Find the points on the graph of y = tan x, 0sxs2n, where the tangent line is parallel to the line 4x - y = 7.
Find the points on the graph of y = tan x, 0sxs2n, where the tangent line is parallel to the line 4x - y = 7.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Points where the tangent line is parallel?
![**Problem Statement:**
Find the points on the graph of \( y = \tan x \), for \( 0 \leq x \leq 2\pi \), where the tangent line is parallel to the line \( 4x - y = 7 \).
**Solution Strategy:**
To solve this problem, we need to find where the derivative of \( y = \tan x \) equals the slope of the line \( 4x - y = 7 \).
1. **Identify the Slope**: The given line \( 4x - y = 7 \) can be rewritten in slope-intercept form as \( y = 4x - 7 \). Hence, the slope is \( 4 \).
2. **Find the Derivative**: The derivative of \( y = \tan x \) is \( y' = \sec^2 x \).
3. **Set the Derivative Equal to the Slope**: Solve \( \sec^2 x = 4 \).
- **Solve for \( x \):**
\[
\sec^2 x = 4 \implies \cos^2 x = \frac{1}{4} \implies \cos x = \pm \frac{1}{2}
\]
- **Find Solutions in the Interval \( 0 \leq x \leq 2\pi \):**
\[
\cos x = \frac{1}{2} \quad \Rightarrow \quad x = \frac{\pi}{3}, \frac{5\pi}{3}
\]
\[
\cos x = -\frac{1}{2} \quad \Rightarrow \quad x = \frac{2\pi}{3}, \frac{4\pi}{3}
\]
4. **Find Corresponding \( y \) Values**: Calculate \( y = \tan x \) for the above values of \( x \).
- \( x = \frac{\pi}{3} \): \( y = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \)
- \( x = \frac{5\pi}{3} \): \( y = \tan\left(\frac{5\pi}{3}\right) = -\sqrt{3} \)
- \( x = \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7616f65-0250-4f00-8be8-7f3ea815d241%2F007c9d53-ba58-46e7-b33f-dfc3ea7a34d5%2Fmi3hbyk_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the points on the graph of \( y = \tan x \), for \( 0 \leq x \leq 2\pi \), where the tangent line is parallel to the line \( 4x - y = 7 \).
**Solution Strategy:**
To solve this problem, we need to find where the derivative of \( y = \tan x \) equals the slope of the line \( 4x - y = 7 \).
1. **Identify the Slope**: The given line \( 4x - y = 7 \) can be rewritten in slope-intercept form as \( y = 4x - 7 \). Hence, the slope is \( 4 \).
2. **Find the Derivative**: The derivative of \( y = \tan x \) is \( y' = \sec^2 x \).
3. **Set the Derivative Equal to the Slope**: Solve \( \sec^2 x = 4 \).
- **Solve for \( x \):**
\[
\sec^2 x = 4 \implies \cos^2 x = \frac{1}{4} \implies \cos x = \pm \frac{1}{2}
\]
- **Find Solutions in the Interval \( 0 \leq x \leq 2\pi \):**
\[
\cos x = \frac{1}{2} \quad \Rightarrow \quad x = \frac{\pi}{3}, \frac{5\pi}{3}
\]
\[
\cos x = -\frac{1}{2} \quad \Rightarrow \quad x = \frac{2\pi}{3}, \frac{4\pi}{3}
\]
4. **Find Corresponding \( y \) Values**: Calculate \( y = \tan x \) for the above values of \( x \).
- \( x = \frac{\pi}{3} \): \( y = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \)
- \( x = \frac{5\pi}{3} \): \( y = \tan\left(\frac{5\pi}{3}\right) = -\sqrt{3} \)
- \( x = \
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