Problem 67 - 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. $$ f(x)=x^{3}+x^{2}-2 x, \quad[-1,2], \quad a=1 $$

Step-by-Step Solution

Answer

The largest \( \delta \) values for \( \epsilon = 0.5, 0.1, 0.01 \) are estimated from the graph.

Step 1: Plot the Function

Using a computer algebra system (CAS), plot the graph of the function \( f(x) = x^3 + x^2 - 2x \) over the interval \([-1, 2]\). This provides a visual understanding of the function's behavior over the specified interval.

Step 2: Calculate the Linearization

The linearization \( L(x) \) of a function at a point \( a \) can be found using the formula \( L(x) = f(a) + f'(a)(x-a) \). First, calculate \( f(a) \):\( f(1) = 1^3 + 1^2 - 2 \times 1 = 0 \). Then, find the derivative \( f'(x) = 3x^2 + 2x - 2 \) and evaluate it at \( a = 1 \): \( f'(1) = 3 \times 1^2 + 2 \times 1 - 2 = 3 \). Thus, the linearization is \( L(x) = 0 + 3(x-1) = 3x - 3 \).

Step 3: Plot Function and Linearization

Plot both the function \( f(x) = x^3 + x^2 - 2x \) and its linearization \( L(x) = 3x - 3 \) on the same graph over the interval \([-1, 2]\). Observing this graph helps visualize how the linearization approximates the function.

Step 4: Plot Absolute Error

Plot the graph of the absolute error \(|f(x) - L(x)|\) over the interval \([-1, 2]\). This shows how closely the linearization approximates \( f(x) \). Use the graph to determine the maximum value of the error over this interval.

Step 5: Estimate \(\delta\) from Error Graph

By analyzing the graph from step 4, estimate the largest value of \( \delta \) such that \( |x - 1| < \delta \) implies \( |f(x) - L(x)| < \epsilon \) for each \( \epsilon = 0.5, 0.1, \) and \( 0.01 \). Verify these estimates using the graph by checking around the point \( a = 1 \).

Key Concepts

Linearization

Linearization is a mathematical technique used to approximate a function near a particular point using a linear equation. If you think of complex functions as curvy roads, linearization acts like a straight tangent line that touches the road at a point of interest. By focusing on this linear segment, we can simplify calculations and make swift estimations.

To perform linearization, we first calculate the function value and its derivative at the point of interest, which in our exercise is at point \(a = 1\). The function is \(f(x) = x^3 + x^2 - 2x\). Evaluating at \(x = 1\), \(f(1) = 0\).

• The derivative \(f'(x)\) = \(3x^2 + 2x - 2\)
• Evaluating the derivative at \(x = 1\) gives, \(f'(1)=3\).

Thus, the linearization formula is \(L(x) = f(a) + f'(a)(x-a)\). Here it becomes \(L(x) = 3x - 3\). This linear equation now provides a simple approximation of the original function around \(x = 1\).
Linearization is helpful when working within small intervals around the point \(a\) as it provides precise estimates without handling the complexities of the original function.

Absolute Error

Absolute error measures how far off an approximation is from the actual value. Think of it like a ruler showing how much an approximation veers from the truth. In this exercise, absolute error is calculated between the original function \(f(x) = x^3 + x^2 - 2x\) and its linear approximation \(L(x) = 3x - 3\).

The formula used is \(|f(x) - L(x)|\), which gives a non-negative value representing how far off a single linear prediction is from the actual function value at any given \(x\). When you plot \(|f(x) - L(x)|\) over the interval \([-1, 2]\), the graph's peaks highlight where the linearization is least accurate.

To quantify this error, find the maximum value of \(|f(x) - L(x)|\) over the chosen interval. This maximum gives a benchmark of the largest possible error within the interval, indicating the overall performance of the linear approximation. The smaller the maximum error, the closer the linear estimation is to the function.

Function Approximation

Function approximation involves finding a simpler function that closely follows the behavior of a more complex one. Here, it is the relationship between the true function \(f(x)\) and its linearization \(L(x)\). This concept is especially valuable in situations where quick computations are vital, or where computing power is limited.

In our exercise, the linearized form \(L(x) = 3x - 3\) serves as an approximation to \(f(x) = x^3 + x^2 - 2x\) over the interval \([-1, 2]\). By plotting both functions on a single graph, you can visualize how closely the linearization follows the original function.

When these approximations are accurate over certain intervals or conditions, they simplify analysis and estimation. Engineers, scientists, and mathematicians often rely on these approximations to solve real-world problems more efficiently. However, the quality of an approximation can vary widely depending on the interval size and the function's complexity.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is software used to perform symbolic mathematics. It's the mathematical equivalent of having a powerful calculator that can handle complex equations and graphs but goes much further.

CAS can generate visual graphs of mathematical functions and execute symbolic computations—like derivatives or integrals—that might take much longer by hand. In exercises like ours, a CAS helps plot the original function \(f(x) = x^3 + x^2 - 2x\) and its linearization \(L(x) = 3x - 3\), over specified intervals like \([-1, 2]\).

This visual representation allows you to assess how well the linear approximation matches the original function. A CAS is also used to compute and visualize the absolute error \(|f(x) - L(x)|\) and to find its maximum over the interval. Such tools show not just numerical results but also how those results translate into graphical insights, enhancing understanding and supporting decision-making for more precise mathematical judgment.