Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Consider the function \(f(x)=\frac{\cos x}{\sin x}\). (a) Where is \(f\) undefined? (b) Where are the zeros of \(f ?\) (c) What is the period of \(f ?\) (d) Sketch the graph of \(f\) on the interval \([0,2 \pi]\).

Step-by-Step Solution

Answer

The function \(f(x)\) is undefined at \(x=n\pi\) where \(n\) is an integer. The zeros of the function are at \(x=(2n+1)\frac{\pi}{2}\), where \(n\) is an integer. The period of the function is \(\pi\). The graph of \(f(x)\) over the interval \([0,2\pi]\) has vertical asymptotes at \(x=n\pi\), is positive between \(0\) and \(\pi\), and negative between \(\pi\) and \(2\pi\).

Step 1: Identify the Undefined Points of the Function

The function will be undefined wherever the denominator \(\sin x\) is equal to zero. The sine function is zero at all integral multiples of \(\pi\), so \(f(x)=\frac{\cos x}{\sin x}\) is undefined at all integral multiples of \(\pi\).

Step 2: Find the Zeros of the Function

The zeros of the function \(f(x)\) are the x-values where \(f(x) = 0\). In this case, \(f(x) = 0\) when \(\cos x = 0\). The cosine function is zero at \(x=(2n+1)\frac{\pi}{2}\) where \(n\) is an integer. So, these are the zeros of the function.

Step 3: Identify the Period of the Function

The cotangent function \(f(x)=\frac{\cos x}{\sin x}\) has the same period as the tangent function, which is \(\pi\). Therefore, the period of this function is \(\pi\).

Step 4: Sketch the Function's Graph

The graph of \(f(x)=\frac{\cos x}{\sin x}\) is the reflection of \(\tan x\) about the line \(y=x\). The function is undefined at \(x=n\pi\) and has zeroes at \(x=(2n+1)\frac{\pi}{2}\). The function is positive in the first and third quadrants (between \(-\pi\) and \(0\), and between \(0\) and \(\pi\)), negative in the second and fourth quadrants (between \(-2\pi\) and \(-\pi\), and between \(\pi\) and \(2\pi\)). Consider these properties while drawing the graph on the interval \([0,2 \pi]\).

Key Concepts

Undefined Points

In trigonometry, undefined points occur when a function involves division by zero. For the function \( f(x) = \frac{\cos x}{\sin x} \), the denominator is \( \sin x \). To find when this function is undefined, determine where \( \sin x = 0 \). Since the sine function is zero at integer multiples of \( \pi \), these are the points where \( f(x) \) is undefined.

Examples of such points include:

• \( x = 0 \)
• \( x = \pi \)
• \( x = 2\pi \)

These values are crucial to consider when analyzing the behavior of trigonometric functions in graphs. They represent vertical asymptotes in the graph. Understanding where a function is undefined helps prevent calculation errors and strengthens comprehension of function limitations.

Zeros of Functions

Zeros of a function are the x-values that make the function equal to zero. For \( f(x) = \frac{\cos x}{\sin x} \), this condition occurs when the numerator \( \cos x \) is zero while the denominator isn't.

To find these zeros, look for where \( \cos x = 0 \). The cosine function equals zero at odd multiples of \( \frac{\pi}{2} \). Specifically, these are:

• \( x = \frac{\pi}{2} \)
• \( x = \frac{3\pi}{2} \)

These points represent where the function crosses the x-axis in its graph.
Understanding zeros helps in identifying function roots and will be valuable when solving equations or graphing.

Period of a Function

The concept of period in trigonometry refers to the distance over which a function repeats itself. The function \( f(x) = \frac{\cos x}{\sin x} \) is essentially the cotangent function, which has the same period as the tangent function.

The period for \( \tan x \) is \( \pi \), meaning that the function repeats its values over every interval of \( \pi \). Therefore, for the cotangent function as well, the period is:

• \( \pi \)

Understanding the period is essential for predicting function behavior over larger intervals. It makes sketching graphs more intuitive and helps in applications involving oscillations and waves.