find the slope of the normal line drawn to the graph of each function at the indicated value of x. h(x)=sin x cos x when x= π/4. 

To find the slope of the normal line to the graph of \( h(x) = \sin(x) \cos(x) \) at \( x = \frac{\pi}{4} \), follow these steps:

### 1. Find the derivative of the function \( h(x) \).
The derivative of \( h(x) \) will give us the slope of the tangent line at a given \( x \). We use the product rule to differentiate \( h(x) = \sin(x) \cos(x) \).

Using the product rule:

\[
h'(x) = \cos(x) \cdot \frac{d}{dx}[\sin(x)] + \sin(x) \cdot \frac{d}{dx}[\cos(x)]
\]

This simplifies to:

\[
h'(x) = \cos(x) \cdot \cos(x) + \sin(x) \cdot (-\sin(x))
\]
\[
h'(x) = \cos^2(x) - \sin^2(x)
\]

### 2. Evaluate the derivative at \( x = \frac{\pi}{4} \).
We substitute \( x = \frac{\pi}{4} \) into the derivative:

\[
h' \left( \frac{\pi}{4} \right) = \cos^2\left(\frac{\pi}{4}\right) - \sin^2\left(\frac{\pi}{4}\right)
\]

Since \( \cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \), we get:

\[
h' \left( \frac{\pi}{4} \right) = \left( \frac{1}{\sqrt{2}} \right)^2 - \left( \frac{1}{\sqrt{2}} \right)^2 = \frac{1}{2} - \frac{1}{2} = 0
\]

### 3. Find the slope of the normal line.
The slope of the normal line is the negative reciprocal of the slope of the tangent line. Since the slope of the tangent line at \( x = \frac{\pi}{4} \) is 0, the normal line is vertical. A vertical line has an undefined slope.

### Final Answer:
The slope of the normal line at \( x = \frac{\pi}{4} \) is **undefined** (since the tangent line is horizontal).

Would you like to explore more details or have any other questions?

### Related Questions:
1. How do we apply the product rule in general for differentiating functions?
2. What is the geometric meaning of a normal line?
3. How would the process change if the slope of the tangent line were not zero?
4. How can we verify that the normal line is vertical when the tangent line is horizontal?
5. What are some examples of other trigonometric functions with horizontal tangents?

### Tip:
Remember that if the tangent line's slope is zero, the normal line will be vertical, leading to an undefined slope.

Find the slope of the normal line drawn to the graph of each function at the indicated value of x. h(x) = sin(x) cos(x) when x = π/4.

Explore how to find the slope of the normal line to the function h(x) = sin(x)cos(x) at x = π/4. This solution uses calculus concepts such as the product rule and trigonometric identities. Learn why the normal line has an undefined slope at this point.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation