https://file.aiquickdraw.com/mathbot/file/1727058089405xcg5pq54.jpg

I have opened the image. Let's proceed with part (a) of the problem.

### Given:
- The function is \( y = 8 + \cot x - 2 \csc x \).
- We need to find the equation of the tangent line to the curve at point \( P \left( \frac{\pi}{2}, 6 \right) \).

### Steps:
1. **Find the derivative** \( \frac{dy}{dx} \) of the function to get the slope of the tangent line.
2. **Evaluate the derivative** at \( x = \frac{\pi}{2} \) to get the slope at \( P \).
3. **Use the point-slope form** of the equation of a line: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) = \left( \frac{\pi}{2}, 6 \right) \).

Let's first compute the derivative and the slope at \( P \).The derivative of the function \( y = 8 + \cot x - 2 \csc x \) is:

\[
\frac{dy}{dx} = -\cot^2(x) + 2 \cot(x) \csc(x) - 1
\]

Evaluating this at \( x = \frac{\pi}{2} \), the slope of the tangent line at point \( P \left( \frac{\pi}{2}, 6 \right) \) is \( m = -1 \).

Now, using the point-slope form of the equation of a line:

\[
y - 6 = -1 \left( x - \frac{\pi}{2} \right)
\]

Simplifying, the equation of the tangent line at \( P \) is:

\[
y = -x + \frac{\pi}{2} + 6
\]

Or:

\[
y = -x + 6 + \frac{\pi}{2}
\] 

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. What is the equation of the tangent line at a different point on the same curve?
2. How does the derivative behave as \( x \to 0 \) for this function?
3. What is the significance of horizontal tangent lines in a graph?
4. How do you find the second derivative of the given function?
5. What is the equation of the normal line at point \( P \)?

**Tip**: Understanding the behavior of the derivative can help you analyze the curve's increasing or decreasing nature.

Find an equation for (a) the tangent line to the curve at P (π/2, 6) and (b) the horizontal tangent line to the curve at Q.

Learn how to find the equation for the tangent line at point P (π/2, 6) on the curve y = 8 + cot(x) - 2csc(x), and determine the horizontal tangent line at point Q using derivatives and trigonometric identities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation