Problem 20 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. Indicate any intercepts and symmetry, and determine whether the function is even, odd, or neither. $$ |\cot x| $$

Step-by-Step Solution

Answer

The function \(|\cot x|\) is even, non-negative, has no intercepts, and is undefined at \(x=k\pi\) for integers \(k\).

Step 1: Understanding the Function

The function given is \(|\cot x|\), which represents the absolute value of the cotangent function. The cotangent function, \(\cot x\), is defined as the reciprocal of the tangent function, \(\cot x = \frac{\cos x}{\sin x}\). It has an undefined value at integer multiples of \(\pi\).

Step 2: Finding Intercepts

The \(y\)-intercept occurs where \(x=0\). However, \(\cot x\) is undefined at \(x = 0\), so there is no \(y\)-intercept. The \(x\)-intercepts occur when \(\cot x = 0\), which is not possible since \(|\cot x|\) is always non-negative and defined outside of \(x = k\pi\), where \(k\) is an integer.

Step 3: Determining Symmetry

Substitute \(-x\) into the function to check symmetry: \(|\cot(-x)| = |\frac{\cos(-x)}{\sin(-x)}| = |\frac{\cos x}{-\sin x}| = |-\cot x| = |\cot x|\), confirming the function is even, as it satisfies \(f(-x) = f(x)\).

Step 4: Sketching the Graph

The graph of \(|\cot x|\) resembles that of \(\cot x\), except all values are non-negative (absolute value taken). It repeats every \(\pi\) units (periodicity). It spikes towards infinity as \(x\) approaches \(k\pi\) from both left and right, with undefined points at each \(x=k\pi\).

Step 5: Summary of Characteristics

- The function \(|\cot x|\) is periodic with period \(\pi\).- It spikes as \(x\) approaches integer multiples of \(\pi\) (undefined there).- The function is even due to its symmetry about the \(y\)-axis.

Key Concepts

Absolute Value

The absolute value function is a mathematical concept that helps in transforming any real number to its non-negative counterpart. It is symbolized by vertical bars around the number or expression. For example, the absolute value of \( a \) is written as \( |a| \). The main feature of absolute values is that \( |a| \) represents the distance between the number \( a \) and zero on a number line.

• For any positive number, the absolute value is the number itself.
• For zero, the absolute value remains zero.
• For any negative number, the absolute value is the positive of that number.

In the context of functions, applying absolute value affects the graph by reflecting any part of the function that dips below the x-axis upwards. This is crucial in \( |\cot x| \) since any negative values from \( \cot x \) become positive, ensuring that \( |\cot x| \) is always non-negative.

Symmetry

Symmetry in functions is a property that indicates a kind of balance or repetitive pattern in the function's graph. It describes how a graph is mirrored or rotated to produce the same shape.

• Even Functions: These symmetrical functions mirror around the y-axis. For any function \( f(x) \), if \( f(-x) = f(x) \), it is even. A strong example is \( |\cot x| \), which remains unchanged when \( x \) is replaced with \(-x \).
• Odd Functions: These functions show symmetry about the origin. They satisfy \( f(-x) = -f(x) \). The original \( \cot x \) function is odd, but its absolute value form, \(|\cot x| \), alters this symmetry.

Understanding symmetry helps when sketching graphs, as you can predict the shape and behavior of the function across certain intervals.

Even Functions

Even functions have a crucial uniqueness—they are symmetrically mirrored across the y-axis. The criterion for a function to qualify as even is that \( f(-x) = f(x) \) for every \( x \) in the function's domain.

• Graphical Interpretation: The right side of the graph (\( x > 0 \)) mirrors the left side (\( x < 0 \)).
• Analytical Insight: When evaluating symmetry, replacing \( x \) with \(-x \) in the function equation should yield the same expression.
• The function \( |\cot x| \) is even, as \( |\cot(-x)| \) indeed equals \( |\cot x| \). This is due to the nature of absolute values making sign irrelevant.

Being aware of a function's parity (whether it's even or odd) greatly aids in understanding its behavior and symmetry.

Cotangent Function

The cotangent function, represented as \( \cot x \), is one of the basic trigonometric functions. It is especially notable for being the reciprocal of the tangent function, where \( \cot x = \frac{\cos x}{\sin x} \). Let's break down some of its key characteristics:

• Periodicity: The \( \cot x \) function is periodic with a period of \( \pi \), meaning it repeats its values every \( \pi \) radians.
• Undefined Points: Cotangent is undefined where the sine function is zero, specifically at integer multiples of \( \pi \). This is because division by zero is undefined. At these points, the graph will have vertical asymptotes.
• Graph Behavior: It decreases steeply from positive infinity to negative infinity within each period.

By understanding the \( \cot x \) function, interpreting \( |\cot x| \) becomes simpler. The \( |\cot x| \) function shares the same periodicity and undefined points yet reflects portions of the graph above the x-axis, creating a sharper visual impact.