Problem 18 - Calculus: One and Several Variables

Question

Assume that \(f\) is a function with \(\left|f^{(n)}(x)\right| \leq 1\) for all \(n\) and all real \(x\). (The sine and cosine functions have this property.) Estimate the maximum possible error if \(P_{7}(-2)\) is used to estimate \(f(-2)\)

Step-by-Step Solution

Answer

The maximum possible error if \(P_{7}(-2)\) is used to estimate \(f(-2)\) is \(\frac{256}{40320}\).

Step 1: Taylor Series Formula

\ \[ f(x) = P_{n}(x) + R_{n}(x) \] Here, \(P_n(x)\) is the Taylor polynomial and \(R_n(x)\) is the remainder term or Taylor series error (the error of our estimate). As we know,

Step 2: Taylor Polynomial

\ \[ P_{n}(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^{k} \] In our case, \(P_7(-2)\) is used to estimate \(f(-2)\), so here n = 7, x = -2, and a = 0. Now let's proceed with finding an estimation of the maximum error. To find the estimation, we'll use the following theorem - Taylor series error bound theorem:

Step 3: Taylor Series Error Bound Theorem

\ \[ \left|R_{n}(x)\right| \leq \frac{M}{(n+1)!}(x-a)^{n+1} \] Here, M is the maximum value of the (n+1)-th derivative in the interval [a, x]. Since \(\left|f^{(n)}(x)\right| \leq 1\) for all \(n\) and all real \(x\), for our case M = 1. Now we can estimate the maximum possible error for the given case using the Taylor series error bound theorem.

Step 4: Estimating Maximum Error

\ \[\left|R_{7}(-2)\right| \leq \frac{1}{(7+1)!}(-2-0)^{7+1}\] Calculating the expression, we have \[\left|R_{7}(-2)\right| \leq \frac{1}{8!} (2)^8 = \frac{256}{40320}\] Hence, the maximum possible error if \(P_{7}(-2)\) is used to estimate \(f(-2)\) is \(\frac{256}{40320}\).

Key Concepts

Taylor Polynomial

Taylor polynomials are an essential tool in calculus used for approximating complex functions with polynomials, which are generally easier to handle. At the core of a Taylor polynomial is its formula:

• For a function \( f(x) \), its Taylor polynomial of degree \( n \) around a point \( a \) is:

\[P_{n}(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^{k}\]This formula builds a polynomial approximation by using the derivatives of \( f \) at a point \( a \). Each term \( \frac{f^{(k)}(a)}{k!} (x-a)^{k} \) contributes to the accuracy of the approximation, accounting for the function's curvature.

• The zeroth term represents constant approximation.
• Higher order terms aim to capture more complex behavior such as slopes and curvatures.

The Taylor polynomial \( P_7(-2) \) in the original exercise is used to approximate \( f(-2) \), where the approximation power increases with higher \( n \). Each increase in \( n \) allows for more accurate reflection of the actual function, provided the required derivatives are both known and manageable.

Taylor Series Error Bound Theorem

When using Taylor polynomials for approximation, it's crucial to measure how good the approximation is. The Taylor Series Error Bound Theorem helps estimate this by bounding the size of the error. The theorem statement is:

• The error \( R_n(x) \) in approximating \( f(x) \) is:

\[\left|R_{n}(x)\right| \leq \frac{M}{(n+1)!}(x-a)^{n+1}\]This formula ensures that our approximation doesn’t stray too far from the true value of the function.

• \( M \) signifies the maximum value of the \((n+1)\)-th derivative of \( f \) within the interval \([a, x]\).
• \( (n+1)! \) is the factorial of \( n+1 \), rapidly increasing with \( n \) and thus contributing to reducing the error as \( n \) grows large.

In the exercise, we're told \( M = 1 \) because \(|f^{(n)}(x)| \leq 1\) for all \( n \), simplifying our task since we don’t have to compute higher derivatives. This makes our calculations more straightforward as we determine error bounds.

Maximum Error Estimation

Estimating maximum error is a pivotal step to ensure the reliability of function approximations using Taylor polynomials. Once the error formula is established, as per the Taylor Series Error Bound Theorem, you can plug in your specific values.In the exercise, we're tasked to find \( \left|R_{7}(-2)\right| \):

• \( a = 0 \) and \( x = -2 \), meaning we're evaluating around zero.
• Substituting these into the error formula, we compute:

\[\left|R_{7}(-2)\right| \leq \frac{1}{8!} (2)^8\]By completing these calculations:

• 8 factorial \( (8!) = 40320 \)
• \( (2)^8 = 256 \)

This gives us the maximum possible error as \( \frac{256}{40320} \), which indicates how much off our polynomial approximation \( P_7(-2) \) might be from the true value \( f(-2) \). Utilizing such estimations helps in understanding the potential deviation and in making informed decisions about whether the approximation is sufficiently precise for particular applications.