Problem 55 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(55 - 58 ,\) complete parts \(( a ) - ( d )\) for the piecewise- definedfunction. \(\quad (\) a) Draw the graph of \(f\) . (b) At what points \(c\) in the domain of \(f\) does \(\lim _ { x \rightarrow c } f ( x )\) exist? (c) At what points \(c\) does only the left-hand limit exist? (d) At what points \(c\) does only the right-hand limit exist? $$f ( x ) = \left\\{ \begin{array} { l l } { \sin x , } & { - 2 \pi \leq x < 0 } \\ { \cos x , } & { 0 \leq x \leq 2 \pi } \end{array} \right.$$

Step-by-Step Solution

Answer

(a) The graph of \(f(x)\) can be sketched by plotting \(\sin x\) for \(-2\pi \leq x < 0\) and \(\cos x\) for \(0 \leq x \leq 2\pi\). (b) \(\lim _ { x \rightarrow c } f ( x )\) exists for every \(x\) in the domain of \(f\) i.e., \(-2\pi \leq x \leq 2\pi\) except at x=0. (c) Only the left-hand limit exists at x=0. (d) Only the right-hand limit exists at x=0.

Step 1: Drawing the graph

To draw the graph of the function, plot \(\sin x\) for interval \(-2\pi \leq x < 0\) and \(\cos x\) for interval \(0 \leq x \leq 2\pi\). The graph of \(\sin x\) for \(-2\pi \leq x < 0\) will start from the point \(( -2\pi, 0 )\) with one complete wave ending at the point \((0,0)\). The graph of \(\cos x\) for \(0 \leq x \leq 2\pi\) will start from \((0,1)\) and forms a full wave ending at the point \((2\pi,1)\).

Step 2: Determining points with existing limits

The limit exists at points c in the domain of f where either both right hand and left-hand limits exist and are equal, or the function is defined at that point. Looking at the graphs, \(\lim _ { x \rightarrow c } f ( x )\) exists for every \(x\) in the domain of \(f\) i.e., \(-2\pi \leq x \leq 2\pi\) except at x=0.

Step 3: Identifying points where only the left-hand limit exists

The left-hand limit of a function at a point exists if the function approaches a certain value as x approaches that point from the left. Looking at the graph, it is observed that only the left-hand limit exists at x=0.

Step 4: Identifying points where only the right-hand limit exists

Similarly, the right-hand limit of a function at a point exists if the function approaches a certain value as x approaches that point from the right. Looking at the graph, it is observed that only the right-hand limit exists at x=0.

Key Concepts

Graphical Representation

Understanding the graph of a piecewise function helps visualize how different parts of the function behave in various intervals. In this case, the function is split into two parts:

• For \(-2\pi \leq x < 0\), the function is \(\sin x\), starting at the point \((-2\pi, 0)\) and forming a full wave that ends at \( (0, 0)\).
• For \(0 \leq x \leq 2\pi\), the function switches to \(\cos x\), beginning at \( (0, 1)\) and creating a complete wave ending at \( (2\pi, 1)\).

These distinct intervals create a continuous graph except at \(x = 0\), where the function switches from \(\sin x\) to \(\cos x\). By plotting both intervals separately, students can see how trigonometric functions behave over a cycle, showing peaks, troughs, and points where they intersect the x-axis. This visual representation aids in comprehending the transitions and behaviors of piecewise functions.

Limits in Calculus

Limits are fundamental in understanding the continuity and behavior of functions at specific points. For piecewise functions, limits can change depending on the side from which you approach a given point. Here’s a breakdown:

• In this exercise, limits exist at all points \(c\) except at \(x = 0\), where the function shifts from \(\sin x\) to \(\cos x\).
• To determine if the limit exists at a point, we need both the left-hand limit \(\lim_{x \to c^-} f(x)\) and the right-hand limit \(\lim_{x \to c^+} f(x)\) to be equal.

At \(x = 0\), the left-hand limit is defined by the approach from \(\sin x\) and equals \(0\), whereas the right-hand limit, defined by \(\cos x\), equals \(1\). Since these limits are not equal, the overall limit does not exist at \(x = 0\). Understanding these aspects helps grasp how piecewise functions behave differently around points of transition.

Trigonometric Functions

The role of trigonometric functions in this piecewise function illustrates their behavior within specific intervals. Here's how it works:

• For the interval \(-2\pi \leq x < 0\), \(\sin x\) is the function that defines the curve, showing a complete sinusoidal wave from negative to zero.
• Switching to the interval \(0 \leq x \leq 2\pi\), \(\cos x\) is used, providing a full cosine wave.

These functions represent cyclical patterns due to their periodic nature:

• Sinusoidal behavior: \(\sin x\) and \(\cos x\) repeat at regular intervals of \(2\pi\).
• Key points: Each function has distinct key points, such as maximums, minimums, and zeros, which are important for graphing.

By understanding these basic properties, students can easily navigate and graph these functions within their respective piecewise intervals, enhancing their comprehension of both trigonometric and piecewise functions.