Problem 40 - 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 \(x \pm|\Delta x|\). $$ f(x)=\sin x, x=-1 \pm 0.05 $$

Step-by-Step Solution

Answer

Compute \(\sin(-1) = -0.84\), and the interval is approximately \([-0.888, -0.792]\).

Step 1: Understand the given problem

We need to find the interval \([f(x) - \Delta f, f(x) + \Delta f]\) that reflects the measurement error in \(x\), given the function \(f(x) = \sin x\) and \(x = -1 \pm 0.05\). This means \(x\) can vary from \(-1.05\) to \(-0.95\).

Step 2: Identify the function and the range of x

Our function is \(f(x) = \sin x\). The given range for \(x\) is \([-1.05, -0.95]\). We will need to evaluate the sine function at both \(x = -1.05\) and \(x = -0.95\).

Step 3: Calculate the values of the function at the range endpoints

Compute \(f(-1.05) = \sin(-1.05)\) and \(f(-0.95) = \sin(-0.95)\). These values will give us the extent of how \(f(x)\) can vary due to the measurement error in \(x\).

Step 4: Compute \(\Delta f\)

To find \(\Delta f\), determine the maximum deviation of \( \sin x\) within the interval. Calculate the absolute differences: \[ |\sin(-1.05) - \sin(-1)| \text{ and } |\sin(-0.95) - \sin(-1)| \]The largest of these will be \(\Delta f\).

Step 5: Construct the interval

We assume the middle of the interval for the function value is \(f(-1) = \sin(-1)\). Thus, the interval is:\[[\sin(-1) - \Delta f, \sin(-1) + \Delta f]\]Compute \( \sin(-1)\) and add/subtract \( \Delta f\) to this value.

Key Concepts

Measurement Error

When measuring a physical quantity, it is common to encounter measurement errors, which are small deviations from the true value. These errors are inevitable and can arise from various sources such as limitations of the measuring instrument or human errors. Measurement errors are typically represented using the symbol \( \Delta x \). Consider an example where you measure a length as 10 meters with an error of \( \pm 0.05 \) meters. This implies that the true length lies between 9.95 and 10.05 meters. Such expression \( x = 10 \pm 0.05 \) indicates that the measurement error is \( 0.05 \), and hence, the actual value of x can be in the interval

• \([x - \Delta x, x + \Delta x] = [9.95, 10.05].\)

In our given exercise, \( x = -1 \pm 0.05 \) tells us the true value of \(x\) is between

• \([-1.05, -0.95]\).

Measurement error impacts the result of functions like the sine function, as seen in the exercise.

Sine Function

The sine function, denoted as \(f(x) = \sin x\), is one of the basic trigonometric functions. It describes the ratio of the length of the side opposite to an angle in a right-angled triangle to the hypotenuse. In a Cartesian plane, the sine function traces a smooth, periodic wave across all real numbers. The range of the sine function lies between -1 and 1, which means \(-1 \leq \sin x \leq 1\). Its periodic nature ensures that it repeats every \(2\pi\). In the context of the problem, when \(x\) varies between

• \([-1.05, -0.95]\)

we need to evaluate \( \sin x \) at these endpoints to understand how the function behaves with the measurement error. Calculations such as \( \sin(-1.05) \) and \( \sin(-0.95) \) inform us about the potential variation range of the function due to small changes in \(x\). This helps gauge the degree of accuracy one can expect from the measurement.

Interval Estimation

Interval estimation provides a range within which we can reasonably expect the true value to lie, considering some degree of uncertainty. This is particularly useful in determining how errors in measurement affect the outcome of function evaluations. For instance, when dealing with the function \( f(x) = \sin(x) \) and with \( x = -1 \pm 0.05 \), the true value of \( x \) is uncertain. Therefore, we must calculate an interval for \( f(x) \) that accounts for these variations.To construct this interval, calculate the function values at the extreme points

• \([-1.05, -0.95]\).

Determine the differences:

• \( |\sin(-1.05) - \sin(-1)| \)
• \( |\sin(-0.95) - \sin(-1)| \)

The largest difference, \( \Delta f \), determines the deviation of \( f(x) \), forming

• the interval \([\sin(-1) - \Delta f, \sin(-1) + \Delta f]\).

This interval estimation responds to the original question, providing the potential range of the sine function's output considering measurement errors in \(x\).