Problem 68 - 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. $$ f(x)=\frac{x-1}{4 x^{2}+1}, \quad\left[-\frac{3}{4}, 1\right], \quad a=\frac{1}{2} $$

Step-by-Step Solution

Answer

Estimate \( \delta \) for \( \epsilon = 0.5, 0.1, 0.01 \) based on error plots, ensuring \( |x-a| < \delta \) implies \( |f(x)-L(x)| < \epsilon \). Monitor graph to confirm.

Step 1: Plot the Function over the Interval

Plot the function \( f(x) = \frac{x-1}{4x^2 + 1} \) over the interval \( I = \left[-\frac{3}{4}, 1\right] \). This gives you a visual representation of the function's behavior within the specified range.

Step 2: Find the Linearization at Point a

Identify the linearization \( L(x) \) of the function \( f(x) \) at \( a = \frac{1}{2} \). First, compute \( f'(x) \), the derivative of \( f \), then evaluate it at \( x = \frac{1}{2} \). The linearization is given by the formula \( L(x) = f(a) + f'(a)(x-a) \). We find \( L(x) \approx -\frac{2}{35} -\frac{16}{35}(x-\frac{1}{2}) \).

Step 3: Graph f and L Together

Plot both \( f(x) \) and \( L(x) \) on the same set of axes over the interval \( I \). This helps compare their respective graphs and visualize the accuracy of the linear approximation.

Step 4: Plot the Absolute Error and Find Its Maximum

Calculate and plot the absolute error \( |f(x) - L(x)| \) over \( I \). Then, determine the maximum value of this error within the interval using a CAS tool for precise computation.

Step 5: Estimate \(\delta\) for Different \(\epsilon\) Values

From the plot of the absolute error, estimate the largest possible \(\delta > 0\) such that for \(\epsilon = 0.5, 0.1,\) and \(0.01\), the condition \(|x-a| < \delta \Rightarrow |f(x)-L(x)| < \epsilon\) holds. Adjust \(\delta\) iteratively and verify graphically.

Key Concepts

Linearization

Linearization is a technique used to approximate a function near a given point. It's like drawing a straight line that closely follows the curve of the function right around that point. We often use linearization to simplify complex calculations and to get a quick estimate of a function's value near a certain point.
To linearize a function at a point, you first need to find the derivative, which indicates the function's rate of change at that point. Next, evaluate this derivative at the given point. Using the formula for linearization, \(L(x) = f(a) + f'(a)(x-a)\), where \(a\) is your point of interest, you can write an equation for the linear function \(L(x)\) that approximates the original function \(f(x)\).

• The goal is to use \(L(x)\) in place of \(f(x)\) when \(x\) is close to \(a\).
• This approximation is most accurate when \(x\) is very near to \(a\).

The key advantage of linearization is its simplicity, providing an effective way to manage complex functions with basic calculations.

Absolute Error

The concept of absolute error helps you understand how far off your approximation is from the actual value. It is the difference between the true value and the estimated value without considering the direction of the error, hence the term 'absolute'.
In mathematical terms, if \(L(x)\) is your linear approximation and \(f(x)\) is the actual function, the absolute error is given by \(|f(x) - L(x)|\). This absolute error provides you a measure of how good your linearization is across a specified interval.

• An absolute error close to zero means your approximation is very close to the actual function.
• A larger absolute error indicates that the approximation may not be as reliable for certain values of \(x\).

When estimating errors, consider both numerical and graphical methods to visualize and adjust for better approximations.

Maximum Error

Maximum error gives us a threshold of the largest possible error across a specific interval. It is the greatest value of the absolute error \(|f(x) - L(x)|\) over the interval of interest. By identifying this point, you get an overview of the worst-case scenario in terms of approximation error.
Finding the maximum error is crucial, especially when you need to determine how accurate your linear approximation is over a larger range.

• Use computational tools to determine and verify the maximum error, ensuring more precise results.
• Graphically, maximum error is where the error plot reaches its peak within the interval studied.

Often, when you account for maximum error, it offers insight into making adjustments for more precise parameter setting, like choosing the right \(\delta\) in error estimates.

Derivative Calculation

Calculating the derivative of a function is essential for linearization and error estimation. The derivative indicates how quickly a function is changing at any point. In the context of linearization, it reflects how steeply the tangent (linear approximation) is sloping at the specific point of interest.
To calculate the derivative, follow these steps:

• Use differentiation rules: power rule, product rule, quotient rule, or chain rule, depending on the function type.
• Evaluate the derivative at the point of interest \(x = a\).
• This derivative value \(f'(a)\) becomes essential in building the linear approximation \(L(x) = f(a) + f'(a)(x-a)\).

Understanding the concept and process of calculating derivatives equips you with the ability to judge the slope and direction of a function accurately, which is crucial for making valid approximations and error judgements.