Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Problem 847

curve \(\mathrm{y}=(3 / 2) \sin 2 \theta, \mathrm{x}=\mathrm{e}^{\theta} \cdot \sin \theta, 0<2\) for which value of \(\theta\) tangent is parallel to X-axis? (a) 0 (b) \((\pi / 2)\) (c) \((\pi / 4)\) (d) \((\pi / 6)\)

Short Answer

Expert verified
The value of θ for which the tangent is parallel to the X-axis is \(\frac{\pi}{4}\), which is option (c).

Step by step solution

01

1. Compute derivatives for x and y

First, let's differentiate x and y with respect to θ: Given: \(y = \frac{3}{2} \sin{2\theta}\) and \(x = e^\theta \sin\theta\) Derivative of y with respect to θ: \(\frac{dy}{d\theta} = \frac{3}{2} (2\cos2\theta)\) Derivative of x with respect to θ: \(\frac{dx}{d\theta} = e^\theta\sin{\theta} + e^\theta\cos{\theta}\)
02

2. Compute dy/dx

Now, let's compute dy/dx: \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\) \(\frac{dy}{dx} = \frac{3\cos{2\theta}}{2(e^\theta\sin{\theta} + e^\theta\cos{\theta})}\)
03

3. Set dy/dx to 0 and solve for θ

Since the tangent is parallel to the X-axis, the slope of the tangent should be equal to 0. Hence, we have to set dy/dx equal to 0 and solve for θ: \(\frac{3\cos{2\theta}}{2(e^\theta\sin{\theta} + e^\theta\cos{\theta})} = 0\) To satisfy the equation, the numerator must be equal to 0. Therefore: \(3\cos{2\theta} = 0\)
04

4. Solve for θ and check the given options

Now let's solve for θ and determine which of the given options satisfies the equation: \(\cos{2\theta} = 0\) \(2\theta = \frac{\pi}{2} + n\pi\) where n is an integer. \(\theta = \frac{\pi}{4} + \frac{n\pi}{2}\) Now let's check which of the given options satisfy the above equation. (a) 0 - No (b) \(\frac{\pi}{2}\) - No (c) \(\frac{\pi}{4}\) - Yes (d) \(\frac{\pi}{6}\) - No Hence, the value of θ for which the tangent is parallel to the X-axis is \(\boxed{\frac{\pi}{4}}\), which is option (c).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If \(\mathrm{A}>0, \mathrm{~B}>0\) and \(\mathrm{A}+\mathrm{B}=(\pi / 3)\) than the maximum value of \(\tan \mathrm{A} \cdot \tan \mathrm{B}\) is (a) \((3 / 2)\) (b) \((1 / 3)\) (c) \((2 / 3)\) (d) 3

\(\mathrm{f}(\mathrm{x})=\mathrm{x} \cdot \sin (1 / \mathrm{x})\) and \(\mathrm{x} \in[-1,1]\). Also \(\mathrm{f}(0)=0\) then. (a) \(\mathrm{f}(\mathrm{x})\) is continuous in \([-1,1]\) (b) Roll's theorem is applicable in \([-1,1]\) (c) First mean value theorem is applicable in \([-1,1]\) (d) none of these

If \(\mathrm{y}=(1 / 3) \log \left[\\{\mathrm{x}+1\\} /\left\\{\mathrm{V}\left(\mathrm{x}^{2}+\mathrm{x}+1\right)\right\\}\right]+(1 / \sqrt{3}) \tan ^{-1}\) \([(2 x-1) /(\sqrt{3})]\) then \((d y / d x)=\) (a) \(\left[1 /\left(1+\mathrm{x}^{3}\right)\right]\) (b) \(\left[\left(x^{2}+x+1\right) /(x-1)\right]\) (c) \(\left[1 /\left(1-\mathrm{x}^{3}\right)\right]\) (d) none of these

Equation of the tangent of the curvey \(\mathrm{y}=1-\mathrm{e}^{(\mathrm{X} / 2)}\) when intersect to \(\mathrm{y}\) -axis than \(=\) (a) \(x+y=0\) (b) \(x+2 y=0\) (c) \(2 \mathrm{x}+\mathrm{y}=0\) (d) \(x-y=0\)

\((\mathrm{d} / \mathrm{d} \mathrm{x})\left[\log (1+\sin \mathrm{x})+\log (\sec \\{(\pi / 4)-(\mathrm{x} / 2)\\})^{2}\right]=\) (a) 0 (b) \(4 \mid[(\cos x-\tan x) /(\sin x+\cos x)]\) (c) \(\log _{\mathrm{e}} 2\) (d) \(-\log _{\mathrm{e}} 2\)
See all solutions

Recommended explanations on English Textbooks

Text Comparison

Read Explanation

English Grammar

Read Explanation

English Language Study

Read Explanation

Global English

Read Explanation

Language Acquisition

Read Explanation

Email

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free