Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Graph each finction in the standand viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then fill in the blank of the following sentence with either increasing or decreasing. $$f(x)=x^{2}-2 x ;(1, \infty)$$

Step-by-Step Solution

Answer

The function is increasing on the interval (1, ∞).

Step 1: Identify the Function and Interval

The given function is \(f(x) = x^2 - 2x\). We are tasked with investigating the behavior of this function over the interval \((1, \infty)\).

Step 2: Understand the Derivative

To determine whether the function is increasing or decreasing, we first find its derivative. The derivative of \(f(x) = x^2 - 2x\) is \(f'(x) = 2x - 2\).

Step 3: Analyze the Derivative on the Interval

Since we are interested in the interval \((1, \infty)\), substitute a value from this interval into the derivative to check its sign. At the lower bound, \(x = 1\), \(f'(x) = 2(1) - 2 = 0\). For any \(x > 1\), \(f'(x) = 2x - 2\) will be positive since 2x will always exceed 2.

Step 4: Conclude Increasing or Decreasing Behavior

Since \(f'(x) > 0\) for \(x > 1\), the function \(f(x) = x^2 - 2x\) is increasing on the interval \((1, \infty)\).

Key Concepts

Derivative Analysis

The derivative of a function helps us understand its rate of change at any given point. For the function \( f(x) = x^2 - 2x \), we calculate the derivative, \( f'(x) = 2x - 2 \). This expression represents the slope of the tangent line to the function at any point \( x \). If \( f'(x) \) is positive, the function is climbing upward at that point, meaning it's increasing. If \( f'(x) \) is negative, the function is descending. Derivative analysis offers a mathematical way to assess whether a function's output (or \( y \)-value) is getting larger or smaller as \( x \) gets larger, without actually having to draw the entire graph. This can be particularly useful for predicting function behavior over an interval, such as \((1, \infty)\).

Increasing and Decreasing Functions

Determining whether a function is increasing or decreasing within a specific interval requires you to look closely at its derivative. For the interval \((1, \infty)\), for our function \( f(x) = x^2 - 2x \), we already derived that \( f'(x) = 2x - 2 \). To see if the function is increasing or decreasing, we focus on the sign of this derivative.

• At the lower bound of the interval, \( x = 1 \), the derivative is \( 0 \).
• For any \( x > 1 \), \( f'(x) = 2x - 2 \) becomes positive. For example, try \( x = 2 \), resulting in \( f'(2) = 2(2) - 2 = 2 \).

This positive sign indicates that as \( x \) increases beyond 1, the function values are also increasing, showing that \( f(x) \) is increasing on \((1, \infty)\). Understanding this process can help predict behavior on larger scales without testing every individual point.

Function Behavior Analysis

Analyzing a function's behavior involves understanding different characteristics like where it increases or decreases, as was done using derivatives. Here, with \( f(x) = x^2 - 2x \), it's clear from the positive derivative \( f'(x) = 2x - 2 \) on the interval \((1, \infty)\) that the function is consistently increasing. Behavioral analysis goes further too:

• Identifying turning points (like where \( f'(x) = 0 \) can be key).
• In our example, at \( x = 1 \), the derivative is zero—this is the threshold between decreasing and increasing.

When \( x = 1 \) and the derivative changes sign, it indicates a potential minimum point, confirming that \( f(x) \) switches behavior from plateauing at \( x = 1 \) and then rising. This deeper look provides a robust picture of the function's tendencies, a crucial piece when sketching or analyzing function graphs. Thus, by understanding the derivative, one comprehends more about the curve's nature beyond simple slope determination.