Problem 20 - Thomas Calculus

Question

Let \(f(t)=1 / t\) for \(t \neq 0.\) a. Find the average rate of change of \(f\) with respect to \(t\) over the intervals (i) from \(t=2\) to \(t=3,\) and (ii) from \(t=2\) to \(t=T\) . b. Make a table of values of the average rate of change of \(f\) with respect to \(t\) over the interval \([2, T],\) for some values of \(T\) approaching \(2,\) say \(T=2.1,2.01,2.001,2.0001,2.00001\) and \(2.000001 .\) c. What does your table indicate is the rate of change of \(f\) with respect to \(t\) at \(t=2 ?\) d. Calculate the limit as \(T\) approaches 2 of the average rate of change of \(f\) with respect to \(t\) over the interval from 2 to \(T=2\) . will have to do some algebra before you can substitute \(T=2.\)

Step-by-Step Solution

Answer

The rate of change of \( f \) at \( t = 2 \) is \(-0.25\).

Step 1: Calculate average rate of change for t from 2 to 3

The average rate of change of a function \( f(t) \) from \( t = a \) to \( t = b \) is given by the formula \( \frac{f(b) - f(a)}{b - a} \). We have \( f(t) = \frac{1}{t} \), \( a = 2 \), and \( b = 3 \). Calculate \( f(2) \) and \( f(3) \):\[ f(2) = \frac{1}{2}, \quad f(3) = \frac{1}{3} \]Now, plug these into the formula:\[ \frac{\frac{1}{3} - \frac{1}{2}}{3 - 2} = \frac{-\frac{1}{6}}{1} = -\frac{1}{6} \]

Step 2: Calculate average rate of change for t from 2 to T

Using the same formula \( \frac{f(b) - f(a)}{b - a} \), with \( a = 2 \) and \( b = T \), we have:\[ f(T) = \frac{1}{T} \]So, the average rate of change is:\[ \frac{\frac{1}{T} - \frac{1}{2}}{T - 2} = \frac{\frac{1}{T} - \frac{1}{2}}{T - 2} \]

Step 3: Create a table for T approaching 2

Calculate the average rate of change \( \frac{\frac{1}{T} - \frac{1}{2}}{T - 2} \) for \( T = 2.1, 2.01, 2.001, 2.0001, 2.00001, \) and \( 2.000001 \):- For \( T = 2.1: \quad \frac{\frac{1}{2.1} - \frac{1}{2}}{2.1 - 2} \approx -0.238 \)- For \( T = 2.01: \quad \frac{\frac{1}{2.01} - \frac{1}{2}}{2.01 - 2} \approx -0.245 \)- For \( T = 2.001: \quad \frac{\frac{1}{2.001} - \frac{1}{2}}{2.001 - 2} \approx -0.2495 \)- For \( T = 2.0001: \quad \frac{\frac{1}{2.0001} - \frac{1}{2}}{2.0001 - 2} \approx -0.24995 \)- For \( T = 2.00001: \quad \frac{\frac{1}{2.00001} - \frac{1}{2}}{2.00001 - 2} \approx -0.249995 \)- For \( T = 2.000001: \quad \frac{\frac{1}{2.000001} - \frac{1}{2}}{2.000001 - 2} \approx -0.2499995 \)

Step 4: Interpret the table

As \( T \) approaches 2, the average rate of change \( \frac{\frac{1}{T} - \frac{1}{2}}{T - 2} \) approaches \(-0.25\). Thus, the rate of change of \( f \) with respect to \( t \) at \( t = 2 \) is \(-0.25\).

Step 5: Compute the limit

To find the limit of the average rate of change as \( T \to 2 \), use the formula:\[ \lim_{{T \to 2}} \frac{\frac{1}{T} - \frac{1}{2}}{T - 2} \]We can simplify:\[ \frac{\frac{1}{T} - \frac{1}{2}}{T - 2} = \frac{2 - T}{2T(T - 2)} = \frac{-1}{2T} \]Taking the limit as \( T \to 2 \):\[ \lim_{{T \to 2}} \frac{-1}{2T} = \frac{-1}{4} \]The limit confirms that the rate of change of \( f \) at \( t = 2 \) is \(-0.25\).

Key Concepts

Understanding Functions

A function is a rule that assigns each input exactly one output. In mathematical terms, it is a relationship where you use an input value to calculate an output value. For example, consider the function \( f(t) = \frac{1}{t} \). Here, \( t \) is the input and \( f(t) \) provides the output by using the rule of dividing 1 by the input \( t \).
Functions can be represented in various forms, such as equations, graphs, or tables. They are fundamental in mathematics because they model relationships between quantities.

Key points to remember about functions:

• A single input gives a single output, ensuring the function is well-defined.
• They can represent physical, economical, or statistical models in real-life scenarios.
• Functions help predict outcomes based on given inputs, making them useful in numerous fields.

In the exercise, the function \( f(t) \) is used to determine how the value changes as \( t \) changes, illustrating how functions have real-world applications and utility.

Limits and Their Significance

The concept of limits is central to calculus and is used to understand the behavior of functions as inputs approach certain values. A limit examines what a function's output gets closer to as the input gets closer to a specific value.
In the exercise, you observed the function \( f(t) = \frac{1}{t} \), particularly as \( t \) approaches specific values. As \( T \) approaches 2, you calculated the average rate of change and noticed it approached a particular number, illustrating the limit concept.

Here's why limits are important:

• They allow us to address cases where a function might not be well-defined at a particular point.
• They help find the exact rate at which a function is changing as it nears a particular input value.
• Limits bridge the transition from average rate of change to instantaneous rate of change.

The exercise shows that determining the limit as \( T \) approaches 2 tells us about the behavior of the function close to that point, crucial for further analysis in calculus.

Derivatives and Instantaneous Rates

A derivative represents the instantaneous rate of change of a function as compared to just the average rate. It describes how a function's output value changes rapidly at a specific point. In shorter words, it is the slope of the tangent line to the function at that point.
In the context of the exercise, by studying the limit of the average rate of change as \( T \) approaches 2, you were led to the concept of the derivative. The limit you calculated, ending up at \(-0.25\), actually represents the derivative of \( f(t) \) at \( t = 2 \).

Essential aspects of derivatives:

• They help determine how a function behaves locally, around particular values.
• Derivatives are useful in various applications, from physics to economics, where understanding how things change immediately is crucial.
• They pave the way for solving many complex problems involving change, using concepts like optimization and curve sketching.

By understanding and applying the derivative through limits, you gain deeper insights into how functions change and interact, which is a foundational skill in calculus.