Problem 324 - Calculus Volume 1

Question

For the following exercises, consider the construction of a pen to enclose an area. A patient’s pulse measures 70 bpm, 80 bpm, then 120 bpm. To determine an accurate measurement of pulse, the doctor wants to know what value minimizes the expression \((x-70)^{2}+(x-80)^{2}+(x-120)^{2} ?\) What value minimizes it?

Step-by-Step Solution

Answer

The value that minimizes the expression is 90.

Step 1: Understand the expression

We need to minimize the expression \((x-70)^2 + (x-80)^2 + (x-120)^2\). This expression represents the sum of the squares of the differences between \(x\) and each of the given pulse measurements (70 bpm, 80 bpm, and 120 bpm).

Step 2: Find the mean of the pulse rates

The expression is minimized when \(x\) is at its average value, known as the mean. Calculate the mean of the pulse measurements: \[\text{Mean} = \frac{70 + 80 + 120}{3} = \frac{270}{3} = 90\]

Step 3: Verify the minimum

Substitute \(x = 90\) back into the original expression to verify it results in a minimum:\[(90 - 70)^2 + (90 - 80)^2 + (90 - 120)^2 = 20^2 + 10^2 + (-30)^2 = 400 + 100 + 900 = 1400\]This is indeed the minimum value of the expression.

Key Concepts

Mean of a data set

Let's dive into the concept of the mean, which is a fundamental part of statistics. The mean, often referred to as the average, is a representation of the central value of a set of numbers. To calculate the mean, simply add up all the numbers in your data set, and then divide that sum by the number of elements in the set.

In this exercise, we have heart rate measurements of 70 bpm, 80 bpm, and 120 bpm. Adding these values gives us 270, and since there are three measurements, we divide 270 by 3 to find that the mean is 90 bpm. By finding the mean, we identify the value that best represents all measurements in the set, helping us make predictions or decisions based on the data.

The mean is significant because it accounts for every element in the data set, providing a balanced view of all values.

Sum of squares

Sum of squares is a mathematical concept often used in statistics to measure the variation or spread of a data set. It is calculated by squaring each deviation from the mean, and then summing those squared deviations.

This technique helps in understanding how much individual measurements deviate from the average value, a crucial aspect in data analysis. In optimization problems like the one in our exercise, the sum of squares indicates the discrepancy between a tentative value, here called \(x\), and the given measurements.

For instance, for our pulse rates, the expression \((x-70)^2 + (x-80)^2 + (x-120)^2\) is the sum of squares. The goal is to find an \(x\) that minimizes this expression, effectively reducing the total discrepancy from the set measurements. Lowering the sum of squares implies a lessened overall difference or error, achieving a more accurate representation of the data set.

Optimization problem

Optimization problems are prevalent in mathematics, where the objective is to find the best possible solution or outcome under given circumstances. Typically, these problems involve either maximizing or minimizing a function.

In this exercise, we're tasked with minimizing the expression \((x-70)^2 + (x-80)^2 + (x-120)^2\), which means finding the value of \(x\) that results in the smallest possible sum of squares. This form of optimization ensures that \(x\) closely aligns with all the measurements, providing the most accurate estimate of the patient's pulse rate.

Optimization is vital in many fields including economics, engineering, and science, as it aids in decision-making and resource allocation. By systematically assessing possible solutions, optimization focuses on achieving the most favorable outcome given the constraints or parameters in the situation.