Problem 186 - Calculus Volume 1

Question

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. $$ f(x)=\csc x, x=\frac{\pi}{2} $$

Step-by-Step Solution

Answer

The equation of the tangent line is \( y = 1 \).

Step 1: Understand the Function and Problem

We're given the function \( f(x) = \csc x \) and need to find the equation of its tangent line at \( x = \frac{\pi}{2} \). \( \csc x \) is the reciprocal of \( \sin x \), meaning \( \csc x = \frac{1}{\sin x} \).

Step 2: Find the Value of f(x)

First, substitute \( x = \frac{\pi}{2} \) into the function \( f(x) = \csc x \) to find the point the tangent line goes through. \( \csc \left( \frac{\pi}{2} \right) = \frac{1}{\sin \left( \frac{\pi}{2} \right)} = 1 \). So the point is \( \left( \frac{\pi}{2}, 1 \right) \).

Step 3: Compute the Derivative

To find the slope of the tangent line, compute the derivative of \( f(x) = \csc x \). We know \( \frac{d}{dx}[\csc x] = -\csc x \cdot \cot x \).

Step 4: Evaluate the Derivative at x = π/2

Evaluate the derivative at \( x = \frac{\pi}{2} \). Since \( \csc \left( \frac{\pi}{2} \right) = 1 \) and \( \cot \left( \frac{\pi}{2} \right) = 0 \), the slope of the tangent line is \( -1 \cdot 0 = 0 \).

Step 5: Find the Equation of the Tangent Line

The tangent line is horizontal with a slope of 0, passing through the point \( \left( \frac{\pi}{2}, 1 \right) \). Thus, the equation is \( y = 1 \).

Step 6: Graphical Verification

Using a calculator, graph \( f(x) = \csc x \) and \( y = 1 \) to check if the equation of the tangent line is correct. The tangent line \( y = 1 \) should touch \( \csc x \) at \( x = \frac{\pi}{2} \).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc x \), is the reciprocal of the sine function. In simpler terms, it is defined as \( \csc x = \frac{1}{\sin x} \). This means that wherever the sine function is zero, the cosecant function is undefined, leading it to have vertical asymptotes at these points. For values of \( x \) where the sine function reaches its maxima (1 or -1), the cosecant gets values of 1 or -1, respectively.

Key characteristics of the cosecant function include:

• Vertical asymptotes at integer multiples of \( \pi \).
• A period of \( 2\pi \).
• Always positive or negative infinity directly above and below where sine is zero.

Understanding these properties is crucial as they help in graphing the function and identifying areas where the function behaves significantly different from the sine wave.

Derivative

The derivative is a fundamental tool in calculus that measures the rate at which a function is changing. To find the tangent line to a curve at any given point, first, it is necessary to compute the derivative at that point.

For the cosecant function \( f(x) = \csc x \), the derivative must account for both the reciprocal nature and the trigonometric identity. The derivative of \( \csc x \) is:

• \( \frac{d}{dx}[\csc x] = -\csc x \cdot \cot x \)

This result combines the chain rule and the fact that the derivative of \( \sin x \) itself is \( \cos x \). Utilizing these tools allows us to calculate the slope of the tangent line, which is crucial in defining its equation.

Horizontal Tangent Line

A horizontal tangent line indicates that the slope of the curve at that particular point is zero. When the derivative of a function, evaluated at a specific point, equals zero, it leads to the conclusion that there's no change in the \( y \)-value at that point, thus forming a horizontal line.

For the problem at hand, after differentiating \( \csc x \) and evaluating it at \( x = \frac{\pi}{2} \), we found:

• The derivative value is \( 0 \).
• The tangent line consequently is \( y = 1 \), indicating a constant function through this point.

This means the tangent is not only horizontal but when graphed, should align perfectly with the point \( (\frac{\pi}{2}, 1) \) of the cosecant function.

Graphical Verification

Graphical verification serves as a visual method to ensure that our calculated equation for the tangent line accurately represents the behavior of the curve at the specified point. With technology such as graphing calculators or software, you can plot both the function and its tangent line.

In this case, when we plot \( f(x) = \csc x \) along with the tangent line \( y = 1 \), we expect the tangent line to intersect the curve exactly at \( x = \frac{\pi}{2} \), without diverging away.

• Graphing calculators are useful tools as they can help visualize complex behaviors of trigonometric functions.
• Observing the alignment of the tangent proves the accuracy of the calculation.

Verifying your algebraic results with a graph helps deepen understanding and ensures proficiency in recognizing the practical application of derivative concepts.