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 ) = ( cos 2 θ , 0 ) ( r , θ ) = ( cos 2 θ , ± π 2 ) ( r , θ ) = ( cos 2 θ , π ) 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 ) = ( cos 2 θ , 0 ) , ( cos 2 θ , π 2 ) , ( cos 2 θ , − π 2 ) , ( cos 2 θ , π ) = ( cos 2 ( 0 ) , 0 ) , ( cos 2 ( π 2 ) , π 2 ) , ( cos 2 ( − π 2 ) , − π 2 ) , ( cos 2 ( π ) , π ) = ( 1 , 0 ) , ( − 1 , π 2 ) , ( − 1 , − π 2 ) , ( 1 , π ) Therefore, the given polar points are ( 1 , 0 ) , ( − 1 , π 2 ) , ( − 1 , − π 2 ) , and ( 1 , π ) . From the give polar points, r is given as follows: r = cos 2 θ Differentiate on both sides of the expression as follows: d r d θ = d d θ ( cos 2 θ ) r ′ = − 2 sin 2 θ Substitute cos 2 θ for r and − 2 sin 2 θ for r ′ in Equation (1) to obtain the slope of the curve. d y d x | ( r , θ ) = ( − 2 sin 2 θ ) sin θ + ( cos 2 θ ) cos θ ( − 2 sin 2 θ ) cos θ − ( cos 2 θ ) sin θ d y d x | ( r , θ ) = − 2 sin 2 θ sin θ + cos 2 θ cos θ − 2 sin 2 θ cos θ − cos 2 θ sin θ (2) Substitute 1 for r and 0 for θ to obtain the slope of the curve at ( 1 , 0 ) . d y d x | ( 1 , 0 ) = − 2 sin [ 2 ( 0 ) ] sin ( 0 ) + cos [ 2 ( 0 ) ] cos ( 0 ) − 2 sin [ 2 ( 0 ) ] cos ( 0 ) − cos [ 2 ( 0 ) ] sin ( 0 ) = − 2 ( 0 ) ( 0 ) + ( 1 ) ( 1 ) − 2 ( 0 ) ( 1 ) − ( 1 ) ( 0 ) = 1 0 Therefore, the slope of the given curve at ( 1 , 0 ) is undefined

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

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

Publisher: PEARSON

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Chapter 10.4, Problem 20E

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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 19E

Chapter 10.4, Problem 21E

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Chapter 10 Solutions

University Calculus: Early Transcendentals (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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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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