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Tangents and normals Let a polar curve be described by r =

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

Problem 50AE Chapter 10.3

Calculus: Early Transcendentals | 1st Edition

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Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

Calculus: Early Transcendentals | 1st Edition

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Problem 50AE

Tangents and normals Let a polar curve be described by r = f(θ) and let ℓ be the line tangent to the curve at the point P(x, y) = P(r, θ) (see figure).

a. Explain why tan α = dy/dx.

b. Explain why tan θ = y/x.

c. Let φ be the angle between ℓ and OP. Prove that tan φ = f(θ)/f′(θ).

d. Prove that the values of θ for which ℓ is parallel to the x-axis satisfy tan θ = −f(θ)/f′(θ).

e. Prove that the values of θ for which ℓ is parallel to the y-axis satisfy tan θ = f′(θ)/f(θ).

Step-by-Step Solution:
Step 1 of 3

Solution 50AE

a. Here α is an angle made by the line with positive x -axis

    Therefore tan α is the slope of the tangent line

    But we know that slope of the tangent to the curve r = f (is

    tan=

b. In the above figure distance from the point to x - axis is y units

    And the distance from the point to x-axis is x units

    From the right angle triangle

     tan=

c. ...

Step 2 of 3

Chapter 10.3, Problem 50AE is Solved
Step 3 of 3

Textbook: Calculus: Early Transcendentals
Edition: 1
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
ISBN: 9780321570567

The answer to “Tangents and normals Let a polar curve be described by r = f(?) and let ? be the line tangent to the curve at the point P(x, y) = P(r, ?) (see figure).a. Explain why tan ? = dy/dx.b. Explain why tan ? = y/x.c. Let ? be the angle between ? and OP. Prove that tan ? = f(?)/f?(?).d. Prove that the values of ? for which ? is parallel to the x-axis satisfy tan ? = ?f(?)/f?(?).e. Prove that the values of ? for which ? is parallel to the y-axis satisfy tan ? = f?(?)/f(?).” is broken down into a number of easy to follow steps, and 98 words. The full step-by-step solution to problem: 50AE from chapter: 10.3 was answered by , our top Calculus solution expert on 03/03/17, 03:45PM. Since the solution to 50AE from 10.3 chapter was answered, more than 238 students have viewed the full step-by-step answer. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321570567. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. This full solution covers the following key subjects: tan, let, prove, explain, curve. This expansive textbook survival guide covers 85 chapters, and 5218 solutions.

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