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 Description:**
Find the equation of the tangent line to the curve \( y = \tan^{-1} x \) at the point \( \left(1, \frac{\pi}{4}\right) \).
**Solution Steps:**
1. **Identify the Function and the Point:**
- Function: \( y = \tan^{-1} x \)
- Point: \( \left(1, \frac{\pi}{4}\right) \)
2. **Find the Derivative:**
- The derivative of \( y = \tan^{-1} x \) is \( y' = \frac{1}{1+x^2} \).
3. **Evaluate the Derivative at \( x = 1 \):**
- Substitute \( x = 1 \) into the derivative:
\[
y'(1) = \frac{1}{1+1^2} = \frac{1}{2}
\]
4. **Use the Point-Slope Form:**
- Point-Slope Form: \( y - y_1 = m(x - x_1) \)
- Here, \( (x_1, y_1) = \left(1, \frac{\pi}{4}\right) \) and \( m = \frac{1}{2} \).
5. **Write the Equation of the Tangent Line:**
- Substitute the point and slope into the point-slope form:
\[
y - \frac{\pi}{4} = \frac{1}{2}(x - 1)
\]
- Simplify if needed:
This yields the equation of the tangent line to the curve at the given point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdeff5ce5-c161-409f-a0ee-bdeaef8b0965%2F3e500982-3235-408e-bfed-386305966e9d%2Fz916_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
Find the equation of the tangent line to the curve \( y = \tan^{-1} x \) at the point \( \left(1, \frac{\pi}{4}\right) \).
**Solution Steps:**
1. **Identify the Function and the Point:**
- Function: \( y = \tan^{-1} x \)
- Point: \( \left(1, \frac{\pi}{4}\right) \)
2. **Find the Derivative:**
- The derivative of \( y = \tan^{-1} x \) is \( y' = \frac{1}{1+x^2} \).
3. **Evaluate the Derivative at \( x = 1 \):**
- Substitute \( x = 1 \) into the derivative:
\[
y'(1) = \frac{1}{1+1^2} = \frac{1}{2}
\]
4. **Use the Point-Slope Form:**
- Point-Slope Form: \( y - y_1 = m(x - x_1) \)
- Here, \( (x_1, y_1) = \left(1, \frac{\pi}{4}\right) \) and \( m = \frac{1}{2} \).
5. **Write the Equation of the Tangent Line:**
- Substitute the point and slope into the point-slope form:
\[
y - \frac{\pi}{4} = \frac{1}{2}(x - 1)
\]
- Simplify if needed:
This yields the equation of the tangent line to the curve at the given point.
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