Problem 69 - 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. $$ f(x)=x^{2 / 3}(x-2), \quad[-2,3], \quad a=2 $$

Step-by-Step Solution

Answer

Maximum error is the peak of the absolute error plot. Delimit \( |x-2| \) such that error remains under \( \epsilon=0.5, 0.1, 0.01 \) for each condition.

Step 1: Plot the Function

Using a Computer Algebra System (CAS), plot the function \[ f(x) = x^{2/3}(x-2) \] over the interval \( I = [-2, 3] \). This visualization helps us understand the behavior of the function over the given range.

Step 2: Find the Linearization

The linearization of the function \( f \) at the point \( a = 2 \) is found using the formula \[ L(x) = f(a) + f'(a)(x-a) \] First, calculate \( f(2) \) and the derivative \( f'(x) \). Then, find \( f'(2) \). Substitute these values into the linearization formula:- \( f(2) = 0 \)- To find \( f'(x) \), use the derivative: \( f'(x) = \frac{2}{3}x^{-1/3}(x-2) + x^{2/3} \)- Evaluate \( f'(2) = \frac{2}{3} \times 2^{-1/3}(2-2) + 2^{2/3} = 2^{2/3} \)Thus, the linearization is:\[ L(x) = 0 + 2^{2/3}(x-2) \]

Step 3: Plot Function and Linearization

Using the CAS, plot both the original function \( f(x) = x^{2/3}(x-2) \) and its linearization \( L(x) = 2^{2/3}(x-2) \) on the same graph. This allows us to compare the two and see how closely they align around \( a = 2 \).

Step 4: Plot Absolute Error

Plot the absolute error function \[ |f(x) - L(x)| \] across the interval \( I = [-2, 3] \). Find the maximum value of this error, which indicates the largest deviation of the linearization from the actual function over that range.

Step 5: Estimate Delta for Various Epsilons

Using the graph from Step 4, estimate the largest possible \( \delta > 0 \) such that for each \( \epsilon \) (0.5, 0.1, and 0.01), the inequality \[ |x-2| < \delta \Rightarrow |f(x) - L(x)| < \epsilon \] holds. This involves identifying the range of \( x \) where the error plot remains below each \( \epsilon \) threshold.

Key Concepts

Linearization

Linearization is a powerful mathematical technique used to approximate more complex functions with simpler linear ones. The key idea involves using the tangent line of a function at a specific point, usually denoted as "a", to represent the function near that point. This linear approximation is represented by the formula\[ L(x) = f(a) + f'(a)(x-a) \]Here,

• \( f(a) \) is the value of the function at the point \( a \)
• \( f'(a) \) is the derivative of the function at that point, which represents the slope of the tangent

The goal of linearization is to make complex calculations more manageable without greatly sacrificing accuracy over small intervals. It's particularly helpful in various real-world applications, such as engineering and physics, where systems often operate close to a specific point. Using linearization can often simplify these problems to a point where manual calculations can suffice.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is specialized software designed for symbolic mathematics tasks. CAS can handle not just numerical calculations but also manipulations of algebraic expressions, which include

• Calculating derivatives and integrals
• Solving equations
• Manipulating polynomials and matrices

In exercises like the one given, a CAS is invaluable for automatically plotting functions and finding derivatives. This not only saves time but reduces the potential for human error in complex calculations. Because of its powerful capabilities, CAS tools like Mathematica, Maple, or even online platforms such as Wolfram Alpha or certain features in Python, are particularly useful in education, making advanced concepts more accessible to students.

Absolute Error

The absolute error in calculations describes the magnitude of the difference between an observed value and the true value. In the context of linearization, it measures how far off the linear approximation \( L(x) \) is from the actual function \( f(x) \) at any point \( x \).The formula to calculate the absolute error is\[ |f(x) - L(x)| \] This value gives a direct sense of how accurate your linear approximation is over the interval you're studying. An understanding of this concept is crucial because it tells you not only how much error exists but also guides decisions on whether a linear approximation is sufficiently accurate for a specific task.For instance, when you set a threshold \( \epsilon \), you can determine a range \( \delta \) where the approximation remains within an acceptable error level, ensuring reliable results in practical situations.

Function Plotting

Function plotting is an essential tool in visualizing mathematical concepts. It involves drawing graphs that represent mathematical functions, allowing us to see insights that might not be obvious from equations alone.When you plot a function, you can:

• Visualize the overall behavior and trends, such as increases and decreases
• Identify key characteristics, including intercepts and asymptotes
• Compare changes between an original function and its approximations

In educational settings, plotting allows students to visually grasp how a linearization \( L(x) \) compares with the original function \( f(x) \) over a certain interval. By plotting the absolute error as well, you can see directly where the approximation diverges from the real function.Many software tools or calculators today feature function plotting capabilities, including CAS tools. They make it easier to run experiments with mathematical functions and gain a more intuitive understanding of concepts, enriching the learning experience.