Problem 16 - Calculus and its applications

Question

Express \(\sum_{i=1}^{4} g\left(x_{i}\right)\) without using summation notation.

Step-by-Step Solution

Answer

The summation is expressed as \( g(x_1) + g(x_2) + g(x_3) + g(x_4) \).

Step 1: Understand the summation notation

The expression \( \sum_{i=1}^{4} g(x_{i}) \) represents the sum of the function \( g(x_i) \) evaluated at certain points \( x_1, x_2, x_3, \) and \( x_4 \). The index \( i \) runs from 1 to 4, indicating that there are 4 terms in the sum.

Step 2: Replace the summation notation with individual terms

To express the summation without the notation, write out each term separately. This means evaluating the function \( g \) at each of the indices from 1 to 4: \( g(x_1), g(x_2), g(x_3), \) and \( g(x_4) \).

Step 3: Write the full expression

Combine the terms from Step 2 into a single expression: \( g(x_1) + g(x_2) + g(x_3) + g(x_4) \). This expression represents the sum as a series of additions without using the summation symbol.

Key Concepts

Index

In summation notation, the index is an essential component that tells you the variable at play. It often appears as a letter placed below the summation symbol \( \Sigma \). This index provides important information about where to start and end the summation process.

• For example, in the notation \( \sum_{i=1}^{4} \), the "i" represents the index.
• Here, the index \( i \) travels from 1 to 4, which means you will evaluate the function at four different points, specifically \( x_1, x_2, x_3, \) and \( x_4 \).
• The index serves as a counter that guides you through each iteration of the sum.

When breaking down a summation, keep an eye on the index as it indicates how many terms you need to account for and ensures each part of the expression gets included.

Function Evaluation

Function evaluation is the process of determining the value of a function given specific inputs. In a summation problem, after identifying the index and its range, you proceed with function evaluation. This step involves inserting each index value into the function.

• For \( g(x_i) \) in the sum \( \sum_{i=1}^{4} g(x_{i}) \), you evaluate "g" at each corresponding \( x_i \).
• For instance, you would find \( g(x_1) \), \( g(x_2) \), \( g(x_3) \), and \( g(x_4) \).
• This evaluation is crucial because it reveals the individual contributions of each term to the total sum.

Each function evaluation results in a numerical value or expression that represents its role within the overall sum. Mastering this step is key for converting summation notation into a detailed algebraic expression.

Mathematical Expression

A mathematical expression is essentially a combination of numbers, variables, and operators that collectively convey a value or relationship. Changing the summation into a mathematical expression involves sequentially displaying each evaluated result.

• From the starting point, create an articulate series that represents each term from the summation.
• For example, the expression \( g(x_1) + g(x_2) + g(x_3) + g(x_4) \) represents the full expansion of the sum.
• Each term in this expression corresponds to the function evaluated at each index value.

This step eliminates the concise summation symbol, instead turning the notation into a clear, detailed list of additions. Expressions like these are more straightforward for some students to understand and work with because they lay out every part of the original notation explicitly.