Problem 50 - College Algebra and Calculus: An Applied Approach

Question

Use the definite integral below to find the required arc length. If \(f\) has a continuous derivative, then the arc length of \(f\) between the points \((a, f(a))\) and \((b, f(b))\) is \(\int_{b}^{a} \sqrt{1+\left[f^{\prime}(x)\right]^{2}} d x\) Arc Length A fleeing hare leaves its burrow \((0,0)\) and moves due north (up the \(y\) -axis). At the same time, a pursuing lynx leaves from 1 yard east of the burrow \((1,0)\) and always moves toward the fleeing hare (see figure). If the lynx's speed is twice that of the hare's, the equation of the lynx's path is \(y=\frac{1}{3}\left(x^{3 / 2}-3 x^{1 / 2}+2\right)\) Find the distance traveled by the lynx by integrating over the interval \([0,1]\).

Step-by-Step Solution

Answer

The exact distance traveled by the lynx is the value of the definite integral over the interval [0,1] for the function \(\sqrt{1+ \left[\frac{3}{2}x^{1/2} - \frac{3}{2}x^{-1/2}\right]^2}\)

Step 1: Find the derivative

First, get the derivative of the provided function. That is \(f'(x)= \frac{3}{2}x^{1/2} - \frac{3}{2}x^{-1/2}\)

Step 2: Calculate the magnitude of the derivative

Calculate the magnitude of the derivative by squaring it, and then take the square root of 1 plus the squared derivative. This will become \(\sqrt{1+ [\frac{3}{2}x^{1/2} - \frac{3}{2}x^{-1/2}]^2}\)

Step 3: Integral Calculation

Calculate the integral over the range [0,1]. This involves integrating the function \(\sqrt{1+ [\frac{3}{2}x^{1/2} - \frac{3}{2}x^{-1/2}]^2}\) from 0 to 1. This could be a complex integration, hence it could be computed using numerical methods or a calculator for simplicity.

Key Concepts

Definite Integral

Let's explore the concept of a definite integral. It is fundamentally a number that represents the area under a curve over a specified interval. In our exercise, we use definite integrals to calculate the arc length of a path traced by an animal.

For the lynx's path, we apply the formula for arc length in terms of definite integral:

• The function is integrated from the starting point to the ending point, here from 0 to 1.
• Inside the integral, the square root of 1 plus the square of the derivative of the function is used.

With these steps, we translate the continuous journey of the lynx into a mathematical form that allows us to compute the total distance traveled.

Derivative

The derivative is all about measuring how a function changes as its input changes. This is crucial when working with arc lengths, as seen in this exercise.

To find the arc length, first, you must find the derivative of the lynx's path:

• Derivatives tell us the rate at which the lynx's path changes with respect to x.
• For the lynx, the derivative is found and calculated as \( f'(x)= \frac{3}{2}x^{1/2} - \frac{3}{2}x^{-1/2} \).

By understanding derivatives, we gain insights into how climbing or falling slopes affect the total path length.

Integral Calculus

Integral calculus is all about accumulation - adding up tiny pieces to find a total. In this problem, it helps us sum up all tiny sections of the lynx's journey to find the complete path length.

We use integral calculus in two main steps in our exercise:

• First, after squaring the derivative, it's added to 1 and put under a square root. This transforms the problem into a manageable form.
• Second, calculate the definite integral from 0 to 1, which can often be complex and may require numerical methods or tools.

This process not only provides the length of the path but also illustrates how integral calculus is a powerful tool for solving real-world problems.

Mathematical Modeling

Mathematical modeling converts real-world situations into mathematical expressions. This is what we've done by creating and solving the lynx's path equation.

Here's how the exercise highlights mathematical modeling:

• The lynx's and hare's movement patterns are expressed as mathematical functions.
• Using these models, we can predict precise details like how far the lynx travels.

This form of modeling is vital in various fields, enabling us to simulate, predict, and analyze. Understanding how to model mathematically allows for solving a range of practical and theoretical challenges.