Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

What points, if any, are the functions discontinuous? $$ h(\theta)=|\sin \theta+\cos \theta| $$

Step-by-Step Solution

Answer

The function \( h(\theta) \) is continuous for all \( \theta \).

Step 1: Understanding the Function

The given function is \( h(\theta) = |\sin \theta + \cos \theta| \). This is an absolute value function involving trigonometric components \( \sin \theta \) and \( \cos \theta \). An absolute value function can potentially be discontinuous where the inside expression changes sign, causing a potential "jump" in values, unless continuity is preserved.

Step 2: Analyzing Trigonometric Expression

Examine \( \sin \theta + \cos \theta \). These are continuous functions individually, and their sum is also continuous across all \( \theta \) because there aren’t any operations (like division) that create breaks in continuity. However, applying the absolute value operator may affect continuity.

Step 3: Checking Absolute Value Discontinuity

To check where \( |\sin \theta + \cos \theta| \) might be discontinuous, investigate where the function inside the absolute value is zero, \( \sin \theta + \cos \theta = 0 \). Since absolute value functions can have discontinuities at points where the expression inside equals zero, these are the points of interest.

Step 4: Solving for Zero Points

Solve the equation \( \sin \theta + \cos \theta = 0 \). Rearrange and factor: \( \sin \theta = -\cos \theta \). This gives \( \tan \theta = -1 \), and the general solution is \( \theta = 3\pi/4 + n\pi \) for integer \( n \).

Step 5: Evaluating Continuity at Zero Points

The function is continuous if the left and right limits as \( \theta \) approaches these zero points are equal to the value at the zero. Consider \( h(\theta) \) as composed of linear segments joining smoothly at these points, which occur at \( \theta = 3\pi/4 + n\pi \). As both segments on either side are a reflection of each other due to the absolute value, continuity is maintained here.

Key Concepts

Trigonometric Functions

Trigonometric functions like sine (\( \sin \)) and cosine (\( \cos \)) are fundamental in calculus and mathematics. They are periodic functions that oscillate between -1 and 1, displaying a smooth wave-like pattern.

• **Sine Function (\( \sin \theta \))** - It starts at zero, reaches 1 at \( \frac{\pi}{2} \), drops back to zero at \( \pi \), and repeats the cycle.
• **Cosine Function (\( \cos \theta \))** - It starts at 1, drops to zero at \( \frac{\pi}{2} \), goes to -1 at \( \pi \), and repeats.

These functions are continuous and differentiable at all points, meaning there are no breaks or holes in their graphs. They are responsible for describing many natural phenomena such as waves and oscillations. In our context, both sine and cosine ensure the smooth continuity of the expression \( h(\theta) = |\sin \theta + \cos \theta| \) before applying the absolute value function.

Absolute Value Function

The absolute value function is crucial for defining magnitude while ignoring the sign. It transforms any real number to its non-negative version, which can be expressed as:\[|a| = \begin{cases} a, & \text{if } a \geq 0 \ -a, & \text{if } a < 0 \end{cases}\]This function can disrupt continuity mainly at points where the argument equals zero. At these points, there can be a 'jump' if the negative side of the function is not symmetric with the positive side. In the context of \( h(\theta) = |\sin \theta + \cos \theta| \), the absolute value modifies \( \sin \theta + \cos \theta \) such that even if the combination of \( \sin \) and \( \cos \) results in zero, the function remains continuous as changes in sign are symmetrically addressed at zero-points. Understanding where the inside expression hits zero helps predict behavior changes in the function but doesn’t necessarily indicate discontinuity if handled symmetrically, as is the case here.

Continuity in Calculus

Continuity is a critical property of a function indicating no interruptions or breaks in its graph. Mathematically, a function \( f \) is continuous at a point \( c \) if:

• \( f(c) \) is defined
• \( \lim_{x \to c} f(x) \) exists
• \( \lim_{x \to c} f(x) = f(c) \)

In calculus, continuity is required for a function to be differentiable, enabling the determination of slopes and rates of change. For the function \( h(\theta) = |\sin \theta + \cos \theta| \), applying these continuity principles shows it is continuous everywhere. At each point where \( \sin \theta + \cos \theta = 0 \), both one-sided limits as \( \theta \) approaches these points equal the value at the point, ensuring continuity. Thus, our function does not break, confirming smooth transitions across all \( \theta \) values.