Problem 13 - Applied Calculus

Question

A quantity \(W\) satisfies the differential equation $$ \frac{d W}{d t}=5 W-20 $$ (a) Is \(W\) increasing or decreasing at \(W=10 ? W=2\) ? (b) For what values of \(W\) is the rate of change of \(W\) equal to zero?

Step-by-Step Solution

Answer

(a) Increasing at \( W = 10 \), decreasing at \( W = 2 \). (b) Zero at \( W = 4 \).

Step 1: Understand the Differential Equation

The given differential equation is \( \frac{dW}{dt} = 5W - 20 \). This equation shows how the quantity \( W \) changes with respect to time \( t \). The term \( 5W \) suggests that \( W \) increases at a rate proportional to its current value, while \( -20 \) is a constant term that may reduce the rate.

Step 2: Evaluate at \(W = 10\)

Substitute \( W = 10 \) into the differential equation: \[ \frac{dW}{dt} = 5(10) - 20 = 50 - 20 = 30 \]. Since \( \frac{dW}{dt} > 0 \), the quantity \( W \) is increasing at \( W = 10 \).

Step 3: Evaluate at \(W = 2\)

Substitute \( W = 2 \) into the differential equation: \[ \frac{dW}{dt} = 5(2) - 20 = 10 - 20 = -10 \]. Since \( \frac{dW}{dt} < 0 \), the quantity \( W \) is decreasing at \( W = 2 \).

Step 4: Determine When the Rate of Change is Zero

Set \( \frac{dW}{dt} = 0 \) and solve for \( W \): \[ 5W - 20 = 0 \]. Thus, \[ 5W = 20 \] leads to \[ W = 4 \]. Hence, the rate of change of \( W \) is zero at \( W = 4 \).

Key Concepts

Rate of Change

When dealing with differential equations like \( \frac{dW}{dt} = 5W - 20 \), the term "rate of change" comes into play when understanding how a quantity changes over time. In this equation, the rate of change of \( W \) with respect to time \( t \) is determined by the expression on the right-hand side. One important aspect of analyzing differential equations is identifying the influence of the terms on the rate.

• The term \( 5W \) signifies a change proportional to the current value of \( W \). This part of the equation indicates how rapidly \( W \) would increase or decrease if no other forces acted upon it.
• The constant term \( -20 \) represents a steady shift in the rate, acting to decrease \( W \) consistently regardless of its present value.

To find when the rate of change is zero, solve \( 5W - 20 = 0 \). Here, \( W = 4 \) shows the moment where growth and decline perfectly balance, halting change temporarily.

Increasing and Decreasing Functions

An important concept in differential equations is determining whether a function is increasing or decreasing at a given point. Through the derivative \( \frac{dW}{dt} \), the behaviour of the function \( W \) can be analyzed. If \( \frac{dW}{dt} > 0 \), \( W \) is increasing, and if \( \frac{dW}{dt} < 0 \), it is decreasing.

• At \( W = 10 \), substituting into the equation yields \( \frac{dW}{dt} = 30 \). Since this is a positive number, the function is increasing here.
• Conversely, at \( W = 2 \), \( \frac{dW}{dt} = -10 \), a negative outcome indicating the function is decreasing.

This clear relationship between the sign of \( \frac{dW}{dt} \) and the behavior of the function enables us to predict \( W \)'s growth pattern without manually plotting each value.

Solving Differential Equations

Solving a differential equation involves determining how a variable changes over time or another variable. The provided equation \( \frac{dW}{dt} = 5W - 20 \) is a first-order linear differential equation and can be approached through several path-based methods:

• **Separation of Variables:** This technique involves isolating terms to one side, simplifying integration to solve for \( W \).
• **Integrating Factor:** Often used for linear equations like this, it transforms the equation into a more easily integrable format.
• **Graphical Analysis:** Observable patterns from solutions offer insights into the behavior of \( W \) over time.

For this specific equation, finding the solutions might require calculus concepts like integration or utilizing initial conditions to track \( W \)'s journey comprehensively. Understanding these solutions lends deeper insights into more complex behaviors, such as limits or steady states like \( W = 4 \) where the system equilibrates.