Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(R\) be the rectangle joining the midpoints of the sides of the quadrilateral \(Q\) having vertices \((\pm x, 0)\) and \((0, \pm 1)\). Calculate $$ \lim _{x \rightarrow 0^{+}} \frac{\text { perimeter of } R}{\text { perimeter of } Q} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify Midpoints of Quadrilateral Sides

To find the midpoints of the sides of the quadrilateral with vertices \((x, 0), (-x, 0), (0, 1), (0, -1)\), we first determine the midpoints of each side: 1. Between \((x, 0)\) and \((0, 1)\), the midpoint is \(\left(\frac{x}{2}, \frac{1}{2}\right)\).2. Between \((x, 0)\) and \((0, -1)\), the midpoint is \(\left(\frac{x}{2}, -\frac{1}{2}\right)\).3. Between \((-x, 0)\) and \((0, 1)\), the midpoint is \(\left(-\frac{x}{2}, \frac{1}{2}\right)\).4. Between \((-x, 0)\) and \((0, -1)\), the midpoint is \(\left(-\frac{x}{2}, -\frac{1}{2}\right)\).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Key Concepts

Rectangles and Quadrilaterals

Understanding the shapes of rectangles and quadrilaterals is crucial when solving geometric problems. A quadrilateral is a four-sided polygon with characteristics that can differ widely, such as squares, rectangles, trapezoids, and others. A rectangle is a specific type of quadrilateral that has four right angles and opposite sides that are equal in length.
In this exercise, we begin with the quadrilateral shaped by the vertices \((x, 0)\), \((-x, 0)\), \((0, 1)\), and \((0, -1)\). By connecting the midpoints of its sides, we aim to form a rectangle. This rectangle has properties that are connected to the geometry of the original quadrilateral. Such transformations help us understand relationships among different types of polygons and how they can be related by their mathematical descriptions.

Midpoint Formula

The midpoint formula helps us find a point that is exactly halfway between two other points on a plane. It is used extensively in geometry to determine the center of a line segment. If you have two points \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint \((M)\) of the line segment joining these points is given by the formula:

• \( M = \, \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

In this problem, applying the midpoint formula is how we determined the vertices of the rectangle formed by the midpoints of the sides of the quadrilateral \(Q\). Each calculated midpoint effectively divides the sides of quadrilateral \(Q\) into two smaller, equal parts. This is an essential step in transforming the quadrilateral into a rectangle \(R\).
The use of the midpoint formula is a fundamental skill that helps you understand how calculations give birth to new geometric shapes based on existing ones.

Limit of a Function

The concept of a limit is central to calculus and involves understanding the behavior of a function as its input approaches a specific value. In simple terms, the limit describes the value that a function approaches as the input approaches some value.

• For example, \( \lim_{x \to a} f(x) \) indicates we're interested in how \( f(x) \) behaves as \( x \) gets closer to \( a \).

In the exercise, we are tasked with calculating the limit of the ratio of perimeters as \( x \) approaches zero from the positive side, denoted as \( x \rightarrow 0^{+} \). This notation means we're considering values of \( x \) that are positive and approaching zero. Understanding this concept allows for analyzing how one quantity changes in relation to another, particularly when a variable diminishes or increases to a particular point.

Perimeter Calculation

Calculating the perimeter involves measuring the distance around the border of a polygon. For both quadrilaterals and rectangles, the perimeter is the sum of the lengths of all sides.

• For example, the perimeter of a rectangle can be found using the formula \( P = 2l + 2w \), where \( l \) and \( w \) are the lengths of the rectangle's sides.

In the given exercise, calculating the perimeter of both the quadrilateral \(Q\) and the resultant rectangle \(R\) is integral in determining the limit expression. By knowing how to compute perimeter, we connect geometric properties to algebraic expressions. This not only aids in solving the problem but also builds a tangible understanding of how transformations like midpoint creation impact overall geometry.