Problem 44 - Calculus for Biology and Medicine

Question

Assume that the measurement of \(x\) is accurate within \(2 \% .\) In each case, determine the error \(\Delta f\) in the calculation of \(f\) and find the percentage error \(100 \frac{\Delta f}{f} .\) The quantities \(f(x)\) and the true value of \(x\) are given. $$ f(x)=\frac{1}{1+x}, x=4 $$

Step-by-Step Solution

Answer

The percentage error in the calculation of \(f\) is 1.6\%.

Step 1: Calculate the Function Value

First, substitute the given value of \(x\) into the function \(f(x)\) to find \(f(4)\). The function is \(f(x) = \frac{1}{1+x}\), so \(f(4) = \frac{1}{1+4} = \frac{1}{5} = 0.2\).

Step 2: Determine the Error in x

The problem states that the measurement of \(x\) is accurate within \(2\%\). This means the error in \(x\), \(\Delta x\), is \(2\%\) of \(4\). Calculate \(\Delta x = 0.02 \times 4 = 0.08\).

Step 3: Derive the Expression for Δf

To find \(\Delta f\), the error in \(f\), we find the derivative of \(f(x)\) with respect to \(x\). The derivative is \(f'(x) = \frac{-1}{(1+x)^2}\).

Step 4: Calculate the Error in f

Use the derivative to compute \(\Delta f\):\[\Delta f = |f'(x) \cdot \Delta x| = \left|\frac{-1}{(1+4)^2} \cdot 0.08 \right| = \left|\frac{-1}{25} \cdot 0.08 \right| = \frac{0.08}{25} = 0.0032.\]

Step 5: Compute the Percentage Error

The percentage error is given by \(100 \, \frac{\Delta f}{f}\). Substitute \(\Delta f = 0.0032\) and \(f = 0.2\) into the formula: \[100 \, \frac{0.0032}{0.2} = 1.6\%.\]

Key Concepts

Error Analysis

Error analysis involves understanding how inaccuracies in measurements affect the overall result of a calculation. For many calculations, the result is dependent on certain input values, which may not be perfectly precise. Analyzing error helps us gauge how these slight variances in input can lead to changes in the outcome. In this context, we look at two different types of errors:

• Absolute Error: The absolute difference between the measured value and the true value.
• Relative Error: The ratio of the absolute error to the true value, often expressed as a percentage, known as percentage error.

Error analysis typically involves the use of derivatives and other calculus tools to approximate and manage these errors, often yielding a more profound understanding of the reliability of measurement and results.

Derivative

The derivative of a function represents the rate at which the function's value changes as its input changes. It's a cornerstone concept in calculus, providing a way to calculate an instantaneous rate of change.For the function given, \[ f(x) = \frac{1}{1+x} \]The derivative, denoted as \( f'(x) \), gives us:\( f'(x) = \frac{-1}{(1+x)^2} \)This expression tells us how the function \( f(x) \) changes with respect to small changes in \( x \). In the context of error analysis, this derivative helps determine the change (or error) in the function value \( f \) when there is a small change in \( x \). By computing \( f'(x) \) at a specific point, we can estimate how much \( f \) will change when \( x \) changes by a given small amount.

Percentage Error

Percentage error quantifies the degree to which the calculated value deviates from the exact value, expressed as a percentage. It's a vital tool for comparing the size of absolute errors in relation to the size of the measured quantity.To find the percentage error, we first need to determine the absolute error \( \Delta f \), which in our case, using the derivative, is calculated as:\[ \Delta f = |f'(x) \cdot \Delta x| \]Once we have \( \Delta f \), the percentage error can be calculated using:\[ 100 \cdot \frac{\Delta f}{f} \]This formula provides the percentage by which the function's value \( f \) deviates due to inaccuracies in the measurement of \( x \). In the example given, this results in a percentage error of 1.6%.

Mathematical Functions

Mathematical functions express the relationship between different quantities and variables. They are fundamental concepts in mathematics used to describe real-world phenomena. Functions take an input, apply a mathematical operation, and produce an output.For this problem, as a simple function, we have:\[ f(x) = \frac{1}{1+x} \]The aim is to understand how variations in the input \( x \) impact the output \( f(x) \). Small errors or changes in \( x \) can lead to errors in \( f(x) \), which is critical in engineering and scientific computations where precision is crucial. Evaluating these impacts using calculus, such as derivatives, allows us to assess and estimate errors that arise naturally from measured data.