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Find an equation of the tangent line to the curve at the given point. y = (x - 1)/(x - 2), (3, 2)

Find an equation of the tangent line to the curve at the given point. y = (x - 1)/(x - 2), (3, 2)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 41E
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2.3 Q5) Hey, I need help with the following calc problem. Thank you!

# Finding the Equation of a Tangent Line to a Curve ## Problem Statement Find an equation of the tangent line to the curve at the given point. \[ y = \frac{(x - 1)}{(x - 2)} \quad \text{at} \quad (3, 2) \] Below the text, a blank rectangle is shown, indicating where additional explanations, calculations, or graphical representations can be added. --- ### Solution Steps 1. **Identify the given function and point:** - Function: \( y = \frac{(x - 1)}{(x - 2)} \) - Point: \( (3, 2) \) 2. **Find the derivative \( y' \) of the function to get the slope of the tangent line at the given point.** 3. **Evaluate the derivative at \( x = 3 \) to find the slope of the tangent line.** 4. **Use the point-slope form of the equation of a line:** - Point-slope form: \( y - y_1 = m(x - x_1) \) - Where \( (x_1, y_1) \) is the given point and \( m \) is the slope. 5. **Substitute the values into the point-slope form to find the equation of the tangent line.** --- This page provides a step-by-step approach to solve for the equation of the tangent line to the given curve at a specific point, including the necessary differentiation and substitution steps.
Transcribed Image Text:# Finding the Equation of a Tangent Line to a Curve ## Problem Statement Find an equation of the tangent line to the curve at the given point. \[ y = \frac{(x - 1)}{(x - 2)} \quad \text{at} \quad (3, 2) \] Below the text, a blank rectangle is shown, indicating where additional explanations, calculations, or graphical representations can be added. --- ### Solution Steps 1. **Identify the given function and point:** - Function: \( y = \frac{(x - 1)}{(x - 2)} \) - Point: \( (3, 2) \) 2. **Find the derivative \( y' \) of the function to get the slope of the tangent line at the given point.** 3. **Evaluate the derivative at \( x = 3 \) to find the slope of the tangent line.** 4. **Use the point-slope form of the equation of a line:** - Point-slope form: \( y - y_1 = m(x - x_1) \) - Where \( (x_1, y_1) \) is the given point and \( m \) is the slope. 5. **Substitute the values into the point-slope form to find the equation of the tangent line.** --- This page provides a step-by-step approach to solve for the equation of the tangent line to the given curve at a specific point, including the necessary differentiation and substitution steps.
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