Problem 51 - 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 \(x_{5}=2\) $$\sum_{i=1}^{3}\left(3 x_{i}-x_{i}^{2}\right)$$

Step-by-Step Solution

Answer

The sum is -14.

Step 1: Understanding the Series

The problem requires us to evaluate the sum \(\sum_{i=1}^{3}(3x_i-x_i^2)\) for given values of \(x_i\). We will substitute the first three values of \(x_i: x_1, x_2, x_3\) into the expression and calculate the result for each term.

Step 2: Calculate First Term

Substitute \(x_1 = -2\) into the expression: \(3x_1 - x_1^2\). Calculate: \[3(-2) - (-2)^2 = -6 - 4 = -10\]

Step 3: Calculate Second Term

Substitute \(x_2 = -1\) into the expression: \(3x_2 - x_2^2\). Calculate:\[3(-1) - (-1)^2 = -3 - 1 = -4\]

Step 4: Calculate Third Term

Substitute \(x_3 = 0\) into the expression: \(3x_3 - x_3^2\). Calculate:\[3(0) - 0^2 = 0\]

Step 5: Summing Up the Terms

Now, add up the results from each individual term obtained in Steps 2 to 4:\[-10 + (-4) + 0 = -14\]

Step 6: Present the Result

The sum of the series \(\sum_{i=1}^{3}(3x_i-x_i^2)\) for the given values is \(-14\).

Key Concepts

Summation

Summation is a mathematical process of adding up multiple numbers, usually represented by the Greek letter \(\Sigma\). It allows us to compactly express the sum of a series of terms without having to write out each addition separately. In this exercise, we were tasked with computing a summation of terms of the form \(3x_i - x_i^2\) for three distinct values of \(x_i\). The index \(i\) in \(\sum_{i=1}^3\) denotes the range over which the terms are summed, which in this case is from 1 to 3. This approach is particularly useful when the expression is complex or involves many terms. It simplifies calculations and provides a clear way to express the addition of potentially countless numbers. Each term in the summation is calculated separately, and the results are then added together to find the total sum.

Series Evaluation

Series evaluation involves calculating the result of a series—another way to describe summation, but it often implies an infinite or finite number of terms that are related to each other in a consistent pattern. In our exercise, the series is finite, with terms limited to \(i = 1\), \(2\), and \(3\). To evaluate this series, each term of the expression \(3x_i - x_i^2\) is substituted with the corresponding \(x_i\) value provided in the problem \(x_1 = -2\), \(x_2 = -1\), and \(x_3 = 0\). By computing each term step by step and then summing them, we derived the total series evaluation. This exercise reinforces the idea of taking methodical steps to handle each part of the process and ensures accuracy in algebraic manipulations.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and arithmetic operations. In this exercise, the expression \(3x_i - x_i^2\) was evaluated multiple times with different values of \(x_i\). This expression features both linear (\(3x_i\)) and quadratic (\(-x_i^2\)) components.

• The linear part, \(3x_i\), suggests a straight-line relationship with \(x_i\), where the change in \(x_i\) results in a proportional change in value.
• The quadratic part, \(-x_i^2\), introduces a parabolic effect, which is non-linear; this means the value changes at a rate proportional to the square of \(x_i\).

Understanding how each part of the expression contributes to the overall result is key in simplifying and evaluating algebraic expressions. When combined in the exercise's expression, they required careful computation to ensure clarity and accuracy in problem-solving.