Problem 27 - Thomas Calculus

Question

Find a polynomial that will approximate \(F(x)\) throughout the given interval with an error of magnitude less than \(10^{-3}.\) \begin{equation} F(x)=\int_{0}^{x} \tan ^{-1} t d t, \quad \text { (a) }[0,0.5] \quad \text { (b) }[0,1] \end{equation}

Step-by-Step Solution

Answer

Use the polynomial from \( x^2/2 - x^4/12 + x^6/30 \) with suitable terms for intervals. Adjust to keep error below \(10^{-3}\) for each.

Step 1: Understand the function

We are given a function defined as an integral, \( F(x) = \int_{0}^{x} \tan^{-1} t \, dt \). Our task is to approximate this function with a polynomial such that the error in the given interval is less than \(10^{-3}\).

Step 2: Consider Taylor Series Expansion

To approximate \( F(x) \), we first find the Taylor series expansion of \( \tan^{-1}(x) \) about 0. The series expansion for \( \tan^{-1}(x) \) is \( x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \).

Step 3: Determine Taylor Series of \( F(x) \)

To find the polynomial that approximates \( F(x) \), integrate the Taylor series of \( \tan^{-1}(x) \): \[ F(x) = \int (x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots) \, dx. \] This gives us: \[ F(x) = \frac{x^2}{2} - \frac{x^4}{12} + \frac{x^6}{30} - \frac{x^8}{56} + \cdots \]

Step 4: Assess Error for Polynomial Degree

The error for the Taylor polynomial is given by the next term in the series. For part (a) \([0,0.5]\), the error when using the term \( x^8/72 \) is \( |\frac{x^8}{72}| \). Solve \( |\frac{0.5^8}{72}| < 10^{-3} \). Similarly, calculate for \([0, 1]\).

Step 5: Determine Required Polynomial Degree

Evaluate to which variable power you need to calculate the term such that the error is less than \(10^{-3}\). Compute the necessary power ensuring it's within max error. Compare the effects at endpoints of both intervals.

Step 6: Finalize Polynomial Expression

For [0,0.5], use enough terms like \(x^6/30\) such that subsequent terms' effects are negligible under \(10^{-3}\). Repeat similarly for [0,1]. If more terms are needed for error control, integrate further terms from the series.

Key Concepts

Integral

An integral is a fundamental concept in calculus, representing the accumulation of quantities. In essence, it adds up infinite tiny quantities to calculate an overall total. For continuous functions, integrals often capture the area under a curve.
In the context of this problem, the function \( F(x) = \int_{0}^{x} \tan^{-1} t \, dt \) represents the area under the curve of \( \tan^{-1}(t) \) from 0 to \( x \).

• This specific integral, called a definite integral, provides the net accumulated value between two points—in this case, from 0 to \( x \).
• It is essential to understand the integral in order to transition to the polynomial approximation via series expansion.

Understanding integrals aids in making accurate approximations by series expansion and assessing how new terms affect the overall function behavior.

Polynomial Approximation

Polynomial approximation involves using a polynomial to closely match the behavior of a more complex function within a certain range. This is beneficial because polynomials are simpler and easier to calculate.
In this exercise, the aim is to approximate the integral function \( F(x) \) using a polynomial derived from the Taylor series of \( \tan^{-1}(t) \).

• Polynomials provide a good approximation when a few terms of the Taylor series are used.
• The higher the degree of the polynomial, the closer the match to the original function.
• For intervals like \([0, 0.5]\) and \([0, 1]\), determining the degree ensures the polynomial is accurate within the desired error margin.

Polynomial approximation not only simplifies computation but also helps in visualizing complex functions using simpler components.

Error Estimation

The concept of error estimation is crucial when approximating functions. It quantifies how off an approximation might be from the actual value of a function.
To gauge the effectiveness of our polynomial approximation for \( F(x) \), we estimate the error using the next term in the series expansion, beyond the final term included in the approximation.

• Each term in the series contributes progressively smaller impacts on the function.
• For the interval \([0,0.5]\), the error term is something like \( |\frac{x^8}{72}| \). To meet criteria, it should be smaller than \(10^{-3}\).
• This step ensures that the polynomial remains a reliable approximation throughout the specified interval.

Having a sound error estimation gives confidence that the polynomial is not just easier to compute but also close in accuracy to the integral's true value.

Series Expansion

Series expansion transforms a function into an infinite sum of terms calculated from that function's derivatives at a single point. In this exercise, the Taylor series expansion is employed to represent \( \tan^{-1}(x) \) as a sum of polynomial terms.
The Taylor series for \( \tan^{-1}(x) \) about zero is:
\[ x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \]

• Each term is critical; the coefficients are inversely related to odd integers.
• Integrating these terms provides the polynomial approximation for the function \( F(x) \).
• The series expansion helps transform a complex function into a usable polynomial form.

Series expansions are powerful tools in simplifying functions, especially when dealing with integrals and other calculus-related tasks.