Explanation Given information : P ( r , θ ) be the point on the polar curve r = f ( θ ) , and α be the angle extending from the radial line to the tangent line. Formula used : tan ( x − y ) = tanx − t a n y 1 + tan x tan y Proof: Let ϕ be the angle between the x-axis and the tangent line. We know that ϕ = θ + α ⇒ α = θ − ϕ The tangent line to the polar curve r = f ( θ ) at the point P ( r , θ ) has slope m = f ( θ ) c o s θ + f ' ( θ ) s i n θ − f ( θ ) sin θ + f ' ( θ ) c o s θ , whenever the denominator is zero The relation between the polar coordinates and the Cartesian coordinates is given by x = r c o s θ y = r s i n θ Differentiating (1), with respect to ‘ θ ’, we have d x d θ = f ' ( θ ) c o s θ − f ( θ ) s i n θ d y d θ = f ' ( θ ) sin θ + f ( θ ) cos θ } Since r = f ( θ ) , tan θ = y x = s i n θ cos θ tan ϕ = d y d x = m Now the slope of the tangent line m is given by m = d y d x = d y d θ d x d θ = f ' ( θ ) sin θ + f ( θ ) cos θ f ' ( θ ) c o s θ − f ( θ ) s i n θ Now t a n α = tan ( θ − ϕ ) = tan θ − t a n ϕ 1 + tan θ tan ϕ t a n α = sin θ cos θ − f ' ( θ ) sin θ + f ( θ ) cos θ f

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ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 6.3, Problem 58PS

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To prove: tanα=f(θ)f'(θ) , f'(θ)≠0

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Chapter 6.3, Problem 57PS

Chapter 6.3, Problem 59PS

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Chapter 6 Solutions

Calculus

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To prove : tan α = f ( θ ) f ' ( θ ) , f ' ( θ ) ≠ 0

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To prove: tanα=f(θ)f'(θ) , f'(θ)≠0

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Chapter 6 Solutions