Problem 2 - Calculus and its applications

Question

Complete each of the following statements. As x approaches_________ , the value of x - 2 approaches 5.

Step-by-Step Solution

Answer

As \( x \) approaches 7.

Step 1: Understand the Expression

The expression given is \( x - 2 \). We are asked to find a value for \( x \) such that \( x - 2 \) approaches 5.

Step 2: Set the Expression Equal to the Given Value

We want \( x - 2 \) to approach 5. Therefore, we can set the expression equal to 5: \( x - 2 = 5 \).

Step 3: Solve for x

Add 2 to both sides of the equation \( x - 2 = 5 \) to isolate \( x \). This gives \( x = 5 + 2 \).

Step 4: Calculate the Value of x

Perform the addition: \( x = 5 + 2 = 7 \).

Step 5: Conclusion

We find that as \( x \) approaches 7, the expression \( x - 2 \) approaches 5.

Key Concepts

Approaching a Value

In calculus, the idea of "approaching a value" is closely linked to the concept of limits. The limit explains how a function behaves as something gets closer to a specific point. When we say that a variable is "approaching" a value, it means that the variable is getting very close to that point, although it might not actually reach it.

Think of it like a person walking towards a door. The closer they get, the more they are "approaching" the door. In mathematical terms, if we say "as \( x \) approaches 7," it means \( x \) is getting closer and closer to 7. Even though \( x \) might not become exactly 7, we're interested in what happens to the function as we get near that value.

In the exercise, as \( x \) approaches 7, the expression \( x - 2 \) approaches 5. This kind of thinking is foundational to understanding limits and how they help us analyze functions in calculus.

Solving Equations

Solving equations is a fundamental skill in mathematics that involves finding the value of variables that make an equation true. The process usually begins with a clear understanding of the equation and proceeds with steps to isolate the variable of interest.

Let's look at the example from the exercise, where we began with the equation \( x - 2 = 5 \). Here, the goal was to find the value of \( x \).

• We start by recognizing the need to balance the equation by performing operations on both sides.
• The main step involves adding 2 to both sides, which simplifies the equation to \( x = 7 \).

Once we've found \( x \), we've "solved" the equation! Being confident in solving such problems helps you tackle more complex mathematical tasks in calculus and beyond.

Basic Algebraic Manipulation

Basic algebraic manipulation is essential for solving equations and involves rearranging and simplifying expressions. It provides the tools needed to transform equations into simpler forms, allowing us to identify solutions.

This typically includes actions such as adding, subtracting, multiplying, or dividing both sides of an equation. In our exercise, we used basic manipulation to isolate the variable \( x \) by adding 2 to both sides of the initial equation \( x - 2 = 5 \).

Another example of algebraic manipulation is like solving a puzzle. We rearrange pieces (or numbers and variables) to "complete the picture" and uncover the solution. Mastering these techniques is key to handling more advanced topics in mathematics effectively.