Problem 425 - Calculus Volume 1

Question

For the following exercises, solve each problem. [T] A chain hangs from two posts \(2 \mathrm{~m}\) apart to form a catenary described by the equation \(y=2 \cosh (x / 2)-1\). Find the slope of the catenary at the left fence post.

Step-by-Step Solution

Answer

The slope at the left fence post is \( \frac{e^{-1/2} - e^{1/2}}{2} \).

Step 1: Understanding the Catenary Equation

The catenary is described by the equation \( y = 2 \cosh(x/2) - 1 \). The hyperbolic cosine function is defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). Our task is to find the slope of the catenary, which means we need to compute the derivative \( \frac{dy}{dx} \) and evaluate it at a specific point.

Step 2: Determine Left Fence Post Position

The posts are 2 meters apart, centered at \( x = 0 \). Therefore, the left fence post will be at \( x = -1 \). We need to find the slope at this point.

Step 3: Find the Derivative of the Catenary Equation

Differentiate the function \( y = 2 \cosh(x/2) - 1 \) with respect to \( x \). Using the chain rule, the derivative is \( \frac{dy}{dx} = 2 \cdot \sinh(x/2) \cdot \frac{1}{2} = \sinh(x/2) \), where \( \sinh(x) \) is the hyperbolic sine function, defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \).

Step 4: Evaluate the Derivative at the Left Post

Substitute \( x = -1 \) into \( \sinh(x/2) \) to find the slope at the left post: \( \sinh(-1/2) = \frac{e^{-1/2} - e^{1/2}}{2} \). Calculate this value to find the slope.

Key Concepts

Hyperbolic Functions and their Role

Hyperbolic functions like hyperbolic cosine (\( \cosh \) ) and hyperbolic sine (\( \sinh \) ) are similar to trigonometric functions but are defined using exponential functions. The hyperbolic cosine function \( \cosh(x) = \frac{e^x + e^{-x}}{2} \) gives us the basic shape of structures like catenaries, which is a smooth, symmetrical curve. They often appear in engineering, especially where forces and equilibrium are concerned. The hyperbolic sine function, \( \sinh(x) = \frac{e^x - e^{-x}}{2} \) , adds variability or tilt to the curve.

In catenary problems, these functions help model the behavior of hanging chains or cables, giving insights about tension and stability. Understanding the symmetry and properties of these functions is crucial for solving real-world problems efficiently.

Applying the Chain Rule

The chain rule is a fundamental technique in calculus used for differentiating composite functions. When you have a function inside another function, like \( y = 2 \cosh(x/2) - 1 \), it tells us how to "unpack" the derivative process.

For our catenary, the outer function is \( 2 \cosh(u) \) where \( u = x/2 \). When differentiating, first take the derivative of the outer function, then multiply by the derivative of the inner function \( u \). This gives us:

• The derivative of \( 2 \cosh(u) \) with respect to \( u \) is \( 2 \sinh(u) \)
• The derivative of \( x/2 \) with respect to \( x \) is \( \frac{1}{2} \)

By combining these results, we get \( 2 \cdot \sinh(x/2) \cdot \frac{1}{2} = \sinh(x/2) \). Using the chain rule simplifies complex differentiation tasks by breaking them down into manageable parts.

Derivative Calculation for Slope

In calculus, the derivative gives us the rate of change of a function. For catenaries, the derivative at a point indicates the slope of the curve at that specific location.

The original problem function, \( y = 2 \cosh(x/2) - 1 \), represents a catenary. To find the slope at a particular point, calculate \( \frac{dy}{dx} \) using differentiation. We already found the derivative to be \( \sinh(x/2) \). Evaluating this derivative at different points on the x-axis will tell us the incline of the catenary at those locations.

When we plug in the position of the left fence post into the derivative, we can directly determine the slope at that exact point. It is essential for visualizing how steep or flat the catenary is where it connects to the fence post.

Fence Post Position on the Catenary

In the given catenary problem, the position of the fence posts plays a crucial role. The posts are placed 2 meters apart centered at \( x = 0 \). Hence, the left fence post is located at \( x = -1 \).

Knowing the position of the fence posts helps us calculate the slope at these specific points. By evaluating the derivative at \( x = -1 \), we determine the tilt of the catenary where it joins the left post.

• Step: Substitute \( x = -1 \) into the derivative \( \sinh(x/2) \)
• Result: The calculation \( \sinh(-1/2) = \frac{e^{-1/2} - e^{1/2}}{2} \)

Finally, we evaluate this expression to find the exact numerical slope, revealing how the chain behaves at the left connection point. Understanding these post positions is vital for designing structures that rely on tension and suspension.