Problem 190 - Calculus Volume 1

Question

For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of \(x .\) Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct. $$[\mathbf{T}] f(x)=5 \cot x x=\frac{\pi}{4}$$

Step-by-Step Solution

Answer

The equation of the tangent line is \( y = -10x + \frac{10\pi}{4} + 5 \).

Step 1: Find the Derivative of the Function

To find the equation of the tangent line, we first need to find the derivative of the function, which gives the slope of the tangent line. The function given is \( f(x) = 5 \cot x \). The derivative of \( \cot x \) is \( -\csc^2 x \). Therefore, the derivative of \( f(x) \) is \( f'(x) = 5 \times (-\csc^2 x) = -5 \csc^2 x \).

Step 2: Evaluate the Derivative at x = \( \pi/4 \)

Now we need to find the slope of the tangent line at \( x = \pi/4 \). Substitute \( x = \pi/4 \) into the derivative: \( f'(\pi/4) = -5 \csc^2(\pi/4) \). Since \( \csc(\pi/4) = \sqrt{2} \), \( \csc^2(\pi/4) = 2 \). Thus, \( f'(\pi/4) = -5 \, \times 2 = -10 \).

Step 3: Find the Function Value at x = \( \pi/4 \)

We also need the function value at \( x = \pi/4 \). Substitute \( x = \pi/4 \) into the original function: \( f(\pi/4) = 5 \cot(\pi/4) \). Since \( \cot(\pi/4) = 1 \), \( f(\pi/4) = 5 \times 1 = 5 \).

Step 4: Use the Point-Slope Form to Write the Equation of the Tangent Line

With the slope \( m = -10 \) and the point \((\pi/4, 5)\), we can use the point-slope form of a line: \( y - y_1 = m(x - x_1) \). Substituting the values, we get \( y - 5 = -10(x - \pi/4) \). Simplifying, the equation of the tangent line is \( y = -10x + \frac{10\pi}{4} + 5 \) or \( y = -10x + \frac{10\pi}{4} + 5 \).

Step 5: Graph the Function and Tangent Line

Using a graphing calculator, graph the function \( f(x) = 5 \cot x \) and the tangent line \( y = -10x + \frac{10\pi}{4} + 5 \). Ensure the tangent line touches the curve only at \( x = \pi/4 \).

Key Concepts

Derivative of Trigonometric Functions

In mathematics, the derivative of a function helps to understand how the function behaves as its input value changes. When dealing with trigonometric functions like cotangent, the derivative provides the slope of the tangent line at any given point of the function.
For the function \( f(x) = 5 \cot x \), we first need to find the derivative, which gives us the slope crucial for defining the tangent line.

• The derivative of \( \cot x \) is \( -\csc^2 x \).
• Thus, for \( f(x) = 5 \cot x \), the derivative becomes \( f'(x) = -5 \csc^2 x \).

Understanding this allows us to predict how sharp or flat the tangent line will be at any particular point. Knowing derivatives helps in many applications, from physics to engineering, by providing instantaneous rates of change and slopes of curves.

Point-Slope Form

When you find the slope of the tangent line at a specific point, you can use it to write the equation of the tangent line using the point-slope form. This special form is practical for constructing a line equation when a point and the slope are known.
The point-slope formula can be expressed as:

• \( y - y_1 = m(x - x_1) \)

Where:

• \( m \) is the slope of the line.
• \( (x_1, y_1) \) is the point through which the line passes.

Using the slope \( m = -10 \) and the point \(( \frac{\pi}{4}, 5 )\), the equation of the tangent line becomes:

• \( y - 5 = -10 \left( x - \frac{\pi}{4} \right) \)

This method provides a straightforward way to connect algebra and geometry, allowing us to visualize how equations of lines work.

Graphing

Graphing is a powerful tool for visualizing mathematical concepts. By graphing functions and their tangent lines, you gain insight into how these lines only touch the curve at a point, showing instant rate of change.
For graphing:

• Use a graphing calculator or software to plot \( f(x) = 5 \cot x \).
• Also plot the tangent line \( y = -10x + \frac{10\pi}{4} + 5 \).

By comparing the graphs, you can verify that the tangent line is accurately placed, touching the curve only at \( x = \frac{\pi}{4} \).
Graphing helps bridge the gap between abstract equations and their real-world representations, making it easier to understand complex relationships and behaviors.