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:**
Find an equation of the tangent line to the curve at the given point.
\[
y = \frac{1 + \sin(x)}{\cos(x)}
\]
at the point \((\pi, -1)\).
**Explanation:**
To find the equation of the tangent line at a specific point on a curve, we need to follow these steps:
1. **Differentiate the Function:**
Calculate the derivative of the function \(y = \frac{1 + \sin(x)}{\cos(x)}\) to determine the slope of the tangent line at any point.
2. **Evaluate the Derivative at the Given Point:**
Substitute \(x = \pi\) into the derivative to find the slope of the tangent line at \((\pi, -1)\).
3. **Use the Point-Slope Form:**
With the slope from step 2 and the given point, use the point-slope form equation \(y - y_1 = m(x - x_1)\) to write the equation of the tangent line, where \(m\) is the slope and \((x_1, y_1)\) is the given point \((\pi, -1)\).
No graphs or diagrams are provided in this text.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fe49cee-acc8-403b-8d12-20ec3514f583%2Ff9e86930-585d-43c1-ad0c-b325f8ce1a5c%2F0lyslpf_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find an equation of the tangent line to the curve at the given point.
\[
y = \frac{1 + \sin(x)}{\cos(x)}
\]
at the point \((\pi, -1)\).
**Explanation:**
To find the equation of the tangent line at a specific point on a curve, we need to follow these steps:
1. **Differentiate the Function:**
Calculate the derivative of the function \(y = \frac{1 + \sin(x)}{\cos(x)}\) to determine the slope of the tangent line at any point.
2. **Evaluate the Derivative at the Given Point:**
Substitute \(x = \pi\) into the derivative to find the slope of the tangent line at \((\pi, -1)\).
3. **Use the Point-Slope Form:**
With the slope from step 2 and the given point, use the point-slope form equation \(y - y_1 = m(x - x_1)\) to write the equation of the tangent line, where \(m\) is the slope and \((x_1, y_1)\) is the given point \((\pi, -1)\).
No graphs or diagrams are provided in this text.
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