Problem 52 - A Graphical Approach to Precalculus with Limits

Question

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\) $$\sum_{i=1}^{3}\left(x_{i}^{2}+1\right)$$

Step-by-Step Solution

Answer

The sum is 8.

Step 1: Identify the Terms to Evaluate

We need to evaluate the terms inside the summation from \(i=1\) to \(i=3\). This summation is represented as \(\sum_{i=1}^{3}(x_i^2 + 1)\). Therefore, we will calculate \(x_1^2 + 1\), \(x_2^2 + 1\), and \(x_3^2 + 1\).

Step 2: Calculate the First Term

To find the first term, substitute \(x_1 = -2\) into the expression \(x_1^2 + 1\). \[ (-2)^2 + 1 = 4 + 1 = 5 \]

Step 3: Calculate the Second Term

Substitute \(x_2 = -1\) into the expression \(x_2^2 + 1\). \[ (-1)^2 + 1 = 1 + 1 = 2 \]

Step 4: Calculate the Third Term

Substitute \(x_3 = 0\) into the expression \(x_3^2 + 1\). \[ 0^2 + 1 = 0 + 1 = 1 \]

Step 5: Add the Terms Together

Now that we have our values: \(5\), \(2\), and \(1\), add them together for the final result of the sum. \[ 5 + 2 + 1 = 8 \]

Key Concepts

Evaluating Expressions

When we talk about evaluating expressions, we're referring to the process of finding the value of a mathematical expression by substituting numbers for variables. It’s like solving a puzzle where each number fits into a specific place. In our exercise, we evaluate expressions like \(x_i^2 + 1\) by plugging in the given values of \(x_i\).
Evaluating involves:

• Identifying what needs to be calculated.
• Substituting each term correctly.
• Following mathematical operations step-by-step.

By substituting the values into specific points, you ensure accuracy in solving the expression. This method is foundational in precalculus and is essential for understanding more complex problems later on.

Step-by-Step Solutions

A step-by-step solution is like a map guiding you through a mathematical problem. Each step carefully builds on the previous one, ensuring clarity and reducing errors. In our original problem, we used a systematic approach to find the sum of squares:

• Identify the terms: Recognize which values need evaluation.
• Substitute carefully: Make sure each value replaces the correct variable.
• Calculate meticulously: Order of operations is key. Follow addition, subtraction, multiplication, and exponentiation as required.
• Add the results: Once individual terms are evaluated, combine them to get the final result.

Following each step precisely paves the way to a successful and clear solution, which becomes particularly important as the problems grow in complexity.

Sum of Squares

The sum of squares is a common concept that appears frequently in mathematics, especially in statistics and algebra. It involves taking each term, squaring it, and then adding them together. In our problem, we looked at an expression involving squares: \(x_i^2 + 1\).
A squared term, like \((x_i)^2\), means multiplying the number by itself. It reveals patterns and relationships in numbers that might not be obvious at first glance.
Below are important aspects of working with sums of squares:

• Calculate each square individually before performing other operations to maintain accuracy.
• Understand that squaring can affect values significantly, especially larger numbers.
• Use the sum of squares in various contexts such as variance in statistics or Pythagorean theorem in geometry.

Understanding the sum of squares can help in analyzing data, solving equations, and interpreting mathematical models.

Precalculus Concepts

Precalculus is a branch of mathematics that prepares students for calculus by offering a comprehensive overview of algebra and trigonometry. Concepts from precalculus, such as summation notation, lay the groundwork for understanding more advanced topics in mathematics. Our exercise involves:

• Summation Notation: Notates the addition of a sequence of terms. For example, \(\sum_{i=1}^{3}(x_i^2 + 1)\), where 'sum' indicates combining several results.
• Functions and Patterns: Recognizing functional relationships between variables, like how \(x^2\) increases as \(x\) increases.
• Expressions Simplification: Break down and resolve expressions step-by-step, reinforcing accuracy and precision.

These concepts are not only crucial for further studies in calculus but also enhance logical thinking and problem-solving skills in various disciplines.