Problem 57 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

The radius of the earth is roughly 4,000 miles, and an object located \(x\) miles from the center of the earth weighs \(w(x)\) lb, where $$ w(x)= \begin{cases}A x & \text { if } x \leq 4,000 \\ \frac{B}{x^{2}} & \text { if } x>4,000\end{cases} $$ and \(A\) and \(B\) are positive constants. Assuming that \(w(x)\) is continuous for all \(x\), what must be true about \(A\) and \(B\) ? Sketch the graph of w(x).

Step-by-Step Solution

Answer

The constants must satisfy \(B = 4000^3 \cdot A\). The graph consists of a line followed by a decreasing curve.

Step 1: Understand the piecewise function

The weight function, \(w(x)\) changes its form depending on whether \(x\) is less than or equal to 4000 miles or greater than 4000 miles. For \(x \leq 4000\), \(w(x) = Ax\). For \(x > 4000\), \(w(x) = \frac{B}{x^2}\).

Step 2: Ensure Continuity

For \(w(x)\) to be continuous at \(x = 4000\), the limit from the left must equal the limit from the right at \(x = 4000\). This gives the equation: \[ \text{lim}_{x \to 4000^-} w(x) = \text{lim}_{x \to 4000^+} w(x) \].

Step 3: Set up the limits for continuity

Compute the limit from the left and the right at \(x = 4000\): Left: \( \text{lim}_{x \to 4000^-} (Ax) = A \cdot 4000 \)Right: \( \text{lim}_{x \to 4000^+} (\frac{B}{x^2}) = \frac{B}{4000^2} \)

Step 4: Equalize the limits

Set the left-hand and right-hand limits equal to each other: \[ A \cdot 4000 = \frac{B}{4000^2} \].

Step 5: Solve for the relationship between A and B

Rearrange the equation to find the relationship between \(A\) and \(B\): Multiply both sides by \(4000^2\) to get: \[ A \cdot 4000^3 = B \].Therefore: \[ B = 4000^3 \cdot A \]

Step 6: Sketch the Graph

The graph of \(w(x)\) has two pieces. For 0 < \(x\) ≤ 4000, it is a straight line with slope \(A\). For \(x > 4000\), it behaves as a decreasing curve \( \frac{B}{x^2} \). At \(x = 4000\), the function transitions smoothly.

Key Concepts

Continuity of Functions

Continuity denotes that a function does not have any abrupt jumps, breaks, or holes. If you can draw a function on a graph without lifting your pen, it is continuous. For a function to be continuous at a point, three conditions must hold:
1. The function must be defined at that point.
2. The limit of the function as it approaches the point from both sides must exist.
3. The limit of the function as it approaches the point must equal the function's value at that point.

In this problem, we are given a piecewise function and asked to find the conditions under which it is continuous over its domain. Specifically, we need to ensure continuity at the point where the function's definition changes, which is at x = 4000.

Piecewise Functions

A piecewise function is one that is defined by different expressions over different intervals. In this problem, the weight function, \(w(x)\), changes its form depending on whether \(x < 4000\) or \(x > 4000\). For \(x \leq 4000\), we have \(w(x) = Ax\), a linear function and for \(x > 4000\), \(w(x) = \frac{B}{x^2}\), a hyperbolic function.

Piecewise functions are often used to model situations where a certain rule does not apply across the entire domain but changes at specific points. This makes them suitable for real-world problems like this, where the weight changes based on the distance from the Earth’s center.

Limits

The concept of a limit is fundamental in calculus and crucial for understanding continuity. When we say the limit of \(f(x)\) as \(x\) approaches a value \(c\), we mean the value that \(f(x)\) gets closer to as \(x\) gets closer to \(c\).

• To find the limit from the left (\text{lim}_{x \to c^-}), we examine values of \(x\) that are less than \(c\).


• To find the limit from the right (\text{lim}_{x \to c^+}), we examine values of \(x\) that are greater than \(c\).

In this exercise, we need to ensure that the limit of \(w(x)\) as \(x\) approaches 4000 from the left (\(w(x) = Ax\)) equals the limit as \(x\) approaches 4000 from the right (\(w(x) = \frac{B}{x^2}\)). By setting these limits equal to each other, we ensure the function is continuous at \(x = 4000\), leading to our continuity condition: \ A \cdot 4000 = \frac{B}{4000^2}. Solving this equation helps us find the necessary relationship between the constants \(A\) and \(B\).