent lines identified in parts (a) and (b) . 28. r = 2 cos 2 θ

Solution Summary: The author explains the slope of the tangent to the curve r=f(theta ) and its slope.

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, Problem 28RE

Slopes of tangent lines

a. Find all points where the following curves have vertical and horizontal tangent lines.
b. Find the slope of the lines tangent to the curve at the origin (when relevant).
c. Sketch the curve and all the tangent lines identified in parts (a) and (b).

28. r = 2 cos 2θ

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Chapter 10, Problem 27RE

Chapter 10, Problem 29RE

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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