Explanation Given: The function, H ( θ ) = θ sin θ . Concept Used: The product rule of derivatives: d d x [ f ( x ) ⋅ g ( x ) ] = f ( x ) ⋅ d d x g ( x ) + g ( x ) ⋅ d d x f ( x ) Power rule of derivatives: d d x ( x n ) = n x n − 1 d d x ( sin x ) = cos x d d x ( cos x ) = − sin x Calculation: In order to find the derivatives of the given function, use the product rule of derivatives. Observe that the given function is a product of two functions θ and ( sin θ ) . So, using the product rule and power rule of derivatives, it gives H ' ( θ ) = d d θ ( θ sin θ ) = θ ⋅ d d θ ( sin θ ) + ( sin θ ) ⋅ d d θ ( θ ) = θ ⋅ ( cos θ ) + ( sin θ ) ⋅ ( 1 ) = θ cos θ

15th Edition

ISBN: 9781133109631

Author: Stewart, JAMES, Day, Troy

Publisher: Cengage Learning,

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Chapter 3.4, Problem 41E

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The derivatives H'(θ) and H''(θ) for the function, H(θ)=θsinθ .

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Chapter 3.4, Problem 40E

Chapter 3.4, Problem 42E

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u, v and w are three coplanar vectors: ⚫ w has a magnitude of 10 and points along the positive x-axis ⚫ v has a magnitude of 3 and makes an angle of 58 degrees to the positive x- axis ⚫ u has a magnitude of 5 and makes an angle of 119 degrees to the positive x- axis ⚫ vector v is located in between u and w a) Draw a diagram of the three vectors placed tail-to-tail at the origin of an x-y plane. b) If possible, find w × (ū+v) Support your answer mathematically or a with a written explanation. c) If possible, find v. (ū⋅w) Support your answer mathematically or a with a written explanation. d) If possible, find u. (vxw) Support your answer mathematically or a with a written explanation. Note: in this question you can work with the vectors in geometric form or convert them to algebraic vectors.

Question 3 (6 points) u, v and w are three coplanar vectors: ⚫ w has a magnitude of 10 and points along the positive x-axis ⚫ v has a magnitude of 3 and makes an angle of 58 degrees to the positive x- axis ⚫ u has a magnitude of 5 and makes an angle of 119 degrees to the positive x- axis ⚫ vector v is located in between u and w a) Draw a diagram of the three vectors placed tail-to-tail at the origin of an x-y plane. b) If possible, find w × (u + v) Support your answer mathematically or a with a written explanation. c) If possible, find v. (ū⋅ w) Support your answer mathematically or a with a written explanation. d) If possible, find u (v × w) Support your answer mathematically or a with a written explanation. Note: in this question you can work with the vectors in geometric form or convert them to algebraic vectors.

Chapter 3 Solutions

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The derivatives H ' ( θ ) and H ' ' ( θ ) for the function, H ( θ ) = θ sin θ .

Solution Summary: The author explains that the given function is a product of two functions thetaandleft.

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The derivatives H'(θ) and H''(θ) for the function, H(θ)=θsinθ .

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