Problem 14 - Calculus of a Single Variable

Question

Think About It In Exercises 13 and \(14,\) determine which value best approximates the area of the region bounded by the graphs of \(f\) and \(g .\) (Make your selection on the basis of a sketch of the region and not by performing any calculations.) $$ \begin{array}{l}{f(x)=2-\frac{1}{2} x, \quad g(x)=2-\sqrt{x}} \\\ {\begin{array}{llll}{\text { (a) } 1} & {\text { (b) } 6} & {\text { (c) }-3} & {\text { (d) } 3} & {\text { (e) } 4}\end{array}}\end{array} $$

Step-by-Step Solution

Answer

The exact answer cannot be provided without visual data. After sketching, it's important to choose the value that best approximates the estimated area of the bounded region.

Step 1: Sketch the Functions

Firstly, sketch the functions \(f(x)=2-\frac{1}{2}x\) and \(g(x)=2-\sqrt{x}\) on the same plane. Notice that \(f(x)\) is a straight line that decreases as \(x\) increases, whereas \(g(x)\) is a square root function that also decreases as \(x\) increases, but at a different rate.

Step 2: Determine the Intersection Points

Secondly, set \(f(x) = g(x)\) and solve for \(x\), giving you the intersection points. These points mark the boundaries of the region of interest.

Step 3: Estimate the Area of the Bounded Region

Thirdly, from the sketch, estimate the area of the region bounded by the two graphs. The estimate won't be exact, but it should give you an idea of whether the area is closer to 1, 3, 4, 6 or -3.

Step 4: Choose the Best Approximation

Finally, compare your estimation to the options given in the problem (1, 3, 4, 6, -3) and choose the one that's closest to your estimate.

Key Concepts

Sketching Functions

Understanding how to sketch functions is crucial when analyzing the graphical relationships between different mathematical expressions.

In the context of the exercise, sketching involves drawing the two functions, namely a linear function represented by \(f(x)=2-\frac{1}{2}x\) and a square root function represented by \(g(x)=2-\sqrt{x}\), on the same coordinate plane. The linear function descends in a straight line as \(x\) increases, illustrating a constant rate of change. Conversely, the square root function also decreases as \(x\) increases but at a variable rate, decelerating as \(x\) grows larger.

Visualizing these on graph paper or using graphing software is an essential step in comparing their behavior and how they interact with each other. A proper sketch will reveal the area between the curves, key to approximating solutions in calculus without actual calculation.

Intersection Points of Functions

Identifying the intersection points of functions is a foundational task in calculus that helps to delineate the region of interest between the curves.

For our example, setting \(f(x)\) equal to \(g(x)\) and solving for \(x\) will reveal the points at which the two functions cross paths. In technical terms, \( x \) values are found when \(2-\frac{1}{2}x = 2-\sqrt{x}\). These points are crucial because they define the horizontal boundaries of the region we're interested in. For visualization, marking these points on our sketch helps to enclose and visually represent the area whose approximation is our ultimate goal.

Estimation Techniques in Calculus

Estimation techniques in calculus are series of methods used to find approximate values of areas, lengths, and other quantities that might be difficult to calculate precisely.

When it comes to estimating the area under a curve—or between two curves, as in our exercise—a common approach is to visually analyze the shadings and shapes formed by the boundaries. Although not exact, these observations can provide a reasonable guess that aligns with possible answers, such as those provided in the exercise: 1, 3, 4, 6, or -3. Estimation techniques prioritize a conceptual understanding over precise calculation, promoting a quicker analysis that is especially useful in the absence of computational tools.

Comparing Functions

Comparing functions is a vital part of understanding how different mathematical expressions relate to each other graphically.

When we compare \(f(x)=2-\frac{1}{2}x\) and \(g(x)=2-\sqrt{x}\), we are contrasting a linear rate of change with a non-linear one. Observing how these functions descend on a graph side by side offers insight into their respective growth rates. The linear function changes at a constant rate, while the square root function changes at a decreasing rate.

By juxtaposing these functions on a graph and looking at the shape of the area they bound together, we gain the necessary perspective to make educated guesses about the size of the area between them, a common type of comparison in calculus studies.