Problem 42
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. $$ f(x)=x^{1 / 4}, x=10 $$
Step-by-Step Solution
Verified Answer
The error \(\Delta f\) is approximately 0.0028, and the percentage error is about 0.157\%.
1Step 1: Differentiate the function
First, we need to determine the differential of the function to find the error in terms of a small change in \(x\). The function given is \(f(x) = x^{1/4}\). The differential is obtained by differentiating the function with respect to \(x\): \(f'(x) = \frac{1}{4}x^{-3/4}\).
2Step 2: Express the error in terms of \(x\)
To find the error \(\Delta f\) in \(f\), we use the differential formula \(\Delta f \approx f'(x)\Delta x\). We know \(f'(x) = \frac{1}{4}x^{-3/4}\), so \(\Delta f \approx \frac{1}{4}x^{-3/4}\Delta x\).
3Step 3: Calculate the small change in \(x\)
The measurement of \(x\) is accurate within \(2\%\), which means \(\Delta x = 0.02x\). With \(x = 10\), \(\Delta x = 0.2\).
4Step 4: Substitute values to find \(\Delta f\)
Using \(x = 10\) and \(\Delta x = 0.2\), substitute these into \(\Delta f \approx \frac{1}{4}(10)^{-3/4}(0.2)\). First, calculate \((10)^{-3/4} = 10^{-0.75} \approx 0.0562\). Now plug in the values: \(\Delta f \approx \frac{1}{4}\times0.0562\times0.2\approx0.0028\).
5Step 5: Calculate the percentage error
To find the percentage error, use the formula \(100\frac{\Delta f}{f(x)}\). First, calculate \(f(10) = 10^{1/4} \approx 1.78\). Substituting the values, \(100\frac{0.0028}{1.78} \approx 0.157\%\).
Key Concepts
Error AnalysisFunction DifferentiationPercentage Error Calculation
Error Analysis
Error analysis is important in understanding how small changes in measurement can impact results. When measuring quantities like length, weight, or time, slight inaccuracies might occur. In the context of calculus, we're focusing on a function of the variable \(x\) and analyzing how errors in \(x\) affect the function's outcome, \(f(x)\). For example, if \(x\) is not measured precisely, then \(f(x) = x^{1/4}\) is also prone to error.
The error in the measurement of \(x\) is denoted as \(\Delta x\) and subsequently leads to an error in \(f(x)\), denoted as \(\Delta f\). This analysis helps us understand the limits of precision and reliability of a given function under practical circumstances.
By determining \(\Delta f \approx f'(x)\Delta x\), we gain insights into the magnitude of an error that could manifest in real-world calculations. Thus, error analysis is a crucial step in assessing the accuracy of measurements and results.
The error in the measurement of \(x\) is denoted as \(\Delta x\) and subsequently leads to an error in \(f(x)\), denoted as \(\Delta f\). This analysis helps us understand the limits of precision and reliability of a given function under practical circumstances.
By determining \(\Delta f \approx f'(x)\Delta x\), we gain insights into the magnitude of an error that could manifest in real-world calculations. Thus, error analysis is a crucial step in assessing the accuracy of measurements and results.
Function Differentiation
Differentiation is a process in calculus used to find the rate at which a function is changing at any given point. In our exercise, we differentiate the function \(f(x) = x^{1/4}\). By doing this, we aim to derive a formula that helps find the error in the function's value when \(x\) is slightly mismeasured.
To differentiate \(f(x) = x^{1/4}\), we apply the power rule which states that the derivative of \(x^n\) is \(nx^{n-1}\). For our function, \(f'(x) = \frac{1}{4}x^{-3/4}\). This derivative represents the slope of the function at any point \(x\).
The derivative \(f'(x)\) is pivotal because it allows us to express the error \(\Delta f\) in terms of \(\Delta x\), the small change or error in \(x\):
To differentiate \(f(x) = x^{1/4}\), we apply the power rule which states that the derivative of \(x^n\) is \(nx^{n-1}\). For our function, \(f'(x) = \frac{1}{4}x^{-3/4}\). This derivative represents the slope of the function at any point \(x\).
The derivative \(f'(x)\) is pivotal because it allows us to express the error \(\Delta f\) in terms of \(\Delta x\), the small change or error in \(x\):
- \(\Delta f \approx f'(x) \Delta x\)
Percentage Error Calculation
Once we have determined the change in the function \(\Delta f\), it's crucial to understand what this error means relative to the actual function value \(f(x)\). This is achieved through calculating the percentage error, which provides a clearer scope of the error's significance.
The percentage error formula is given by:
After carrying out the calculation, we found a percentage error of approximately 0.157%.
This percentage indicates how significant or negligible the error is in the overall context of the function's application, providing an understanding of accuracy for scientists, engineers, and mathematicians alike.
The percentage error formula is given by:
- \(100 \frac{\Delta f}{f(x)}\)
After carrying out the calculation, we found a percentage error of approximately 0.157%.
This percentage indicates how significant or negligible the error is in the overall context of the function's application, providing an understanding of accuracy for scientists, engineers, and mathematicians alike.
Other exercises in this chapter
Problem 41
Differentiate the functions with respect to the independent variable. $$ f(x)=\exp [x-\ln x] $$
View solution Problem 41
Differentiate $$ R(T)=\frac{2 \pi^{5}}{15} \frac{k^{4}}{c^{2} h^{3}} T^{4} $$
View solution Problem 42
4Suppose a particle moves along a straight line. The position at time \(t\) is given by $$ s(t)=3 t-t^{2}, \quad t \geq 0 $$ where \(t\) is measured in seconds
View solution Problem 42
Find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. \(y=\left(1+2(x+3)^{4}\right)^{2}\)
View solution