4) Find the equation of the tangent line to the function f(x) = 3 at x = 1. Round the slope to one decimal place.

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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**Problem 4: Finding the Equation of a Tangent Line**

Objective: Find the equation of the tangent line to the function \( f(x) = 3^x \) at \( x = 1 \). Round the slope to one decimal place.

To solve this problem, follow these steps:

1. **Find the Derivative**: The first step is to determine the derivative of the function \( f(x) = 3^x \). Use the rule for the derivative of an exponential function, which involves natural logarithms.

2. **Evaluate the Derivative at the Point**: Substitute \( x = 1 \) into the derivative to find the slope of the tangent line at that specific point.

3. **Calculate the Equation of the Tangent Line**:
   - Use the point-slope form of the equation of a line: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is the point on the curve.
   - Substitute the slope and the point \((1, f(1))\) into the equation.

4. **Round the Slope**: Ensure the slope is rounded to one decimal place for precision.

By following these steps, you will find the equation of the tangent line to the function at the specified point.
Transcribed Image Text:**Problem 4: Finding the Equation of a Tangent Line** Objective: Find the equation of the tangent line to the function \( f(x) = 3^x \) at \( x = 1 \). Round the slope to one decimal place. To solve this problem, follow these steps: 1. **Find the Derivative**: The first step is to determine the derivative of the function \( f(x) = 3^x \). Use the rule for the derivative of an exponential function, which involves natural logarithms. 2. **Evaluate the Derivative at the Point**: Substitute \( x = 1 \) into the derivative to find the slope of the tangent line at that specific point. 3. **Calculate the Equation of the Tangent Line**: - Use the point-slope form of the equation of a line: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is the point on the curve. - Substitute the slope and the point \((1, f(1))\) into the equation. 4. **Round the Slope**: Ensure the slope is rounded to one decimal place for precision. By following these steps, you will find the equation of the tangent line to the function at the specified point.
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