Problem 35

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 $x \pm|\Delta x|$. $$ f(x)=2 x, x=1 \pm 0.1 $$

Step-by-Step Solution

Verified

Answer

The interval for $ f(x) $ is $[1.8, 2.2]$.

1Step 1: Understand the Problem

We have a function $ f(x) = 2x $ and each value of $ x $ is affected by a measurement error $ \Delta x = 0.1 $. We need to find an interval for $ f(x) $ expressed as $[f(x)-\Delta f, f(x)+\Delta f]$ reflecting the measurement error.

2Step 2: Calculate the Range of x

Given $ x = 1 \pm 0.1 $, this means $ x $ can range from 0.9 to 1.1. So, $ x $ is between $ 0.9 \leq x \leq 1.1 $.

3Step 3: Apply the Function to the x Values

Use the function $ f(x) = 2x $ on the lower and upper limits of $ x $: 1. For $ x = 0.9 $, $ f(0.9) = 2 \times 0.9 = 1.8 $.2. For $ x = 1.1 $, $ f(1.1) = 2 \times 1.1 = 2.2 $.

4Step 4: Determine the Interval for f(x)

The function values $ f(x) $ range from 1.8 to 2.2 when considering the measurement error in $ x $. Therefore, the interval is $[1.8, 2.2]$.

Key Concepts

Measurement ErrorFunction IntervalFunction Range

Measurement Error

Measurement error is crucial in understanding how inaccuracies in inputs can affect the output of a function. When taking measurements, errors can come from various sources: improper tools, human error, or even environmental conditions.

In our scenario, we refer to a specific error in measuring the variable $ x $.
This error is denoted by $ \Delta x $, which represents a small deviation from the true value of $ x $.
In the exercise, the value of $ x $ is given as $ 1 \pm 0.1 $, meaning $ x $ might be slightly less than or greater than 1 due to this measurement error.

Understanding the nature of measurement errors helps in assessing how much our calculated outputs might differ from reality. Accounting for these errors in calculations ensures the results are as reliable as possible despite these uncertainties.

Function Interval

The function interval refers to the range of input values over which a function is evaluated. When there's measurement error, this interval represents the possible real-world values.

In our example, $ x $ spans an interval due to the measurement error: $ x = 1 \pm 0.1 $.
This translates to $ 0.9 \leq x \leq 1.1 $, as $ x $ can be as low as 0.9 or as high as 1.1.
This interval of $ x $ is crucial for understanding how $ f(x) $ will behave across its range.

Using intervals is key to applying calculus functions in real-world scenarios where precision varies. By calculating a function over an interval, instead of a fixed point, any potential errors in inputs are naturally incorporated into predictions and results.

Function Range

The function range is the set of all possible outputs a function can have, given an interval of input values. It highlights how changes in inputs due to errors influence the overall values of the function outcome.

Given the function $ f(x) = 2x $, when $ x $ varies within the interval $ 0.9 \leq x \leq 1.1 $, we compute the corresponding range for $ f(x) $.
By substituting the lower limit of $ x $, $ f(0.9) = 1.8 $, and using the upper limit, $ f(1.1) = 2.2 $.
Thus, the function range, depending on the measurement error in $ x $, is $ [1.8, 2.2] $.

Determining the function range based on input intervals helps in providing a comprehensive understanding of all possible output values, which is essential in maintaining accuracy and reliability in results derived from mathematical models.

Problem 34

Problem 35

Other exercises in this chapter

Problem 34

Differentiate the functions with respect to the independent variable. $$ f(x)=\left(\ln \left(1-x^{2}\right)\right)^{3} $$

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Problem 34

Differentiate $$ h(s)=a^{4} s^{2}-a s^{4}+\frac{s^{2}}{a^{4}} $$

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Problem 35

The following limit represents the derivative of a function $f$ at the point $(a, f(a))$ : $$ \lim _{h \rightarrow 0} \frac{2(a+h)^{2}-2 a^{2}}{h} $$

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Problem 35

Suppose that $f^{\prime}(x)=\frac{1}{x}$. Find the following: (a) $\frac{d}{d x} f\left(x^{2}+3\right)$ (b) $\frac{d}{d x} f(\sqrt{x-1})$

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