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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Question
![**Find an equation for the tangent to the curve at P.**
**Equation of the Curve:**
\[ y = 1 + \sec x \]
**Point P on the Curve:**
\[ P \left( \frac{\pi}{4}, 1 + \sqrt{2} \right) \]
**Graph Explanation:**
- The graph shows the curve of \( y = 1 + \sec x \).
- The x-axis ranges from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).
- The y-axis is labeled with integers from -2 to 6.
- The curve appears to have a vertical asymptote at \( x = -\frac{\pi}{2} \) and \( x = \frac{\pi}{2} \).
- The point \( P \left( \frac{\pi}{4}, 1 + \sqrt{2} \right) \) is marked on the curve.
**Possible Tangent Equations:**
1. \[ y = \sqrt{2}x - \frac{\pi \sqrt{2}}{4} + 1 + \sqrt{2} \]
2. \[ y = -\sqrt{2}x + \frac{\pi \sqrt{2}}{4} - 1 - \sqrt{2} \]
3. \[ y = -\sqrt{2}x - 1 - \sqrt{2} \]
4. \[ y = \sqrt{2}x + 1 + \sqrt{2} \]
**Task:** Determine which equation is the correct one for the tangent to the curve at the given point \( P \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2fab4e6d-dfba-4263-8692-88b0f2add278%2F01def11d-864b-4b2f-82f4-563fde3ea611%2Ffhmp5g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Find an equation for the tangent to the curve at P.**
**Equation of the Curve:**
\[ y = 1 + \sec x \]
**Point P on the Curve:**
\[ P \left( \frac{\pi}{4}, 1 + \sqrt{2} \right) \]
**Graph Explanation:**
- The graph shows the curve of \( y = 1 + \sec x \).
- The x-axis ranges from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).
- The y-axis is labeled with integers from -2 to 6.
- The curve appears to have a vertical asymptote at \( x = -\frac{\pi}{2} \) and \( x = \frac{\pi}{2} \).
- The point \( P \left( \frac{\pi}{4}, 1 + \sqrt{2} \right) \) is marked on the curve.
**Possible Tangent Equations:**
1. \[ y = \sqrt{2}x - \frac{\pi \sqrt{2}}{4} + 1 + \sqrt{2} \]
2. \[ y = -\sqrt{2}x + \frac{\pi \sqrt{2}}{4} - 1 - \sqrt{2} \]
3. \[ y = -\sqrt{2}x - 1 - \sqrt{2} \]
4. \[ y = \sqrt{2}x + 1 + \sqrt{2} \]
**Task:** Determine which equation is the correct one for the tangent to the curve at the given point \( P \).
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