Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(\Delta y\) for the given values of \(x_{1}\) and \(x_{2}(\) see Example 7). $$ y=\frac{3}{x+1}, x_{1}=2.34, x_{2}=2.31 $$

Step-by-Step Solution

Answer

The change \(\Delta y\) is approximately 0.008.

Step 1: Calculate y at x1

First, find the function value at \(x_1 = 2.34\). Substitute \(x_1\) into the equation \(y = \frac{3}{x+1}\): \[ y_1 = \frac{3}{2.34 + 1} = \frac{3}{3.34} \approx 0.898 \]

Step 2: Calculate y at x2

Next, find the function value at \(x_2 = 2.31\). Substitute \(x_2\) into the equation \(y = \frac{3}{x+1}\):\[ y_2 = \frac{3}{2.31 + 1} = \frac{3}{3.31} \approx 0.906 \]

Step 3: Compute Delta y

Find \(\Delta y\) by subtracting \(y_1\) from \(y_2\):\[ \Delta y = y_2 - y_1 = 0.906 - 0.898 = 0.008 \]

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input. In this exercise, we have the function \( y = \frac{3}{x+1} \). To evaluate this function, we substitute specific values for \( x \) into the function to find \( y \).

• First, we substitute \( x_1 = 2.34 \) into the equation to find \( y_1 \).
• Then, we substitute \( x_2 = 2.31 \) to find \( y_2 \).

The purpose of function evaluation is to determine what the function's output is for specific values of \( x \). This can help us understand how the function behaves with different inputs.

Change in Output

In calculus, the change in output of a function when the input changes is often denoted as \( \Delta y \). This exercise illustrates how to calculate the change in output from one input value to another.

To find \( \Delta y \), we first calculate the output at two different input values (\( x_1\) and \( x_2\)). In our example:

• We found \( y_1 \) by evaluating the function at \( x_1 = 2.34 \).
• Next, \( y_2 \) was found by evaluating the function at \( x_2 = 2.31 \).

The change in output \( \Delta y \) is then calculated as \( y_2 - y_1 \). This measure helps us understand how much the output of the function changes in response to a slight change in the input.

Calculus Problem Solving

Calculus problem solving often involves determining how quantities change and establishing relationships between variables. This involves using tools like derivatives and function evaluations.

When solving calculus problems, there are several steps that can streamline the process:

• Identify the function and the values at which you need to evaluate it.
• Understand what is being asked—such as finding a specific change or rate of change.
• Break down the problem into smaller, manageable steps, like evaluating the function and finding differences.

This approach helps simplify complex problems using logical and structured methods.

Analytical Methods

Analytical methods in calculus involve systematic approaches to finding and interpreting values and changes within functions. By using these methods, we can solve complex mathematical problems logically and accurately.

Here, the analytical method was applied by:

• Substituting specific values into a given function to evaluate its output.
• Calculating the difference to find \( \Delta y \), representing the change in output.

Using these analytical techniques allows us to assess the behavior of functions over different intervals or segments, providing insights into larger mathematical concepts.