Problem 88 - Calculus Volume 1

Question

For the following exercises, use a calculator to graph \(f(x)\) . Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$f(x)=1+x+\frac{1}{x}$$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 1 - \frac{1}{x^2} \).

Step 1: Find the Derivative of f(x)

Given the function \( f(x) = 1 + x + \frac{1}{x} \), we need to find \( f'(x) \), the derivative of \( f(x) \). The derivative of \( 1 \) is \( 0 \), the derivative of \( x \) is \( 1 \), and the derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \). Thus, \( f'(x) = 0 + 1 - \frac{1}{x^2} = 1 - \frac{1}{x^2} \).

Step 2: Graph f(x) Using a Calculator

Use a graphing calculator or graphing software to plot the function \( f(x) = 1 + x + \frac{1}{x} \). Observe the general shape and behavior of the graph, particularly noting any asymptotes or intercepts.

Step 3: Graph f'(x) Using a Calculator

Now graph the derivative \( f'(x) = 1 - \frac{1}{x^2} \) using a calculator. This graph will show where the function \( f(x) \) is increasing or decreasing, and help identify any critical points where the slope changes from positive to negative or vice versa.

Key Concepts

Graphing Functions

Graphing functions is an essential skill in calculus. It helps visualize how a function behaves over different intervals of its domain. In this exercise, we focus on the function:

• \( f(x) = 1 + x + \frac{1}{x} \)

Plotting this function requires a graphing calculator or software due to its complexity. When graphing \( f(x) \), observe the following:

• **Intercepts**: Where the graph crosses the x-axis and y-axis.
• **Asymptotes**: Notice any vertical asymptotes, which occur at values of \( x \) that make the function undefined, such as \( x = 0 \) in this case.
• **Intervals**: Identify which intervals the function increases or decreases.

By understanding these features, you gain insights into the overall behavior of the function, crucial for interpreting real-life situations described by such functions.

Computational Tools in Mathematics

In modern mathematics, computational tools like graphing calculators or computer software play a significant role. They allow students and professionals to visualize complex functions quickly. For this problem:

• Plotting functions using a calculator aids in validating manual computations.
• It simplifies the task of identifying graph features, such as intercepts and asymptotes.
• Such tools provide an interactive means to manipulate functions and observe real-time changes.

By relying on these tools, you can quickly explore various mathematical scenarios, which saves time and enhances your understanding. When working with derivatives, using a calculator helps see the change in slopes and helps in solving problems related to optimization.

Derivative Calculation

Derivatives are a fundamental concept in calculus, representing the rate of change of a function. To comprehend the derivative of the function \( f(x) = 1 + x + \frac{1}{x} \), we follow:

• The constant \(1\) has a derivative of \(0\).
• The derivative of \(x\) is \(1\), as it's a linear function whose slope is constant.
• The term \(\frac{1}{x}\) differentiates to \(-\frac{1}{x^2}\), a common derivative encountered in calculus.

Thus, the derivative of \( f(x) \) becomes: \[ f'(x) = 1 - \frac{1}{x^2} \]This derivative, \( f'(x) \), allows us to evaluate where the original function \( f(x) \) increases or decreases. By graphing \( f'(x) \), we can spot critical points where changes occur, such as turning points. Understanding this derivative transformation process helps solve various calculus problems, especially those involving motion and growth.