Explanation Given: The polar curve is as below. r = 1 + cos θ . Calculation: Horizontal tangents is obtained when the condition d y d θ = 0 is true, and the vertical tangents is obtained with the condition d x d θ = 0 . Differentiate the curve equation ( r = 1 + cos θ ) with respect to θ . r = 1 + cos θ d r d θ = − sin θ Write the chain rule for the condition d y d x . d y d x = d y d θ d x d θ = d r d θ sin θ + r cos θ d r d θ cos θ − r sin θ For horizontal tangents, the condition d y d θ = 0 is true. Substitute ( − sin θ ) for ( d r d θ ) and ( 1 + cos θ ) for ( r ) in equation ( d r d θ cos θ − r sin θ ) . d x d θ = d r d θ cos θ − r sin θ = ( − sin θ ) cos θ − ( 1 + cos θ ) sin θ = − sin θ cos θ − sin θ − cos θ sin θ = − 2 sin θ cos θ − sin θ = − sin θ ( 2 cos θ + 1 ) Substitute ( − sin θ ) for ( d r d θ ) and ( 1 + cos θ ) for ( r ) in equation ( d r d θ sin θ + r cos θ ) . d y d θ = d r d θ sin θ + r cos θ d y d θ = ( − sin θ ) sin θ + ( 1 + cos θ ) cos θ = sin 2 θ + cos θ + cos 2 θ = cos 2 θ + cos θ = 2 cos 2 θ − 1 + cos θ = ( 2 cos θ − 1 ) ( cos θ + 1 ) Horizontal tangents is obtained by the condition d y d θ = 0 . Substitute ( 2 cos θ − 1 ) ( cos θ + 1 ) for ( d y d θ ) . d y d θ = 0 ( 2 cos θ − 1 ) ( cos θ + 1 ) = 0 cos θ = 1 2 or − 1 θ = π 3 , π , 5 π 3 Substitute ( π 3 ) for θ in equation r = 1 + cos θ . r = 1 + cos θ = 1 + cos ( π 3 ) = 3 2 Substitute ( π ) for θ in equation r = 1 + cos θ . r = 1 + cos θ = 1 − cos ( π ) = 0 Substitute ( 5 π 3 ) for θ in equation r = 1 + cos θ

8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Chapter 10.3, Problem 63E

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To find: The tangent line is horizontal or vertical with the point for the curve following relation r=1+cosθ .

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Chapter 10.3, Problem 62E

Chapter 10.3, Problem 64E

Chapter 10 Solutions

Single Variable Calculus

Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E 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 - Prob. 10E

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 - Describe the motion of a particle with position...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 - Prob. 27E Ch. 10.1 - Match the parametric equations with the graphs...Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - Use a graphing device and the result of Exercise...Ch. 10.1 - Prob. 33E Ch. 10.1 - Prob. 34E Ch. 10.1 - Prob. 35E Ch. 10.1 - Prob. 36E Ch. 10.1 - Prob. 37E Ch. 10.1 - Prob. 38E Ch. 10.1 - Prob. 39E Ch. 10.1 - Prob. 40E Ch. 10.1 - Prob. 41E Ch. 10.1 - Prob. 42E Ch. 10.1 - Prob. 43E Ch. 10.1 - (a) Find parametric equations for the set of all...Ch. 10.1 - Suppose that the position of one particle at time...Ch. 10.1 - Prob. 46E Ch. 10.1 - Prob. 47E Ch. 10.1 - Prob. 48E Ch. 10.1 - Prob. 49E Ch. 10.1 - Prob. 50E Ch. 10.1 - Prob. 51E Ch. 10.1 - Prob. 52E Ch. 10.2 - Prob. 1E Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - Prob. 5E Ch. 10.2 - Prob. 6E Ch. 10.2 - Prob. 7E Ch. 10.2 - Prob. 8E Ch. 10.2 - Prob. 9E Ch. 10.2 - Find an equation of the tangent to the curve at...Ch. 10.2 - Prob. 11E Ch. 10.2 - Prob. 12E Ch. 10.2 - Prob. 13E Ch. 10.2 - Prob. 14E Ch. 10.2 - Find dy/dx and d2y/dx2. For which values of t is...Ch. 10.2 - Prob. 16E Ch. 10.2 - Prob. 17E Ch. 10.2 - Prob. 18E Ch. 10.2 - Prob. 19E Ch. 10.2 - Prob. 20E Ch. 10.2 - Use a graph to estimate the coordinates of the...Ch. 10.2 - Prob. 22E Ch. 10.2 - Prob. 23E Ch. 10.2 - Prob. 24E Ch. 10.2 - Prob. 25E Ch. 10.2 - Prob. 26E Ch. 10.2 - (a) Find the slope of the tangent line to the...Ch. 10.2 - Prob. 28E Ch. 10.2 - At what point(s) on the curve x = 3t2 + 1, y = t3 ...Ch. 10.2 - Prob. 30E Ch. 10.2 - Use the parametric equations of an ellipse, x = a...Ch. 10.2 - Find the area enclosed by the curve x = t2 2t,...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 - Find the exact length of the curve. 41. x = 1 +...Ch. 10.2 - Find the exact length of the curve. 42. x = et t,...Ch. 10.2 - Prob. 43E Ch. 10.2 - Prob. 44E Ch. 10.2 - Prob. 45E Ch. 10.2 - Prob. 46E Ch. 10.2 - Prob. 47E Ch. 10.2 - Prob. 48E Ch. 10.2 - Prob. 49E Ch. 10.2 - In Exercise 10.1.43 you were asked to derive the...Ch. 10.2 - Prob. 51E Ch. 10.2 - Prob. 52E Ch. 10.2 - Prob. 53E Ch. 10.2 - Prob. 54E Ch. 10.2 - (a) Graph the epitrochoid with equations...Ch. 10.2 - Prob. 57E Ch. 10.2 - Set up an integral that represents the area of the...Ch. 10.2 - Prob. 59E Ch. 10.2 - Prob. 60E Ch. 10.2 - Prob. 61E Ch. 10.2 - Prob. 62E Ch. 10.2 - Prob. 63E Ch. 10.2 - Prob. 64E Ch. 10.2 - Prob. 65E Ch. 10.2 - Prob. 66E Ch. 10.2 - Prob. 67E Ch. 10.2 - Prob. 68E Ch. 10.2 - Prob. 69E Ch. 10.2 - (a) Use the formula in Exercise 69(b) to find the...Ch. 10.2 - Prob. 71E Ch. 10.2 - Prob. 72E Ch. 10.2 - A string is wound around a circle and then unwound...Ch. 10.2 - A cow is tied to a silo with radius r by a rope...Ch. 10.3 - Prob. 1E Ch. 10.3 - Prob. 2E Ch. 10.3 - Plot the point whose polar coordinates are given....Ch. 10.3 - Plot the point whose polar coordinates are given....Ch. 10.3 - The Cartesian coordinates of a point are given....Ch. 10.3 - Prob. 6E Ch. 10.3 - 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Prob. 8E Ch. 10.4 - Prob. 9E Ch. 10.4 - Prob. 10E Ch. 10.4 - Prob. 11E Ch. 10.4 - Prob. 12E Ch. 10.4 - Prob. 13E Ch. 10.4 - Prob. 14E Ch. 10.4 - Prob. 15E Ch. 10.4 - Prob. 16E Ch. 10.4 - Prob. 17E Ch. 10.4 - Prob. 18E Ch. 10.4 - Prob. 19E Ch. 10.4 - Prob. 20E Ch. 10.4 - Prob. 21E Ch. 10.4 - Prob. 22E Ch. 10.4 - Prob. 23E Ch. 10.4 - Prob. 24E Ch. 10.4 - Find the area of the region that lies inside the...Ch. 10.4 - Prob. 26E 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 - Prob. 33E Ch. 10.4 - Prob. 34E Ch. 10.4 - Prob. 35E 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 - The points of intersection of the cardioid r = 1 +...Ch. 10.4 - When recording live performances, sound engineers...Ch. 10.4 - Prob. 45E Ch. 10.4 - Find the exact length of the polar curve. 46. r =...Ch. 10.4 - Prob. 47E Ch. 10.4 - Prob. 48E Ch. 10.4 - Find the exact length of the curve. Use a graph to...Ch. 10.4 - Find the exact length of the curve. Use a graph to...Ch. 10.4 - Prob. 51E Ch. 10.4 - Prob. 52E Ch. 10.4 - Prob. 53E Ch. 10.4 - Use a calculator to find the length of the curve...Ch. 10.4 - (a) Use Formula 10.2.6 to show that the area of...Ch. 10.4 - Prob. 56E Ch. 10.5 - Prob. 1E Ch. 10.5 - Prob. 2E Ch. 10.5 - Prob. 3E Ch. 10.5 - Prob. 4E Ch. 10.5 - Prob. 5E Ch. 10.5 - Prob. 6E Ch. 10.5 - Prob. 7E Ch. 10.5 - Prob. 8E Ch. 10.5 - Prob. 9E Ch. 10.5 - Prob. 10E Ch. 10.5 - Prob. 11E Ch. 10.5 - Prob. 12E Ch. 10.5 - Prob. 13E Ch. 10.5 - Prob. 14E Ch. 10.5 - Prob. 15E Ch. 10.5 - Prob. 16E Ch. 10.5 - Prob. 17E Ch. 10.5 - Prob. 18E Ch. 10.5 - Prob. 19E Ch. 10.5 - Prob. 20E Ch. 10.5 - Prob. 21E Ch. 10.5 - Prob. 22E Ch. 10.5 - Prob. 23E Ch. 10.5 - Prob. 24E Ch. 10.5 - Prob. 25E Ch. 10.5 - Prob. 26E Ch. 10.5 - Prob. 27E Ch. 10.5 - Prob. 28E Ch. 10.5 - Prob. 29E Ch. 10.5 - Prob. 30E Ch. 10.5 - Prob. 31E Ch. 10.5 - Prob. 32E Ch. 10.5 - Prob. 33E Ch. 10.5 - Prob. 34E Ch. 10.5 - Prob. 35E Ch. 10.5 - Prob. 36E Ch. 10.5 - Prob. 37E Ch. 10.5 - Prob. 38E Ch. 10.5 - Prob. 39E Ch. 10.5 - Prob. 40E Ch. 10.5 - Prob. 41E Ch. 10.5 - Prob. 42E Ch. 10.5 - Prob. 43E Ch. 10.5 - Prob. 44E Ch. 10.5 - Prob. 45E Ch. 10.5 - Prob. 46E Ch. 10.5 - Prob. 47E Ch. 10.5 - Prob. 48E Ch. 10.5 - Prob. 49E Ch. 10.5 - Prob. 50E Ch. 10.5 - Prob. 51E Ch. 10.5 - Prob. 52E Ch. 10.5 - Show that the function defined by the upper branch...Ch. 10.5 - Prob. 54E Ch. 10.5 - Determine the type of curve represented by the...Ch. 10.5 - Prob. 56E Ch. 10.5 - Show that the tangent lines to the parabola x2 =...Ch. 10.5 - Show that if an ellipse and a hyperbola have the...Ch. 10.5 - Prob. 59E Ch. 10.5 - Prob. 60E Ch. 10.5 - Find the area of the region enclosed by the...Ch. 10.5 - Prob. 62E Ch. 10.5 - Prob. 63E Ch. 10.5 - (a) Calculate the surface area of the ellipsoid...Ch. 10.5 - Prob. 65E Ch. 10.5 - Prob. 66E Ch. 10.6 - Prob. 1E Ch. 10.6 - Prob. 2E Ch. 10.6 - Prob. 3E Ch. 10.6 - Prob. 4E Ch. 10.6 - Prob. 5E Ch. 10.6 - Prob. 6E Ch. 10.6 - Prob. 7E Ch. 10.6 - Prob. 8E Ch. 10.6 - Prob. 9E Ch. 10.6 - Prob. 10E Ch. 10.6 - Prob. 11E Ch. 10.6 - Prob. 12E Ch. 10.6 - Prob. 13E Ch. 10.6 - Prob. 14E Ch. 10.6 - Prob. 15E Ch. 10.6 - Prob. 16E Ch. 10.6 - Prob. 17E Ch. 10.6 - Prob. 18E Ch. 10.6 - Prob. 19E Ch. 10.6 - Prob. 20E Ch. 10.6 - Prob. 21E Ch. 10.6 - Prob. 22E Ch. 10.6 - Prob. 23E Ch. 10.6 - Prob. 24E Ch. 10.6 - Prob. 25E Ch. 10.6 - Prob. 26E Ch. 10.6 - The orbit of Halleys comet, last seen in 1986 and...Ch. 10.6 - Prob. 28E Ch. 10.6 - The planet Mercury travels in an elliptical orbit...Ch. 10.6 - The distance from the dwarf planet Pluto to the...Ch. 10.6 - Prob. 31E Ch. 10 - Prob. 1RCC Ch. 10 - Prob. 2RCC Ch. 10 - Prob. 3RCC Ch. 10 - Prob. 4RCC Ch. 10 - Prob. 5RCC Ch. 10 - Prob. 6RCC Ch. 10 - Prob. 7RCC Ch. 10 - Prob. 8RCC Ch. 10 - Prob. 9RCC Ch. 10 - Prob. 1RQ Ch. 10 - Prob. 2RQ Ch. 10 - Prob. 3RQ Ch. 10 - Prob. 4RQ Ch. 10 - Determine whether the statement is true or false....Ch. 10 - Prob. 6RQ Ch. 10 - Prob. 7RQ Ch. 10 - Prob. 8RQ Ch. 10 - Prob. 9RQ Ch. 10 - Prob. 10RQ Ch. 10 - Prob. 1RE 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 - Prob. 9RE Ch. 10 - Prob. 10RE Ch. 10 - Prob. 11RE Ch. 10 - Prob. 12RE Ch. 10 - Prob. 13RE Ch. 10 - Sketch the polar curve. 14. r = 2 cos(/2)Ch. 10 - Prob. 15RE Ch. 10 - Prob. 16RE Ch. 10 - Prob. 17RE Ch. 10 - Prob. 18RE Ch. 10 - Prob. 19RE Ch. 10 - Prob. 20RE Ch. 10 - Prob. 21RE Ch. 10 - Prob. 22RE Ch. 10 - Prob. 23RE Ch. 10 - Prob. 24RE Ch. 10 - Prob. 25RE Ch. 10 - Prob. 26RE Ch. 10 - Prob. 27RE Ch. 10 - Prob. 28RE Ch. 10 - Prob. 29RE Ch. 10 - Prob. 30RE Ch. 10 - Prob. 31RE Ch. 10 - Prob. 32RE Ch. 10 - Find the points of intersection of the curves r =...Ch. 10 - Prob. 34RE Ch. 10 - Find the area of the region that lies inside both...Ch. 10 - Find the area of the region that lies inside the...Ch. 10 - Prob. 37RE Ch. 10 - Prob. 38RE Ch. 10 - Prob. 39RE Ch. 10 - Prob. 40RE Ch. 10 - Find the area of the surface obtained by rotating...Ch. 10 - Prob. 42RE Ch. 10 - Prob. 43RE Ch. 10 - Prob. 44RE Ch. 10 - Prob. 45RE Ch. 10 - Prob. 46RE Ch. 10 - Prob. 47RE Ch. 10 - Prob. 48RE 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 - Prob. 55RE Ch. 10 - Prob. 56RE Ch. 10 - Prob. 57RE Ch. 10 - A curve called the folium of Descartes is defined...Ch. 10 - Prob. 1P Ch. 10 - Prob. 2P Ch. 10 - What is the smallest viewing rectangle that...Ch. 10 - Four bugs are placed at the four comers of a...Ch. 10 - Prob. 5P Ch. 10 - A circle C of radius 2r has its center at the...

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The tangent line is horizontal or vertical with the point for the curve following relation r = 1 + cos θ .

Solution Summary: The author explains how the tangent line is horizontal or vertical with the relation r=1+mathrmcostheta .

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To find: The tangent line is horizontal or vertical with the point for the curve following relation r=1+cosθ .

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Chapter 10 Solutions