3. (a) („ Consider the curve defined by ry = In(3r +sin(y)). Use implicit differenti-ation to find Solve for but DO NOT simplify the final answer.(b) (кShow that the point (1/3.0) is on the curve in part (a).(c)Find the slope of the line tangent to the curve at the point (1/3,0).

Name: 3. (a) („ Consider the curve defined by ry = In(3r +sin(y)). Use impl
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Description: 3. (a) („ Consider the curve defined by ry = In(3r +sin(y)). Use implicit differenti-ation to find Solve for but DO NOT simplify the final answer.(b) (кShow that the point (1/3.0) is on the curve in part (a).(c)Find the slope of the line tangent to

Answered: (a) (., Consider the curve defined by…

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...

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3. (a) („ Consider the curve defined by ry = In(3r +sin(y)). Use implicit differenti-
ation to find Solve for but DO NOT simplify the final answer.
(b) (к
Show that the point (1/3.0) is on the curve in part (a).
(c)
Find the slope of the line tangent to the curve at the point (1/3,0).

Expert Solution

Step 1

Consider the given equation of the curve,

$x y = \ln (3 x + \sin y)$

Use the implicit differentiation,

$\begin{matrix} \frac{d}{d x} [x y] = \frac{d}{d x} [\ln (3 x + \sin y)] \\ x \frac{d y}{d x} + y (1) = \frac{1}{3 x + \sin y} \frac{d}{d x} [3 x + \sin y] \\ x \frac{d y}{d x} + y = \frac{1}{3 x + \sin y} (3 + \cos y \frac{d y}{d x}) \\ x \frac{d y}{d x} - \frac{\cos y}{3 x + \sin y} \frac{d y}{d x} = \frac{3}{3 x + \sin y} - y \\ (x - \frac{\cos y}{3 x + \sin y}) \frac{d y}{d x} = \frac{3}{3 x + \sin y} - y \\ \frac{d y}{d x} = \frac{\frac{3}{3 x + \sin y} - y}{x - \frac{\cos y}{3 x + \sin y}} \\ \frac{d y}{d x} = \frac{3 - 3 x y - y \sin y}{3 x^{2} + x \sin y - \cos y} \end{matrix}$

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