Problem 37 - Calculus for Biology and Medicine

Question

A measurement error in \(x\) affects the accuracy of the value \(f(x) .\) In each case, determine an interval of the form $$[f(x)-\Delta f, f(x)+\Delta f]$$ that reflects the measurement error \(\Delta x .\) In each problem, the quantities given are \(f(x)\) and \(x=\) true value of \(x \pm|\Delta x| .\) . \(f(x)=3 x^{2}, x=2 \pm 0.1\)

Step-by-Step Solution

Answer

The interval of \( f(x) \) given the error in \( x \) is \([10.83, 13.23]\).

Step 1: Identify the Function and Error

The function given is \( f(x) = 3x^2 \) and the true value of \( x \) is \( 2 \pm 0.1 \). This means \( x \) ranges from 1.9 to 2.1. We need to determine the effect of this range on \( f(x) \).

Step 2: Evaluate the Function at the Boundaries

Calculate \( f(x) \) at the boundary values of \( x \). For \( x = 1.9 \), \( f(1.9) = 3(1.9)^2 \). For \( x = 2.1 \), \( f(2.1) = 3(2.1)^2 \).

Step 3: Calculate Specific Function Values

- For \( x = 1.9 \), \( f(1.9) = 3 \times 1.9^2 = 3 \times 3.61 = 10.83 \).- For \( x = 2.1 \), \( f(2.1) = 3 \times 2.1^2 = 3 \times 4.41 = 13.23 \).

Step 4: Determine the Interval of \( f(x) \)

The interval for \( f(x) \) considering the error in \( x \) is \([10.83, 13.23]\) because these are the values of \( f(x) \) calculated at the extremes of the possible \( x \) values.

Key Concepts

Understanding Measurement Error

Measurement error refers to the uncertainty or deviation in the values obtained from measuring a particular quantity. In calculus, when measuring an input value \(x\), small errors can affect the results computed from a function \(f(x)\). For example, if \(x\) is given as \(2 \pm 0.1\), this shows that \(x\) might range between 1.9 and 2.1. The presence of measurement error necessitates consideration of how \(f(x)\) changes within this range. When dealing with functions, it is crucial to understand how these minor variations impact the resulting function values. This highlights the importance of acknowledging and addressing potential errors that could affect predictions or results. Understanding and deterministically incorporating measurement error ensures conclusions drawn from calculations are reliable. This consideration is vital for rigorous error analysis in calculus, which can avoid significant inaccuracies resulting from overlooking measurement errors.

Interval Estimation and Its Role

When dealing with functions and their computations, interval estimation provides a range within which the values of \(f(x)\) are expected to fall, based on the possible range of \(x\). Instead of providing a single value, interval estimation offers a range \([f(x)-\Delta f, f(x)+\Delta f]\) that represents the possible outcomes given the input variability.In our example with \( f(x) = 3x^2 \) and \(x = 2 \pm 0.1\), the range of \(x\) demands us to consider how the function behaves at its boundaries \(x = 1.9\) and \(x = 2.1\). By computing \(f(1.9)\) and \(f(2.1)\), we estimate the error range in \(f(x)\) from the provided \(x\) range.This type of estimation is crucial when an exact value is difficult to obtain or prone to inaccuracies due to measurement or observed variability. Providing a confidence range helps mitigate the impact of potential inaccuracies and informs better decision-making.

Function Evaluation with Errors

Function evaluation involves calculating specific outputs \(f(x)\) from given inputs \(x\). When faced with input errors or uncertainties, it becomes essential to determine how these affect the computed function values. Evaluating a function at various possible values of \(x\), especially at its extremities, helps understand the full impact of input errors.In the example \(f(x) = 3x^2\) where \(x = 2 \pm 0.1\), conducting evaluations at \(x = 1.9\) and \(x = 2.1\) gives results that frame the function's behavior over this error span. We calculate \(f(1.9) = 10.83\) and \(f(2.1) = 13.23\), thus obtaining an interval estimate.These evaluations visualize the practical effect of input variations and provide insights into managing measurement errors. This approach ensures that prediction and modeling in calculus remain as accurate as possible even when uncertainties are present.