Problem 52 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Deal with topics developed in Chapter \(7 .\) POLARIZED LIGHT A polarized light wave travels in such a way that its vertical displacement \(y\) at time \(t\) is a function of both \(t\) and its horizontal displacement \(x\) according to the formula $$ y(x, t)=0.27 \sin \left(10 \pi t-3 \pi x+\frac{\pi}{4}\right) $$ a. Find \(\frac{\partial y}{\partial x}\) and \(\frac{\partial y}{\partial t}\). b. For what points \((x, t)\) is \(y(x, t)\) maximized? For what points is \(y(x, t)\) minimized?

Step-by-Step Solution

Answer

Maximized when \( 10 t - 3 x = k + \frac{1}{4} \). Minimized when \( 10 t - 3 x = m - \frac{1}{4} \).

Step 1 - Compute the partial derivative with respect to x

To find \( \frac{\partial y}{\partial x} \), differentiate the given function \( y(x, t) = 0.27 \sin \left(10 \pi t - 3 \pi x + \frac{\pi}{4}\right) \) with respect to \( x \). Use the chain rule: \( \frac{\partial}{\partial x} \sin(u) = \cos(u) \cdot \frac{\partial u}{\partial x} \), where \( u = 10 \pi t - 3 \pi x + \frac{\pi}{4} \).\[ \frac{\partial y}{\partial x} = 0.27 \cos \left(10 \pi t - 3 \pi x + \frac{\pi}{4}\right) \cdot (-3 \pi) \ = -0.81 \pi \cos \left(10 \pi t - 3 \pi x + \frac{\pi}{4}\right) \]

Step 2 - Compute the partial derivative with respect to t

To find \( \frac{\partial y}{\partial t} \), differentiate the given function \( y(x, t) = 0.27 \sin \left(10 \pi t - 3 \pi x + \frac{\pi}{4} \right) \) with respect to \( t \). Use the chain rule: \( \frac{\partial}{\partial t} \sin(u) = \cos(u) \cdot \frac{\partial u}{\partial t} \), where \( u = 10 \pi t - 3 \pi x + \frac{\pi}{4} \).\[ \frac{\partial y}{\partial t} = 0.27 \cos \left( 10 \pi t - 3 \pi x + \frac{\pi}{4} \right) \cdot (10 \pi) \ = 2.7 \pi \cos \left( 10 \pi t - 3 \pi x + \frac{\pi}{4} \right) \]

Step 3 - Find points where y(x, t) is maximized or minimized

The function \( y(x, t) \) achieves its maximum value when the sine function equals 1 and its minimum value when the sine function equals -1. Therefore, solve the equations:\[ 0.27 \sin \left(10 \pi t - 3 \pi x + \frac{\pi}{4} \right) = 0.27 \] and\[ 0.27 \sin \left(10 \pi t - 3 \pi x + \frac{\pi}{4} \right) = -0.27 \] To determine these points:Maximum: When \( \sin \left(10 \pi t - 3 \pi x + \frac{\pi}{4} \right) = 1 \), \[ 10 \pi t - 3 \pi x + \frac{\pi}{4} = \frac{\pi}{2} + 2k\pi \quad (k \in \mathbb{Z}) \ 10 \pi t - 3 \pi x = \frac{\pi}{4} + 2k\pi - \frac{\pi}{2} \ 10 t - 3 x = k + \frac{1}{4} \] Minimum: When \( \sin \left(10 \pi t - 3 \pi x + \frac{\pi}{4} \right) = -1 \), \[ 10 \pi t - 3 \pi x + \frac{\pi}{4} = \frac{3\pi}{2} + 2m\pi \quad (m \in \mathbb{Z}) \ 10 \pi t - 3 \pi x = \frac{\pi}{4} + 2m\pi - \frac{3\pi}{2} \ 10 t - 3 x = m - \frac{1}{4} \]

Key Concepts

Partial Derivatives

Partial derivatives are used to measure the rate of change of a function with respect to one variable while keeping other variables constant. In the given function, y(x, t), we can find the partial derivative with respect to x and t to understand how y changes with changes in x and t respectively.
Here's a quick summary on how to calculate partial derivatives:

• Identify the function \(y(x, t) = 0.27 \sin(10 \pi t - 3 \pi x + \frac{\pi}{4})\).
• For \(\frac{\partial y}{\partial x}\), keep t constant. Use the chain rule: \( \frac{\partial}{\partial x} \sin(u) = \cos(u) \cdot \frac{\partial u}{\partial x} \), where u is the inner function.
• For \(\frac{\partial y}{\partial t}\), keep x constant. Use the chain rule: \( \frac{\partial}{\partial t} \sin(u) = \cos(u) \cdot \frac{\partial u}{\partial t} \).

By applying these principles, we find that: \( \frac{\partial y}{\partial x} = -0.81 \pi \cos(10 \pi t - 3 \pi x + \frac{\pi}{4}) \) and \( \frac{\partial y}{\partial t} = 2.7 \pi \cos(10 \pi t - 3 \pi x + \frac{\pi}{4}) \). Understanding these derivatives helps in evaluating how the function y responds to variations in x and t.

Sinusoidal Functions

Sinusoidal functions like sine and cosine are fundamental in describing oscillatory or wave-like behaviors. The given function, \(y(x, t) = 0.27 \sin(10 \pi t - 3 \pi x + \frac{\pi}{4})\), represents a polarized light wave and is a perfect example of a sinusoidal function.
Key characteristics of sinusoidal functions:

• They oscillate between a maximum and minimum value. For the sine function, these values are +1 and -1 respectively.
• They have a regular periodic pattern, repeating every fixed interval.
• The amplitude determines the height of the wave. In our function, the amplitude is 0.27.

The argument of the sine function, \(10 \pi t - 3 \pi x + \frac{\pi}{4}\), determines the shape and position of the wave. When this argument changes, the value of the sine function oscillates, resulting in the wave's continuous motion. Sinusoidal functions are crucial in many fields like physics, engineering, and even in music due to their repetitive and predictable nature.

Maxima and Minima

Maxima and minima refer to the highest and lowest values a function can achieve. For the function \(y(x, t) = 0.27 \sin(10 \pi t - 3 \pi x + \frac{\pi}{4})\), the sine function achieves its maximum value of 1 and minimum value of -1.
To find when y(x, t) is maximized or minimized:

• Maximized when \(\sin(10 \pi t - 3 \pi x + \frac{\pi}{4}) = 1\). Solve \(10 \pi t - 3 \pi x + \frac{\pi}{4} = \frac{\pi}{2} + 2k \pi\) (where k is an integer).
• Minimized when \(\sin(10 \pi t - 3 \pi x + \frac{\pi}{4}) = -1\). Solve \(10 \pi t - 3 \pi x + \frac{\pi}{4} = \frac{3\pi}{2} + 2m \pi\) (where m is an integer).

These simplified conditions help locate points \((x, t)\) where the function reaches its peak values. For maxima: \(10 t - 3 x = k + \frac{1}{4}\). For minima: \(10 t - 3 x = m - \frac{1}{4}\). Knowing where maxima and minima occur is vital in optimization problems and analyzing periodic motions in various practical applications.