Problem 37 - Precalculus: Mathematics for Calculus

Question

A Function from a Description Find a formula from a function \(f\) that satisfies the following conditions. Asymptote Behavior How close to \(-3\) do we have to take \(x\) so that $$\frac{1}{(x+3)^{2}}>10,000$$

Step-by-Step Solution

Answer

\( x \) must be between \( -3.01 \) and \( -2.99 \).

Step 1: Understanding the Problem

We need to look at the condition \( \frac{1}{(x+3)^{2}} > 10,000 \) and determine how close \( x \) needs to be to \( -3 \) to satisfy it.

Step 2: Solving the Inequality

Rewrite the inequality \( \frac{1}{(x+3)^{2}} > 10,000 \) as \( (x+3)^{2} < \frac{1}{10,000} \).

Step 3: Simplifying the Square

Take the square root of both sides of \( (x+3)^{2} < \frac{1}{10,000} \) to get \( |x+3| < \frac{1}{100} \).

Step 4: Understanding the Absolute Value

The inequality \( |x+3| < \frac{1}{100} \) means that the distance between \( x \) and \(-3\) is less than \( \frac{1}{100} \).

Step 5: Finding the Range of x

Solve \( -\frac{1}{100} < x+3 < \frac{1}{100} \) by subtracting 3 from all parts, giving \( -3.01 < x < -2.99 \).

Key Concepts

Inequality Solutions

Inequalities can seem tricky, but once you know the basic steps, they become much easier to handle. In our exercise, the inequality given is \( \frac{1}{(x+3)^{2}} > 10,000 \). To find the solution, we begin by manipulating the inequality to make it simpler to solve.
First, we rewrite the inequality as \( (x+3)^{2} < \frac{1}{10,000} \). By rewriting it, we shift our focus from dividing to multiplying, which is more straightforward. The central goal in solving inequalities is to find the range of values that make the inequality true.

• Step 1: Restructure the inequality into a more manageable form.
• Step 2: Make sure to perform the same operation on both sides of the inequality to keep it balanced. In this case, we multiplied both sides by \((x+3)^2\) and took the reciprocal.
• Step 3: Continue transforming the inequality until it is in a form that is easy to interpret.

This process ultimately revealed that the function's behavior is controlled around the asymptote at \( x = -3 \), indicating the range for \( x \) is carefully constrained.

Absolute Value

Absolute value is a mathematical tool that helps express the magnitude of a number, regardless of its sign. In essence, it measures how far a number is from zero. For example, \(|-5|\) and \(|5|\) both equal 5.
In the exercise, we encounter \(|x+3|<\frac{1}{100}\). This statement describes the absolute value, or magnitude, of \(x+3\), informing us about the distance of \(x\) from \(-3\).
When dealing with absolute values in inequalities, the notation \(|x| < a \) can be transformed into a compound inequality: \(-a < x < a\). This method allows us to directly see the bounds within which \(x\) resides.

• \(|x + 3| < \frac{1}{100}\) indicates that \(x\) must be within \(\frac{1}{100}\) units of \(-3\).
• The absolute value captures all possible values within this interval.
• Use absolute values when the equation involves distances or when direction is irrelevant.

Essentially, absolute values simplify the process of finding a range of solutions without worrying about positive or negative directions.

Distance Between Numbers

The concept of distance in mathematics often involves measuring how far apart two numbers are on the number line. It’s a crucial aspect in various fields, ranging from basic math to advanced calculus.
In this case, the inequality \(|x+3| < \frac{1}{100}\) represents the distance between \(x\) and \(-3\). It essentially tells us how close \(x\) needs to be to \(-3\) for the inequality condition to hold true.

• To visualize, think of a number line where \(-3\) is the central point.
• The inequality states that \(x\) must reside within \(\frac{1}{100}\) units on either side of \(-3\).
• This range is crucial for ensuring the condition is satisfied given the problem's constraints.

Understanding mathematical distances helps in interpreting and solving inequalities, allowing us to locate and define precise solution sets. This understanding is beneficial not only in math but also in real-world applications, where measuring and defining parameters is often required.