Problem 36
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 \(x \pm|\Delta x|\). $$ f(x)=1-3 x, x=-2 \pm 0.3 $$
Step-by-Step Solution
Verified Answer
The interval for \( f(x) \) is \([6.1, 7.9]\).
1Step 1: Understand the Given Function and Error Interval
We are given a function \( f(x) = 1 - 3x \) and an interval for \( x \) as \( x = -2 \pm 0.3 \). This means the true value of \( x \) is between \( -2 - 0.3 = -2.3 \) and \( -2 + 0.3 = -1.7 \). We need to determine the interval for \( f(x) \) based on this range of \( x \).
2Step 2: Calculate \( f(x) \) at the Endpoints
To find the interval of \( f(x) \) that accounts for the error \( \Delta x \), we first calculate \( f(x) \) at the endpoints of the \( x \) interval. Calculate \( f(x) \) when \( x = -2.3 \) and when \( x = -1.7 \).- When \( x = -2.3 \), \[ f(-2.3) = 1 - 3(-2.3) = 1 + 6.9 = 7.9 \]- When \( x = -1.7 \), \[ f(-1.7) = 1 - 3(-1.7) = 1 + 5.1 = 6.1 \]
3Step 3: Determine the Interval for \( f(x) \)
With the values calculated for the endpoints, we can establish that the range for \( f(x) \) is between the minimum and maximum values computed.Thus, the interval is \( [6.1, 7.9] \).
4Step 4: Express the Interval in Terms of \( \Delta f \)
To express the error interval \([f(x)-\Delta f, f(x)+\Delta f]\), we identify from the range that the center of the interval is \( f(x) = 7 \) (midpoint of 6.1 and 7.9) and \( \Delta f = 0.9 \) (half the difference of 7.9 and 6.1). Therefore, the interval is \([7 - 0.9, 7 + 0.9]\) or \([6.1, 7.9]\).
Key Concepts
Measurement ErrorInterval EstimationFunction Error Analysis
Measurement Error
Measurement error occurs when there is a deviation in the actual measurement from the true value. In this context, the measurement error in the variable \( x \) is expressed as \( x = -2 \pm 0.3 \). This notation indicates that the true value of \( x \) could lie anywhere from \(-2.3\) to \(-1.7\). The possible range of error, \( \Delta x \), is 0.3 units either side of the measured value \(-2\). This uncertainty represents the lack of precision in the measurement of \( x \), due to factors such as equipment inaccuracies or human error.
Understanding measurement error is crucial when analyzing functions, as any inaccuracies in \( x \) directly impact the computation of \( f(x) \). Such errors can lead to misinterpretations if not properly evaluated. When measurement error is quantified accurately, it gives us a bound within which the true results are likely to fall, improving the reliability of any subsequent error analysis performed on the calculations.
Understanding measurement error is crucial when analyzing functions, as any inaccuracies in \( x \) directly impact the computation of \( f(x) \). Such errors can lead to misinterpretations if not properly evaluated. When measurement error is quantified accurately, it gives us a bound within which the true results are likely to fall, improving the reliability of any subsequent error analysis performed on the calculations.
Interval Estimation
Interval estimation is a statistical technique used to determine the range in which a parameter lies with a certain level of confidence. In our exercise, the function value \( f(x) \) is affected by the variability in \( x \), prompting the need to create an interval that reflects this uncertainty.
By calculating \( f(x) \) at the endpoints of the given interval for \( x \), we create possible bounds for the function value. For instance, when \( x \) ranges from \(-2.3\) to \(-1.7\), \( f(x) \) correspondingly ranges from 6.1 to 7.9, leading to the interval \([6.1, 7.9]\). This interval is our interval estimation for \( f(x) \), encompassing potential variations due to errors in \( x \).
In practice, such interval estimates provide a "buffer" around the function value, acknowledging that the exact value might not be known precisely. This method is fundamental in many fields, from engineering to finance, whenever precise measurements are not possible or cost-effective.
By calculating \( f(x) \) at the endpoints of the given interval for \( x \), we create possible bounds for the function value. For instance, when \( x \) ranges from \(-2.3\) to \(-1.7\), \( f(x) \) correspondingly ranges from 6.1 to 7.9, leading to the interval \([6.1, 7.9]\). This interval is our interval estimation for \( f(x) \), encompassing potential variations due to errors in \( x \).
In practice, such interval estimates provide a "buffer" around the function value, acknowledging that the exact value might not be known precisely. This method is fundamental in many fields, from engineering to finance, whenever precise measurements are not possible or cost-effective.
Function Error Analysis
Function error analysis examines how errors in input values affect the output of a function. It's vital in assessing the reliability and precision of numerical results, especially in calculus where functions often involve operations that can amplify errors.
In our example, the function \( f(x) = 1 - 3x \) represents a linear transformation of \( x \). A measurement error in \( x \) also leads to an error in \( f(x) \), calculated by checking how changes in \( x \) affect \( f(x) \'s \) outputs. After calculating \( f(x) \) across the range \([-2.3, -1.7]\), we found the interval \([6.1, 7.9]\), with \( \Delta f = 0.9 \) and a midpoint \( f(x) = 7 \).
This analysis highlights an essential aspect of managing errors in calculus — each function has its way of propagating errors based on its form. For linear functions like \( 1 - 3x \), errors in \( x \) produce proportional errors in \( f(x) \), which can be readily calculated through basic algebra. However, for more complex non-linear functions, additional techniques such as differential calculus may be required to quantify these effects accurately. By understanding and applying function error analysis, industries can make informed decisions, managing risks associated with imprecision in measurements and predictions.
In our example, the function \( f(x) = 1 - 3x \) represents a linear transformation of \( x \). A measurement error in \( x \) also leads to an error in \( f(x) \), calculated by checking how changes in \( x \) affect \( f(x) \'s \) outputs. After calculating \( f(x) \) across the range \([-2.3, -1.7]\), we found the interval \([6.1, 7.9]\), with \( \Delta f = 0.9 \) and a midpoint \( f(x) = 7 \).
This analysis highlights an essential aspect of managing errors in calculus — each function has its way of propagating errors based on its form. For linear functions like \( 1 - 3x \), errors in \( x \) produce proportional errors in \( f(x) \), which can be readily calculated through basic algebra. However, for more complex non-linear functions, additional techniques such as differential calculus may be required to quantify these effects accurately. By understanding and applying function error analysis, industries can make informed decisions, managing risks associated with imprecision in measurements and predictions.
Other exercises in this chapter
Problem 35
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Assume that \(f(x)\) and \(g(x)\) are differentiable. Find \(\frac{d}{d x} \sqrt{f(x)+g(x)}\).
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