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Find the equation of the tangent line to the curvef (x) = x- + 2x at x = 1. Enter the answer in the box.

Find the equation of the tangent line to the curvef (x) = x- + 2x at x = 1. Enter the answer in the box.

Calculus: Early Transcendentals
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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**Instructions:** Use the graph of \( f'(x) \) shown to identify a possible graph of \( f(x) \). **Graphs Explanation:** - **Top Graph (f'(x)):** - The graph depicted is of the derivative \( f'(x) \) shown in red. - The graph appears to increase steeply and then levels off as \( x \) increases beyond a certain point. - It shows distinct behavior changes particularly around the origin. **Possible Graphs for f(x):** 1. **Bottom Left (Green Graph):** - Exhibits a decrease until around \( x = -3 \), then steadily increases. - Negligible variation around the origin suggests local extremum behavior. 2. **Bottom Middle (Yellow Graph):** - Displays an initial decrease, transitioning to a gradual increase as \( x \) becomes positive. - Features a more gradual slope transition compared to the first option. 3. **Bottom Right (Purple Graph):** - Starts with an increase, steeply rises past the origin, and continues to increase as \( x \) increases. - Represents a consistent positive slope behavior across the range. 4. **Top Right (Blue Graph):** - Generally exhibits a downward concave parabola-like shape. - Shows reciprocal behavior to the derivative’s decreasing trend to zero. **Considerations:** - Analyze the zero crossings and slope changes in \( f'(x) \) to correspond these with local minimums and maximums in \( f(x) \). - The relation between areas where \( f'(x) \) is positive or negative, indicating where \( f(x) \) is increasing or decreasing, will aid in identifying the correct graph.
Transcribed Image Text:**Instructions:** Use the graph of \( f'(x) \) shown to identify a possible graph of \( f(x) \). **Graphs Explanation:** - **Top Graph (f'(x)):** - The graph depicted is of the derivative \( f'(x) \) shown in red. - The graph appears to increase steeply and then levels off as \( x \) increases beyond a certain point. - It shows distinct behavior changes particularly around the origin. **Possible Graphs for f(x):** 1. **Bottom Left (Green Graph):** - Exhibits a decrease until around \( x = -3 \), then steadily increases. - Negligible variation around the origin suggests local extremum behavior. 2. **Bottom Middle (Yellow Graph):** - Displays an initial decrease, transitioning to a gradual increase as \( x \) becomes positive. - Features a more gradual slope transition compared to the first option. 3. **Bottom Right (Purple Graph):** - Starts with an increase, steeply rises past the origin, and continues to increase as \( x \) increases. - Represents a consistent positive slope behavior across the range. 4. **Top Right (Blue Graph):** - Generally exhibits a downward concave parabola-like shape. - Shows reciprocal behavior to the derivative’s decreasing trend to zero. **Considerations:** - Analyze the zero crossings and slope changes in \( f'(x) \) to correspond these with local minimums and maximums in \( f(x) \). - The relation between areas where \( f'(x) \) is positive or negative, indicating where \( f(x) \) is increasing or decreasing, will aid in identifying the correct graph.
**Problem:** Find the equation of the tangent line to the curve \( f(x) = x^2 + 2x \) at \( x = 1 \). Enter the answer in the box. \[ T(x) = \square \] **Solution Approach:** To find the equation of the tangent line, follow these steps: 1. **Differentiate the function**: Find the derivative \( f'(x) \) to determine the slope of the tangent line at any point \( x \). 2. **Evaluate the derivative at \( x = 1 \)**: This will give the slope of the tangent line at the specific point \( x = 1 \). 3. **Calculate the y-coordinate of the point**: Substitute \( x = 1 \) into the original function to find \( f(1) \). 4. **Use point-slope form to find the equation**: The tangent line's equation can be written in the form \( y - f(a) = f'(a)(x - a) \), where \( a = 1 \) in this case. Following these steps will yield the equation for \( T(x) \).
Transcribed Image Text:**Problem:** Find the equation of the tangent line to the curve \( f(x) = x^2 + 2x \) at \( x = 1 \). Enter the answer in the box. \[ T(x) = \square \] **Solution Approach:** To find the equation of the tangent line, follow these steps: 1. **Differentiate the function**: Find the derivative \( f'(x) \) to determine the slope of the tangent line at any point \( x \). 2. **Evaluate the derivative at \( x = 1 \)**: This will give the slope of the tangent line at the specific point \( x = 1 \). 3. **Calculate the y-coordinate of the point**: Substitute \( x = 1 \) into the original function to find \( f(1) \). 4. **Use point-slope form to find the equation**: The tangent line's equation can be written in the form \( y - f(a) = f'(a)(x - a) \), where \( a = 1 \) in this case. Following these steps will yield the equation for \( T(x) \).
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