Problem 39 - 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 \(x \pm|\Delta x| .\) \(f(x)=e^{x}, x=2 \pm 0.2\)

Step-by-Step Solution

Answer

\([6.0496, 9.0250]\) is the interval reflecting the error.

Step 1: Identify the function and error

The given function is \( f(x) = e^x \). The true value of \( x \) is 2, with a measurement error \( \Delta x = 0.2 \). We will use this information to determine the interval \([f(x)-\Delta f, f(x)+\Delta f]\).

Step 2: Calculate the function value at nominal x

First, calculate \( f(x) \) at \( x = 2 \):\[ f(2) = e^2. \]

Step 3: Consider the maximum and minimum x values

Account for the measurement error by considering \( x = 2 \pm 0.2 \). This gives the interval for \( x \) as \([1.8, 2.2]\).

Step 4: Calculate the maximum and minimum function values

Evaluate \( f(x) \) at the bounds of the interval for \( x \): - For \( x = 1.8, f(1.8) = e^{1.8} \), and - For \( x = 2.2, f(2.2) = e^{2.2} \). Determine these values using a calculator.

Step 5: Compute the interval for f(x)

Using the calculations:- \( f(1.8) \approx 6.0496 \), and - \( f(2.2) \approx 9.0250 \).The interval for \( f(x) \) is \([6.0496, 9.0250]\).

Key Concepts

Measurement Error

Measurement error is a common issue in any form of data collection or experimentation. It refers to the difference between the true value of a measurement and the value obtained in practice. In this exercise, the measurement error in question affects the variable \( x \), which in turn impacts the accuracy of the calculated value \( f(x) \).
When we encounter measurement error, it's vital to quantify this uncertainty. This is done through interval estimation. By knowing the true value of \( x \) and the error \( \Delta x \), we can calculate an interval \([1.8, 2.2]\). This interval represents the range in which the true value of \( x \) might be located.
The concept of measurement error is essential for understanding the reliability and validity of data. It helps us determine how much trust we can place in the results and how sensitive our outcomes are to changes in the input variables.

Exponential Function

Exponential functions are mathematical functions characterized by a constant base raised to a variable exponent. In this case, the function \( f(x) = e^x \) uses the mathematical constant \( e \) (approximately 2.718) as its base—a number important in calculus due to its unique properties.
Exponential functions grow rapidly, which has significant implications when considering errors or changes in the input. For instance, small variations in \( x \) can lead to substantial changes in \( f(x) \). In this exercise, the function was evaluated at \( x = 2 \) and as a result of the error, the interval was considered between \( x = 1.8 \) to \( x = 2.2 \).
Understanding exponential functions is crucial in many fields such as finance, biology, and physics, where they are used to model growth processes, decay, and many natural phenomena.

Interval Estimation

Interval estimation is a statistical technique that provides a range of values within which a parameter is expected to lie. It's particularly useful when dealing with measurement errors, as it accommodates uncertainty by defining a margin around a calculated value.
In our exercise, the interval for the exponential function \( f(x) = e^x \) was determined as \([6.0496, 9.0250]\). This interval takes into account the possible values \( x \) might assume, factoring in its error \( \Delta x \). This means that while \( f(2) = e^2 \) gives us a nominal value, our interest in the range reflects our understanding that data and measurements are rarely perfect.
By providing both upper and lower boundaries, interval estimation offers a fuller picture of possible outcomes, which is particularly valuable in decision-making and predictive analyses.