Problem 20 - Calculus - AP Edition

Question

Confirm that the formula is the local linear approximation at \(x_{0}=0,\) and use a graphing utility to estimate an interval of \(x\) -values on which the error is at most ±0.1. $$\frac{1}{(1+2 x)^{5}} \approx 1-10 x$$

Step-by-Step Solution

Answer

The approximation is valid near \( x = 0 \) for \( |x| \leq 0.01 \) with error ±0.1.

Step 1: Understand the Formula

The given formula is \( \frac{1}{(1+2x)^5} \approx 1-10x \). This is in the form of a Taylor expansion at \( x_0 = 0 \), where we want to confirm if \( 1 - 10x \) is a good approximation.

Step 2: Local Linear Approximation

A linear approximation at \( x_0 = 0 \) for \( f(x) = \frac{1}{(1+2x)^5} \) is obtained from its Taylor series around \( x_0 \). Calculate \( f(0) = \frac{1}{1^5} = 1 \) and the derivative \( f'(x) = -10(1+2x)^{-6} \). At \( x_0 = 0 \), \( f'(0) = -10 \). Thus, \( f(x) \approx 1 - 10x \), confirming the formula.

Step 3: Estimate Error using Taylor's Remainder

The error for the linear approximation can be estimated using the Lagrange remainder, \( R_1(x) = \frac{f''(c)}{2}(x-x_0)^2 \). Calculate \( f''(x) \) and use \( c \) between 0 and the current \( x \) to bound the error.

Step 4: Use Graphing Utility

Graph the function \( g(x) = f(x) - (1 - 10x) \) to see where \( |g(x)| \leq 0.1 \). This helps find the range of \( x \) values where the approximation error is within ±0.1.

Step 5: Determine Interval with Bounded Error

The graph will show that for small values of \( x \), such as \( |x| \leq 0.01 \), the error is within the limit. Adjust the graph zoom level and testing points to confirm the appropriate interval of \( x \)-values.

Key Concepts

Taylor series

The Taylor series is a powerful mathematical tool that allows us to represent complex functions as infinite sums of their derivatives at a specific point. This concept is often used to approximate functions, especially when we only need a simple representation, like a polynomial, around a point of interest.

In our problem, we deal with a function, and we're looking at its behavior near the point where the variable, usually expressed as \(x\), is zero. For the function \(f(x) = \frac{1}{(1+2x)^5}\), the Taylor series provides a step-by-step way to approximate this function using the first few derivative terms. This process simplistically transforms it from a complex function into a simpler linear approximation, \(1 - 10x\), around \(x = 0\).

• First, we calculate the function value at \(x_0 = 0\).
• Then, we find the derivative values such as \(f'(x)\), evaluating these at the same point.
• By substituting these into a polynomial format, we get the local linear approximation.

These steps are the essence of using Taylor series for function approximation.

error estimation

Error estimation is crucial in understanding how accurate an approximation is. When we use a Taylor series to find a linear approximation of a function, this method gives us not just the approximation but also an idea of how closely it resembles the real function.

In this context, we estimated error by using different mathematical tools to gauge the deviation from the true function value. For example, we can use the Lagrange remainder, a part of the Taylor's theorem, to quantify how much our polynomial (here, \(1-10x\)) differs from the actual function over an interval around our point of expansion \(x_0\).

• Lagrange remainder formula is key to estimating the error.
• It involves calculating extra higher-order derivatives and evaluating them.
• This remainder predicts whether the difference between the approximated and actual function values is within acceptable bounds.

Through careful computation, step by step, we can refine our interval size where the error remains manageable, ensuring that users relying on these approximations can trust their precision for small values of \(x\).

Lagrange remainder

The Lagrange remainder is a technical yet fascinating aspect of the Taylor series, providing an upper bound for the approximation error. This remainder tells us how much our approximation might be off at various points in the interval.

The Lagrange remainder is expressed mathematically as:\[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1}.\]Here, \(c\) is some value in the interval between \(x_0\) and \(x\), \(f^{(n+1)}(c)\) represents the \((n+1)\)th derivative of the function, and \(n+1\) signifies the degree of approximation.

• This expression helps find the possible error magnitude based on subsequent derivatives not included in the approximation.
• By testing possible values in this interval, we locate the maximum error.
• For simpler approximations, like linear ones, we check the second derivatives to evaluate this remainder.

Effectively, this tool assures us that even beyond the included polynomial terms, the deviation from the actual function remains within a tolerable error range, particularly for smaller intervals and values close to \(x_0\).