Problem 56 - Calculus with Concepts in Calculus

Question

Graph each pair of functions. Shade the region(s) the graphs enclose. $$ \begin{aligned} &g(t)=t-\cos t \text { for }-\pi / 4 \leq t \leq 7 \pi / 4 \text { and } k(t)=t+\sin t \text { for } \\ &-\pi / 4 \leq t \leq 7 \pi / 4 \end{aligned} $$

Step-by-Step Solution

Answer

Plot both functions, determine intersection points at \( t = \frac{3\pi}{4} \) and \( \frac{7\pi}{4} \), and shade the enclosed region accordingly on the graph.

Step 1: Understand the Functions

We have two functions of the variable \( t \) to graph: \( g(t) = t - \cos t \) and \( k(t) = t + \sin t \). Both functions are defined over the interval \(-\pi/4 \leq t \leq 7\pi/4\). Our task is to find the common region they enclose.

Step 2: Graph the Functions

Using a graphing tool or graph paper, plot the functions \( g(t) = t - \cos t \) and \( k(t) = t + \sin t \) over the interval \(-\pi/4 \leq t \leq 7\pi/4\). For \( g(t) \), note that it is a line with the wave of the cosine function modifying its path and for \( k(t) \), it is affected periodically by the sine wave. Make sure to mark the points where both functions intersect.

Step 3: Determine Intersection Points

Set the equations of \( g(t) \) and \( k(t) \) equal to each other to find intersection points: \( t - \cos(t) = t + \sin(t) \). Simplify to \( \cos(t) + \sin(t) = 0 \). Solve this equation within the given interval. The solutions will help identify the boundaries of enclosed regions.

Step 4: Solve for Intersection Points

Solve \( \cos(t) + \sin(t) = 0 \) by using the identity \( \tan(t) = -1 \). The general solutions are of the form \( t = \frac{3\pi}{4} + n\pi \). Check the solutions that fall within the interval \(-\pi/4 \leq t \leq 7\pi/4\). These are \( t = \frac{3\pi}{4} \) and \( t = \frac{7\pi}{4} \).

Step 5: Shade the Enclosed Region

Identify the segments between intersection points where one function is above the other. Shade the region below \( g(t) \) and above \( k(t) \) from \(-\pi/4\) to \(3\pi/4\), as well as between \(3\pi/4\) and \(7\pi/4\). Ensure consistent shading to indicate the enclosed region.

Key Concepts

Intersection Points

When dealing with graphs of functions, finding intersection points is crucial, as these points help us identify where two curves meet. To find intersection points between two functions, set them equal to each other. For our specific exercise, we have:

• For function \( g(t) = t - \cos t \)
• And function \( k(t) = t + \sin t \)

We set \( g(t) = k(t) \) to find where they intersect: \( t - \cos t = t + \sin t \). Simplifying this equation yields \( \cos t + \sin t = 0 \).
To solve, observe that \( \tan t = -1 \), giving solutions \( t = \frac{3\pi}{4} + n\pi \) where \( n \) is an integer. Within our interval \(-\pi/4 \leq t \leq 7\pi/4\), we check these solutions and find that the intersection points are at \( t = \frac{3\pi}{4} \) and \( t = \frac{7\pi}{4} \). Not only do these points provide boundaries, but they also guide us in shading the enclosed region.

Trigonometric Functions

Trigonometric functions like \( \sin t \) and \( \cos t \) are periodic and have specific properties that affect graphs. They repeat their values in predictable cycles, making them ideal for modeling waves. In this exercise, these functions modify otherwise linear graphs:

• \( g(t) = t - \cos t \) is a slanted line subjected to the cosine wave fluctuations.
• \( k(t) = t + \sin t \) similarly features the sine wave's deviations.

Both functions oscillate around their linear counterparts. The cosine and sine components shift the main line up or down periodically. It is essential to understand these changes to accurately plot the functions across the specified interval. This understanding aids in visualizing and identifying the regions where the graphs meet or cross each other, influencing the solution to where intersections occur.

Enclosed Regions

An enclosed region on a graph is formed where the paths of two functions meet and create a distinct area bounded by their intersection points. In this case, once we identify the intersection points, the goal is to determine where one graph lies above or below the other:

• From \(-\pi/4\) to \(3\pi/4\), plot reveals that \( g(t) \) resides above \( k(t) \).
• From \(3\pi/4\) to \(7\pi/4\), \( k(t) \) does not surpass \( g(t) \).

To shade these regions, note the segments where the graphs intersect. The visual representation helps comprehend where the different parts of each function's curve interact, signifying the changes in their respective heights. Successful shading emphasizes these particular segments, reinforcing understanding of areas these function pairs confine.