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: Finding the Tangent Line**
**Given Function:**
\[ y = \sin(x) \cos(x) \]
**Point of Tangency:**
\[ x = \frac{\pi}{4} \]
---
In this problem, we are tasked with finding the formula for the tangent line to the function \( y = \sin(x) \cos(x) \) at the point where \( x \) is equal to \( \frac{\pi}{4} \).
### Steps to Find the Tangent Line:
1. **Evaluate the Function at \( x = \frac{\pi}{4} \):**
- \[ y\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) \cos\left(\frac{\pi}{4}\right) \]
2. **Compute the Derivative \( y' \):**
- Use the product rule for differentiation if necessary.
3. **Evaluate the Derivative at \( x = \frac{\pi}{4} \):**
- \[ y'\left(\frac{\pi}{4}\right) \]
4. **Construct the Tangent Line Equation:**
- Using the point-slope form of the line: \[ y - y_1 = m(x - x_1) \]
Where \( y_1 \) is the function value at \( x = \frac{\pi}{4} \) and \( m \) is the slope obtained from the derivative evaluated at the same point.
By following these steps, you will find the equation of the tangent line to \( y = \sin(x) \cos(x) \) at \( x = \frac{\pi}{4} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2c26efb-761a-4d98-b451-c4611947b84e%2Ff05a0798-7c1b-4252-b026-804598cb3ee7%2Fq16bpla_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem: Finding the Tangent Line**
**Given Function:**
\[ y = \sin(x) \cos(x) \]
**Point of Tangency:**
\[ x = \frac{\pi}{4} \]
---
In this problem, we are tasked with finding the formula for the tangent line to the function \( y = \sin(x) \cos(x) \) at the point where \( x \) is equal to \( \frac{\pi}{4} \).
### Steps to Find the Tangent Line:
1. **Evaluate the Function at \( x = \frac{\pi}{4} \):**
- \[ y\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) \cos\left(\frac{\pi}{4}\right) \]
2. **Compute the Derivative \( y' \):**
- Use the product rule for differentiation if necessary.
3. **Evaluate the Derivative at \( x = \frac{\pi}{4} \):**
- \[ y'\left(\frac{\pi}{4}\right) \]
4. **Construct the Tangent Line Equation:**
- Using the point-slope form of the line: \[ y - y_1 = m(x - x_1) \]
Where \( y_1 \) is the function value at \( x = \frac{\pi}{4} \) and \( m \) is the slope obtained from the derivative evaluated at the same point.
By following these steps, you will find the equation of the tangent line to \( y = \sin(x) \cos(x) \) at \( x = \frac{\pi}{4} \).
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