Book edition 7th Edition

Author(s) James Stewart, Lothar Redlin, Saleem Watson

Pages 948 pages

ISBN 9781337067508

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Short Answer

Find the equation of the tangent line to the curve at the given point. Graph the curve and tangent line.
. $y = \frac{x}{x - 1}, a t (2, 2)$

The equation of the tangent line to the curve at a given point is $y = - x + 4$ .

The graph of the curve and tangent line:

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

The function here given is,

$\begin{array}{l} y = \frac{x}{x - 1} \\ p o int = (2, 2) \end{array}$

Step 2. Formula used.

The tangent line to the curve $y = f (x)$ at the point $P (a, f (a))$ is the line through p with slope,

$m = \lim_{x \to a} \frac{f (x) - f (a)}{x - a}$ .

provided that this limit exists.

Equation of tangent line:

$y = m (x - a) + f (a)$

Step 3. Finding the slope of tangent line at given point.

Let $y = \frac{x}{x - 1}$ , then the slope of the tangent line at $(2, 2)$ is,

$\begin{matrix} m = \lim_{x \to 2} \frac{f (x) - f (2)}{x - 2} \\ = \lim_{x \to 2} \frac{(\frac{x}{x - 1}) - (\frac{2}{2 - 1})}{x - 2} \\ = \lim_{x \to 2} \frac{\frac{x}{x - 1} - \frac{2}{1} \cdot \frac{x - 1}{x - 1}}{x - 2} \\ = \lim_{x \to 2} \frac{\frac{x - 2 x + 2}{x - 1}}{x - 2} \\ = \lim_{x \to 2} \frac{- (x - 2)}{(x - 2) (x - 1)} \\ = - \frac{1}{2 - 1} \\ = - 1 \end{matrix}$

The slope of the tangent line is -1.

Step 4. Finding the equation of the tangent line at a given point.

Here, the slope of the tangent line is $m = - 1$ . Equation of the tangent line is written as,

$\begin{array}{l} y = - 1 (x - 2) + 2 \\ y = - x + 2 + 2 \\ y = - x + 4 \end{array}$ ,

Therefore, the equation of the tangent line is $y = - x + 4$ .

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Precalculus Mathematics for Calculus

Answers without the blur.

Short Answer

Find the equation of the tangent line to the curve at the given point. Graph the curve and tangent line.
. $y = \frac{x}{x - 1}, a t (2, 2)$

Step by Step Solution

Step 1. Given information.

Step 2. Formula used.

Step 3. Finding the slope of tangent line at given point.

Step 4. Finding the equation of the tangent line at a given point.

Step 5. Graphing the curve and the tangent line.

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Precalculus Mathematics for Calculus

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Short Answer

Find the equation of the tangent line to the curve at the given point. Graph the curve and tangent line..y=xx−1, at (2,2)

Step by Step Solution

Step 1. Given information.

Step 2. Formula used.

Step 3. Finding the slope of tangent line at given point.

Step 4. Finding the equation of the tangent line at a given point.

Step 5. Graphing the curve and the tangent line.

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Find the equation of the tangent line to the curve at the given point. Graph the curve and tangent line.
. $y = \frac{x}{x - 1}, a t (2, 2)$