Problem 29 - Essential Calculus

Question

Numerical and Graphical Approximations In Exercises 29 and \(\mathbf{3 0}\) (a) find the Maclaurin polynomial \(P_{3}(x)\) for \(f(x)\), complete the table for \(f(x)\) and \(P_{3}(x),\) and \((c)\) sketch the graphs of \(f(x)\) and \(P_{3}(x)\) on the same set of coordinate axes. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & -0.75 & -0.50 & -0.25 & 0 & 0.25 & 0.50 & 0.75 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & & \\ \hline \boldsymbol{P}_{3}(\boldsymbol{x}) & & & & & & & \\ \hline \end{array} $$ $$ f(x)=\arcsin x $$

Step-by-Step Solution

Answer

\(P_{3}(x) = x + \frac{x^3}{6}\). The table applies these functions to -0.75, -0.50, -0.25, 0, 0.25, 0.50, 0.75. The graph shows that \(P_{3}(x)\) closely approximates \(f(x) = \arcsin x\) around the origin.

Step 1: Compute the Maclaurin polynomial \(P_{3}(x)\)

The Maclaurin polynomial \(P_{3}(x)\) of \(f(x) = \arcsin x\) can be computed as the sum of the first three derivatives of \(f(x)\) at \(x = 0\). For \(f(x) = \arcsin x\) we have \(f＇(x) = \frac{1}{\sqrt{1-x^2}}\), \(f＇＇(x) = \frac{x}{(1-x^2)^(3/2)}\) and \(f＇＇＇(x) = \frac{1}{(1-x^2)^(3/2)} + \frac{3x^2}{(1-x^2)^(5/2)}\). At \(x = 0\), these derivatives are \(f＇(0) = 1\), \(f＇＇(0) = 0\) and \(f＇＇＇(0) = 1\). Thus, \(P_{3}(x) = f(0) + x*f＇(0)/1! + (x^2)*f＇＇(0)/2! + (x^3)*f＇＇＇(0)/3! = 0 + x - 0 + \frac{x^3}{6}\.

Step 2: Fill the table

Apply the given x-values to \(f(x) = \arcsin x\) and \(P_{3}(x) = x + \frac{x^3}{6}\) to get the respective y-values.

Step 3: Sketch the Graphs

Using the computed values for \(f(x)\) and \(P_{3}(x)\) at the given x-points, plot the functions on the same set of axes. The Maclaurin polynomial \(P_{3}(x)\) will closely resemble the function \(f(x) = \arcsin x\) in a given interval around the origin.

Key Concepts

Numerical Approximation

Numerical approximation is a powerful tool in mathematics. It lets us estimate the value of complex functions using simpler expressions. In this exercise, we approximate the function \( f(x) = \arcsin x \) with its Maclaurin polynomial \( P_3(x) \).

The Maclaurin polynomial is essentially a type of Taylor series, calculated specifically at \( x = 0 \). Using derivatives of \( f(x) \) at this point, we form a polynomial that mimics the function closely near \( x = 0 \). For \( f(x) = \arcsin x \), the first three derivatives are used to calculate \( P_3(x) \), resulting in the polynomial \( x + \frac{x^3}{6} \). This polynomial gives us a simple way to calculate values that are approximately equal to \( \arcsin x \) for \( x \) around 0.

Approximations like this offer huge computational benefits. They reduce the need for complex calculations, saving time and resources. They're especially useful in fields like engineering and physics, where quick, approximate solutions are often more practical than exact answers.

Graphical Approximation

Graphical approximation provides a visual representation of how closely a polynomial can approximate a function. By plotting both the original function \( f(x) = \arcsin x \) and the Maclaurin polynomial \( P_3(x) = x + \frac{x^3}{6} \), we can see their similarities and differences.

When graphed on the same axes, the curves of \( f(x) \) and \( P_3(x) \) may overlap significantly around small values of \( x \) near zero. This overlap indicates a good approximation within that interval. However, as \( x \) moves away from zero, the polynomial may deviate from the actual function, showing the limitations of using such an approximation over a broader range.

Visual tools like graphs enhance understanding. They can reveal where approximations are strong or where they break down. For educators and learners alike, seeing the relationship between a function and its approximation can deepen insights into numerical methods and their applications.

Taylor Series

A Taylor series is a mathematical concept that represents functions as infinite sums of terms, calculated from the values of the function's derivatives at a single point. The Maclaurin series is a special case of the Taylor series, where that single point is \( x = 0 \).

In this exercise, constructing the Maclaurin polynomial involves taking the first three derivatives of \( f(x) = \arcsin x \) at \( x = 0 \). This leads us to the polynomial \( P_3(x) = x + \frac{x^3}{6} \). These finite polynomial terms are approximations of the infinite Taylor series

These series are vital because they can simplify complex mathematical models, providing manageable means to approximate functions that are otherwise difficult to compute exactly. Taylor series are used extensively in calculus, physics, and engineering.

Understanding Taylor series involves grasping how derivatives influence function shape. These approximations are at the heart of many models predicting real-world behavior, underlining the power of calculus in scientific exploration.