Problem 70 - Thomas Calculus

Question

In Exercises \(67-70\) , use a CAS to estimate the magnitude of the error in using the linearization in place of the function over a specified interval \(I\) . Perform the following steps: a. Plot the function \(f\) over \(I .\) b. Find the linearization \(L\) of the function at the point \(a\) . c. Plot \(f\) and \(L\) together on a single graph. d. Plot the absolute error \(|f(x)-L(x)|\) over \(I\) and find its maximum value. e. From your graph in part (d), estimate as large a \(\delta > 0\) as you can, satisfying $$ |x-a|<\delta \quad \Rightarrow \quad|f(x)-L(x)|<\epsilon $$ for \(\epsilon=0.5,0.1,\) and 0.01 . Then check graphically to see if your \(\delta\) -estimate holds true. $$ f(x)=\sqrt{x}-\sin x, \quad[0,2 \pi], \quad a=2 $$

Step-by-Step Solution

Answer

Maximum error occurs when estimating \(\delta\) for \(\epsilon = 0.5, 0.1, 0.01\); use graph to find \(\delta\).

Step 1: Plot the function over the interval

We start by plotting the function \( f(x) = \sqrt{x} - \sin x \) over the interval \([0, 2\pi]\). This gives us a visual idea of how the function behaves over this range.

Step 2: Find the linearization at the point

The linearization \( L(x) \) of a function at a point \( a \) is given by \( L(x) = f(a) + f'(a)(x - a) \). For our function, first compute \( f(2) = \sqrt{2} - \sin(2) \) and find the derivative \( f'(x) = \frac{1}{2\sqrt{x}} - \cos x \). Then, compute \( f'(2) \).

Step 3: Plot both function and linearization

We now plot both the original function \( f(x) \) and the linearization \( L(x) \) on the same graph over \([0, 2\pi]\). This will allow us to compare the linear approximation with the actual function.

Step 4: Plot and analyze absolute error

The absolute error is given by \( |f(x) - L(x)| \). We plot \( |f(x) - L(x)| \) over \([0, 2\pi]\) to observe how it varies and identify the maximum error.

Step 5: Estimate \(\delta\) for given \(\epsilon\) values

To find the largest \( \delta > 0 \) such that \( |x - 2| < \delta \Rightarrow |f(x) - L(x)| < \epsilon \), we analyze the error graph. For \( \epsilon = 0.5, 0.1, \) and \( 0.01 \), find the maximum \( \delta \) where the error is less than each \( \epsilon \). Check graphically to confirm each \( \delta \) estimate.

Key Concepts

Linearization

Linearization is a method used in calculus to approximate a function near a given point. It converts a non-linear function into a simpler linear form. This method is particularly useful for estimating function values that lie close to the point of interest. The linearization of a function at a point \( a \) is given by the formula:

• \( L(x) = f(a) + f'(a)(x - a) \)

This formula uses both the function value and its derivative at the point \( a \) to create a tangent line that serves as the best linear approximation of the function. Linearization is widely employed as it provides a simple way to handle complex functions within a small range, giving us an easy-to-understand model while still reflecting the behavior of the original function closely.

Error estimation

Error estimation in calculus involves analyzing how much an approximate solution deviates from the true function. For linearization, the error can be thought of as the difference between the actual function and its linear approximation. The error helps identify how accurate the approximation is.
In this context, we use the absolute error, which quantifies the estimation error as:

• \( |f(x) - L(x)| \)

To assess the reliability of a linear approximation over an interval, we plot this absolute error. By doing so, we can inspect the regions where the approximation might deviate significantly from the actual function. This plot helps in understanding the maximum possible deviation (or error) that one might expect, thus aiding in making informed decisions about when a linear approximation could be trusted or not.

Absolute error

The absolute error provides a precise measure of how far an estimate is from the actual value. In the context of linearization, it can be expressed using the following expression for any \( x \):

• \( |f(x) - L(x)| \)

This quantity tells us the difference in value between the function \( f(x) \) and its linear approximation \( L(x) \). A smaller absolute error means the linear approximation closely matches the true function, making it a reliable approximation.
Detecting the maximum absolute error within a given interval allows us to quantify the worst-case scenario deviation when using the linearization instead of the original function. Having this information empowers individuals to determine the usability of the approximation for practical applications, thus improving decision-making.

Derivative

The derivative of a function is a core concept in calculus, representing the rate at which the function’s value changes as its input changes. It is a crucial component in linearization since it not only helps in understanding function behavior but also in constructing the linear approximation itself.
The derivative, when evaluated at a specific point \( a \), gives us the slope of the tangent line to the function at that point. In the case of the given exercise, for a function \( f(x) = \sqrt{x} - \sin x \), the derivative \( f'(x) \) is:

• \( f'(x) = \frac{1}{2\sqrt{x}} - \cos x \)

Using this derivative, we calculate \( f'(2) \) to form the linearization \( L(x) \). The derivative directly influences how the linearization approximates the function, showcasing its central role in the approximation process. Understanding derivatives facilitates a deeper grasp of both the behavior of functions and the nuances of approximating them.