Problem 43 - Essential Calculus

Question

In Exercises \(43-46,\) determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of \(x\) to be less than \(0.0001 .\) Use a computer algebra system to obtain and evaluate the required derivatives. $$ f(x)=\ln (x+1), \text { approximate } f(0.5) $$

Step-by-Step Solution

Answer

The degree of the Maclaurin polynomial required to approximate the function \(f(x)=\ln (x+1)\) at \(x = 0.5\) such that the error is less than 0.0001 can be obtained by solving the aforementioned inequality. The resulting value for \(n\) is the degree that ensures the approximation error to be smaller than 0.0001.

Step 1: Expression of the function as a Maclaurin series

The Maclaurin series expansion of \(f(x)\) is given by: \[f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \cdot\cdot\cdot\] For \(f(x)=\ln (x+1)\), its derivatives at x = 0 are \(f(0)=0, f'(0)=1, f''(0)=-1, f'''(0)=2, f''''(0)=-6\), and so on. The pattern for the nth derivative of the function at x = 0 is given as \(f^n(0) = (-1)^{n+1}(n-1)!\). This can be used to construct the Maclaurin series.

Step 2: Using Taylor’s Theorem to find error

From Taylor’s Theorem, the remainder or the error \(R_n\) of using the nth degree Taylor polynomial is given by: \[R_n = \frac{f^{(n+1)}(c)x^{n+1}}{(n+1)!}\] where c is between 0 and x. By noting that f is decreasing for x > 0 and applying the alternating series estimation theorem, we can replace c by 0. To get an error less than 0.0001, we need to find n for which the absolute value of \(R_n\) is less than 0.0001.

Step 3: Estimate the Degree of the Polynomial

Replace \(R_n\) in the inequality \(|R_n| \lt 0.0001\) with the value obtained in step 2 to derive a relationship for n. Solving the inequality \[\left|\frac{(-1)^{n+1}(n)!(0.5)^{n+1}}{(n+1)!}\right| \lt 0.0001\] will give the smallest integer value of \(n\) which ensures the error in the approximation would be less than 0.0001.

Step 4: Conclusion

Obtain the numerical solution of the inequality derived in Step 3 using a calculator or a computational tool. The resultant value of \(n\) is the minimal degree of the Maclaurin polynomial that ensures an error less than 0.0001.

Key Concepts

Taylor Series

A Taylor Series offers a powerful method to approximate complex functions using polynomials. The foundation of this series is built upon derivatives of a function evaluated at a specific point. It provides a way to express a function as an infinite sum of terms. Each term involves higher derivatives of the function, evaluated at a specific point, often denoted as "a."

For the Maclaurin series, which is a special case of the Taylor series where "a" is 0, the function expressed is centered at this point. This makes it especially useful for functions like the natural logarithm, which can be challenging to compute directly for every input value.

The general form of a Taylor series is given by:

• \(f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)(x-a)^2}{2!} + \cdots \)

For the problem where the function is \(\ln(x+1)\), the derivatives at \(x=0\) are calculated and used to build the Maclaurin series. This approximates \(f(0.5)\) effectively using several terms of the series.

Error Estimation

When using a Taylor series to approximate a function, estimating the error is crucial to understand how close the approximation is to the actual function. This is where the concept of the error term, or remainder, comes into play. The goal is to control this error to meet specific precision requirements.

Taylor's Theorem provides us with an error formula, which helps determine how accurate our approximation is. The error \(R_n\) after using an \(n\)-degree polynomial is given by the formula:

• \(R_n = \frac{f^{(n+1)}(c)x^{n+1}}{(n+1)!}\)

where \(c\) is some value between 0 and \(x\). For practical computation, especially when trying to find approximations within a small error margin, we approximate \(c\) to simplify calculations. For decreasing functions like \(\ln(x+1)\), \(c\) can often be approximated to zero, aiding in easier computations. By solving the inequality \(|R_n| < 0.0001\), we ensure that our approximation does not deviate beyond a predefined acceptable error threshold.

Polynomial Approximation

Polynomial approximation through Taylor and Maclaurin series is a strategic technique for estimating function values. After deriving the degree of the polynomial required to keep the error within a specific limit, we can use it to efficiently estimate the function's value at any given point.

In this context, the polynomial approximation is achieved by summing up a finite number of terms from the series expansion. This sum provides a polynomial function that closely mimics the behavior of the original function near a specific point. Since polynomials are much simpler to work with computationally than many complex functions, this method is widely applied in practical computational scenarios.

To determine the minimal number of terms needed for our approximation in the given exercise, we compute the number of terms required for the error to stay below a certain threshold, such as 0.0001. This process involves substituting values and solving inequalities until the smallest suitable degree of the polynomial that satisfies the error condition is found. This number will tell you how many terms to sum to achieve the desired approximation accuracy.