Problem 21 - Precalculus

Question

The population of a small town can be modeled by the function \(p(t)=1.2 \sqrt{t}+40,\) where \(p\) is measured in thousands and \(t\) is the number of years after \(2008 .\) Find the limit of the difference quotient for \(p\) to obtain a function \(p(t)\) that represents the instantaneous rate of change of population at time \(t.\)

Step-by-Step Solution

Answer

The instantaneous rate of change is \( p'(t) = \frac{1.2}{2\sqrt{t}} \).

Step 1: Understand the Difference Quotient

The difference quotient is used to find the average rate of change of a function over a small interval. It is given by the formula \( \frac{p(t+h) - p(t)}{h} \), where \( h \) is a small increment in \( t \). Our goal is to find the limit as \( h \to 0 \) to obtain the instantaneous rate of change.

Step 2: Write Expression for Difference Quotient

Substitute the given population function into the difference quotient: \( \frac{(1.2\sqrt{t+h} + 40) - (1.2\sqrt{t} + 40)}{h} \). Simplify the expression to \( \frac{1.2\sqrt{t+h} - 1.2\sqrt{t}}{h} \).

Step 3: Simplify the Numerator

Factor out 1.2 from the numerator: \( 1.2 (\sqrt{t+h} - \sqrt{t}) \). The expression is now \( \frac{1.2(\sqrt{t+h} - \sqrt{t})}{h} \).

Step 4: Rationalize the Numerator

Multiply the numerator and the denominator by the conjugate of the numerator, \( \sqrt{t+h} + \sqrt{t} \). This gives \( \frac{1.2(\sqrt{t+h} - \sqrt{t})(\sqrt{t+h} + \sqrt{t})}{h(\sqrt{t+h} + \sqrt{t})} \).

Step 5: Simplify the Expression

Use the identity \( (a-b)(a+b) = a^2 - b^2 \) to simplify the numerator, resulting in \( (t+h) - t = h \). The expression now becomes \( \frac{1.2h}{h(\sqrt{t+h} + \sqrt{t})} \). Cancel \( h \) from the numerator and denominator.

Step 6: Calculate the Limit as h approaches 0

After canceling, the expression simplifies to \( \frac{1.2}{\sqrt{t+h} + \sqrt{t}} \). As \( h \to 0 \), \( \sqrt{t+h} \to \sqrt{t} \), simplifying the fraction to \( \frac{1.2}{2\sqrt{t}} \).

Step 7: Determine the Instantaneous Rate of Change

The instantaneous rate of change of the population function \( p(t) \) is given by the derivative \( p'(t) = \frac{1.2}{2\sqrt{t}} \).

Key Concepts

Difference Quotient

The difference quotient is an essential concept in calculus and functions as a tool for finding the average rate of change of a function over a small interval. This is often a first step in understanding how a function behaves over time. When you have a function, like our population model, the difference quotient is given by the formula:
\[ \frac{p(t+h) - p(t)}{h} \]This formula compares the function values at two points separated by a small interval \(h\).

• The goal is to evaluate the function's behavior as this interval becomes extremely small (approaches zero).
• This shrinking interval helps move from an average rate of change to an instantaneous one.

When you calculate this for a specific function, you're preparing to explore its instantaneous rate of change through limits.

Limit Calculation

Calculating the limit of the difference quotient is the next step to find the instantaneous rate of change. This involves taking the limit as \( h \to 0 \):
\[ \lim_{h \to 0} \frac{1.2(\sqrt{t+h} - \sqrt{t})}{h} \]By rationalizing and simplifying this expression, you effectively calculate how fast the population is changing at any specific moment.

• This step transforms the average rate of change to the instantaneous rate of change.
• Limits pave the way to derivatives, capturing a snapshot of the function's behavior.

Understanding limits is crucial to grasp how functions behave in an infinitely small time step.

Population Model

Our population model describes the population growth over time using the equation:
\[ p(t)=1.2 \sqrt{t}+40 \]Here, \(p\) is the population in thousands, and \(t\) is years since 2008.

• This model predicts changes in population, providing an overall understanding of growth patterns.
• Applying calculus concepts like difference quotients and limits reveals specific growth rates at given instances.

Using mathematical models, we can make sense of natural phenomena and predict future trends.

Derivative

The derivative of a function is a powerful tool that gives the instantaneous rate of change. In this context, it tells us how quickly the population changes at any given time \(t\). After calculating the limit of the difference quotient, the derivative of our population model is:
\[ p'(t) = \frac{1.2}{2\sqrt{t}} \]This derivative gives a precise measurement of the rate at which the population grows or shrinks.

• Derivatives allow us to understand the dynamics of changing phenomena.
• In this case, they offer insights into population increments year by year.

By understanding the derivative, you can predict changes accurately and gain insights into the underlying function's behavior.