Find all points (if any) of horizontal and vertical tangency to the curve x = cos2 θ, y = cos θ. Use a graphing utility to confirm your results.

Name: Find all points (if any) of horizontal and vertical tangency to the cu
Uploaded: 2024-03-20T06:46:55.000Z
Channel: Bartleby
Description: Find all points (if any) of horizontal and vertical tangency to the curve x = cos2 θ, y = cos θ. Use a graphing utility to confirm your results.

Answered: Find all points (if any) of horizontal…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Related questions

Concept explainers

Topic Video

Question

Find all points (if any) of horizontal and vertical tangency to the curve x = cos² θ, y = cos θ. Use a graphing utility to confirm your results.

Expert Solution

Step 1

Given:

$x = \cos^{2} (θ) y = \cos (θ)$

To Find:

$Find all points (if any) of horizontal and vertical tangency to the curve.$

Concept:

$(i) . Parametric curve (x (θ), y (θ)) has a horizontal tangent if it's slope \frac{d y}{d x} = 0, i.e. when \frac{d y}{d θ} = 0 and \frac{d x}{d θ} \neq 0 . (i i) . Parametric curve (x (θ), y (θ)) has a vertical tangent if it's slope appraches infinity, i.e. when \frac{d y}{d θ} \neq 0 and \frac{d x}{d θ} = 0 .$

Step 2

Explanation:

$We have: x = \cos^{2} (θ) and y = \cos (θ) Now, differentiating x and y with respect to θ, we get: \frac{d x}{d θ} = - 2 \cos (θ) \sin (θ) and \frac{d y}{d θ} = - \sin (θ) \Rightarrow \frac{d x}{d θ} = - \sin (2 θ) and \frac{d y}{d θ} = - \sin (θ) Now, for horizontal tangent, put \frac{d y}{d θ} = 0 \Rightarrow - \sin (θ) = 0 \Rightarrow θ = n π, \forall n \in ℤ i . e . θ = 0, \pm π, \pm 2 π, \pm 3 π, . . . . . . and for vertical tangent, put \frac{d x}{d θ} = 0 \Rightarrow - \sin (2 θ) = 0 \Rightarrow θ = \frac{n π}{2}, \forall n \in ℤ i . e . θ = 0, \pm \frac{π}{2}, \pm π, \pm \frac{3 π}{2}, . . . . . . .$

Step by step

Solved in 4 steps with 1 images

Knowledge Booster

$Fundamentals of Trigonometric Identities$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Similar questions