Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**Finding the Equation of the Tangent Line to a Graph** To find the equation for the line tangent to the graph of the function \[ f(x) = \frac{2x}{x + 4} \] at the point \((2, 0.66666666666667)\). Below the instruction, there is an empty input box where students can input the equation of the tangent line in the form \( y = \). ### Explanation: 1. **Identify the Function**: The function given is \( f(x) = \frac{2x}{x + 4} \). 2. **Given Point on the Graph**: The point at which the tangent line is to be found is \((2, 0.66666666666667)\). 3. **Objective**: Find the equation of the tangent line to the graph of \( f(x) \) at the given point. ### Steps to Find the Tangent Line Equation: 1. **Find the derivative of the function** \( f(x) \) to determine the slope of the tangent line. 2. **Evaluate the derivative at \( x = 2 \) to find the slope at the given point**. 3. **Use the point-slope form of the equation of a line**: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the point \((2, 0.66666666666667)\) and \( m \) is the slope found in step 2. ### Input Your Answer: Students will input their final equation for the tangent line in the designated box provided. --- This structured explanation helps guide learners through the process of finding the tangent line step-by-step.
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Transcribed Image Text:**Finding the Equation of the Tangent Line to a Graph** To find the equation for the line tangent to the graph of the function \[ f(x) = \frac{2x}{x + 4} \] at the point \((2, 0.66666666666667)\). Below the instruction, there is an empty input box where students can input the equation of the tangent line in the form \( y = \). ### Explanation: 1. **Identify the Function**: The function given is \( f(x) = \frac{2x}{x + 4} \). 2. **Given Point on the Graph**: The point at which the tangent line is to be found is \((2, 0.66666666666667)\). 3. **Objective**: Find the equation of the tangent line to the graph of \( f(x) \) at the given point. ### Steps to Find the Tangent Line Equation: 1. **Find the derivative of the function** \( f(x) \) to determine the slope of the tangent line. 2. **Evaluate the derivative at \( x = 2 \) to find the slope at the given point**. 3. **Use the point-slope form of the equation of a line**: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the point \((2, 0.66666666666667)\) and \( m \) is the slope found in step 2. ### Input Your Answer: Students will input their final equation for the tangent line in the designated box provided. --- This structured explanation helps guide learners through the process of finding the tangent line step-by-step.
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