To determine Find the slope of the curve at the given polar points and sketch the curve along with its tangent lines at the points. Explanation Given data: The given polar points are written as follows: ( r , θ ) = ( sin 2 θ , ± π 4 ) ( r , θ ) = ( sin 2 θ , ± 3 π 4 ) Formula used: Write the expression for slope of the curve as follows: d y d x | ( r , θ ) = r ′ sin θ + r cos θ r ′ cos θ − r sin θ (1) Calculation: Rewrite the given polar points as follows: ( r , θ ) = ( sin 2 θ , π 4 ) , ( sin 2 θ , − π 4 ) , ( sin 2 θ , 3 π 4 ) , ( sin 2 θ , − 3 π 4 ) = ( sin 2 ( π 4 ) , π 4 ) , ( sin 2 ( − π 4 ) , − π 4 ) , ( sin 2 ( 3 π 4 ) , 3 π 4 ) , ( sin 2 ( − 3 π 4 ) , − 3 π 4 ) = ( 1 , π 4 ) , ( − 1 , − π 4 ) , ( − 1 , 3 π 4 ) , ( 1 , − 3 π 4 ) Therefore, the given polar points are ( 1 , π 4 ) , ( − 1 , − π 4 ) , ( − 1 , 3 π 4 ) , and ( 1 , − 3 π 4 ) . From the give polar points, r is given as follows: r = sin 2 θ Differentiate on both sides of the expression as follows: d r d θ = d d θ ( sin 2 θ ) r ′ = 2 cos 2 θ Substitute sin 2 θ for r and 2 cos 2 θ for r ′ in Equation (1) to obtain the slope the curve. d y d x | ( r , θ ) = ( 2 cos 2 θ ) sin θ + ( sin 2 θ ) cos θ ( 2 cos 2 θ ) cos θ − ( sin 2 θ ) sin θ d y d x | ( r , θ ) = 2 cos 2 θ sin θ + sin 2 θ cos θ 2 cos 2 θ cos θ − sin 2 θ sin θ (2) Substitute 1 for r and π 4 for θ to obtain the slope of the curve at ( 1 , π 4 ) . d y d x | ( 1 , π 4 ) = 2 cos 2 ( π 4 ) sin ( π 4 ) + sin 2 ( π 4 ) cos ( π 4 ) 2 cos 2 ( π 4 ) cos ( π 4 ) − sin 2 ( π 4 ) sin ( π 4 ) = 2 ( 0 ) ( 1 2 ) + ( 1 ) ( 1 2 ) 2 ( 0 ) ( 1 2 ) − ( 1 ) ( 1 2 ) = − 1 Substitute − 1 for r and − π 4 for θ in Equation (2) to obtain the slope of the curve at ( − 1 , − π 4 )

University Calculus: Early Transcendentals, Loose-leaf Edition (4th Edition)

4th Edition

ISBN: 9780135164860

Author: Joel R. Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 10.4, Problem 19E

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Find the slope of the curve at the given polar points and sketch the curve along with its tangent lines at the points.

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Chapter 10.4, Problem 18E

Chapter 10.4, Problem 20E

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Chapter 10 Solutions

University Calculus: Early Transcendentals, Loose-leaf Edition (4th Edition)

Ch. 10.1 - Finding Cartesian from Parametric...Ch. 10.1 - Finding Cartesian from Parametric...Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Finding Cartesian from Parametric...Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Finding Cartesian from Parametric Equations...Ch. 10.1 - Finding Cartesian from Parametric...

Ch. 10.1 - Finding Cartesian from Parametric Equations...Ch. 10.1 - Prob. 12E Ch. 10.1 - Finding Cartesian from Parametric Equations...Ch. 10.1 - Prob. 14E Ch. 10.1 - Finding Cartesian from Parametric Equations...Ch. 10.1 - Finding Cartesian from Parametric Equations...Ch. 10.1 - Finding Cartesian from Parametric Equations...Ch. 10.1 - Finding Cartesian from Parametric...Ch. 10.1 - Prob. 19E Ch. 10.1 - Prob. 20E Ch. 10.1 - In Exercises 19–24, match the parametric equations...Ch. 10.1 - In Exercises 19–24, match the parametric equations...Ch. 10.1 - In Exercises 19–24, match the parametric equations...Ch. 10.1 - In Exercises 19–24, match the parametric equations...Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - In Exercises 25–28, use the given graphs of x =...Ch. 10.1 - Finding Parametric Equations Find parametric...Ch. 10.1 - Find parametric equations and a parameter interval...Ch. 10.1 - Prob. 31E Ch. 10.1 - In Exercises 31–36, find a parametrization for the...Ch. 10.1 - 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Prob. 2E Ch. 10.3 - Prob. 3E Ch. 10.3 - Prob. 4E Ch. 10.3 - Prob. 5E Ch. 10.3 - Prob. 6E Ch. 10.3 - Find the polar coordinates, 0 = ? = 2p and r = 0,...Ch. 10.3 - Prob. 8E Ch. 10.3 - Prob. 9E Ch. 10.3 - Find the polar coordinates, and , of the...Ch. 10.3 - Graph the sets of points whose polar coordinates...Ch. 10.3 - Prob. 12E Ch. 10.3 - Prob. 13E Ch. 10.3 - Prob. 14E Ch. 10.3 - Graph the sets of points whose polar coordinates...Ch. 10.3 - Prob. 16E Ch. 10.3 - Prob. 17E Ch. 10.3 - Prob. 18E Ch. 10.3 - Prob. 19E Ch. 10.3 - Prob. 20E Ch. 10.3 - Prob. 21E Ch. 10.3 - Prob. 22E Ch. 10.3 - Prob. 23E Ch. 10.3 - Prob. 24E Ch. 10.3 - Prob. 25E Ch. 10.3 - Prob. 26E Ch. 10.3 - Prob. 27E Ch. 10.3 - Prob. 28E Ch. 10.3 - Replace the polar equations in Exercises 2752 with...Ch. 10.3 - Prob. 30E Ch. 10.3 - Replace the polar equations in Exercises 2752 with...Ch. 10.3 - Prob. 32E Ch. 10.3 - Prob. 33E Ch. 10.3 - Prob. 34E Ch. 10.3 - Prob. 35E Ch. 10.3 - Prob. 36E Ch. 10.3 - Replace the polar equations in Exercises 27–52...Ch. 10.3 - 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Finding Polar Areas Find the areas of the regions...Ch. 10.5 - Prob. 5E Ch. 10.5 - Prob. 6E Ch. 10.5 - Prob. 7E Ch. 10.5 - Prob. 8E Ch. 10.5 - Find the areas of the regions in Exercises...Ch. 10.5 - Find the areas of the regions in Exercises...Ch. 10.5 - Find the areas of the regions in Exercises...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 - Find the areas of the regions in Exercises...Ch. 10.5 - Find the areas of the regions in Exercises...Ch. 10.5 - Prob. 19E Ch. 10.5 - Prob. 20E Ch. 10.5 - Find the lengths of the curves in Exercises 2128....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 - Find the lengths of the curves in Exercises 2128....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 - Prob. 1GYR Ch. 10 - Prob. 2GYR Ch. 10 - Prob. 3GYR Ch. 10 - Prob. 4GYR Ch. 10 - Prob. 5GYR Ch. 10 - 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