Problem 28 - Calculus for Biology and Medicine

Question

Use the following form of the error term $$ \left|R_{n+1}(x)\right| \leq \frac{K|x|^{n+1}}{(n+1) !} $$ where \(K=\) largest value of \(\left|f^{(n+1)}(t)\right|\) for \(0 \leq t \leq x\), to determine in advance the degree of Taylor polynomial at \(a=0\) that would achieve the indicated accuracy in the interval \([\mathbf{0}, \boldsymbol{x}] .\) (Do not compute the Taylor polynomial.) $$ f(x)=\cos x, x=1, \text { error }<10^{-2} $$

Step-by-Step Solution

Answer

The required degree of the Taylor polynomial is 4.

Step 1: Understand the Taylor Polynomial Error Formula

We are given the error term \( |R_{n+1}(x)| \leq \frac{K|x|^{n+1}}{(n+1)!} \). This formula provides an estimate for the error in approximating the function \( f(x) \) using a Taylor polynomial of degree \( n \) about \( a=0 \). Here, \( K \) is the maximum value of \( |f^{(n+1)}(t)| \) for \( 0 \leq t \leq x \). Our goal is to find the degree \( n \) such that the error is less than \( 10^{-2} \) when \( x=1 \).

Step 2: Determine the Function and its Derivatives

The function given is \( f(x) = \cos x \). The derivatives follow a pattern: \( f'(x) = - ext{sin}(x) \), \( f''(x) = - ext{cos}(x) \), \( f^{(3)}(x) = ext{sin}(x) \), and \( f^{(4)}(x) = ext{cos}(x) \), repeating every four derivatives.

Step 3: Identify the Largest Derivative Value

Since \( f^{(n+1)}(t) \) is either \( ext{sin}(t) \) or \( ext{cos}(t) \) depending on \( n \), the maximum absolute value over \( 0 \leq t \leq 1 \) is 1 (as both sine and cosine have maximum values of 1). Therefore, \( K = 1 \).

Step 4: Set Up the Inequality for the Error Term

Substitute \( K = 1 \), \( |x| = 1 \) (since we evaluate at \( x=1 \)), and the desired error bound into the inequality: \( \frac{1 \cdot 1^{n+1}}{(n+1)!} < 10^{-2} \). This simplifies to \( \frac{1}{(n+1)!} < 10^{-2} \).

Step 5: Solve for the Degree of Polynomial

To find the smallest \( n \) that satisfies the inequality, try successive values of \( (n+1)! \). First check \( n+1 = 1 \): \( 1! = 1 \) which does not satisfy. Continue until \( n+1 = 5: 5! = 120 \), which satisfies \( \frac{1}{120} < 10^{-2} \). Therefore, \( n = 4 \), the degree of the polynomial needed is 4.

Key Concepts

Error Term

An important part of Taylor polynomials is the error term, which helps us understand how accurately the polynomial approximates a function. In the equation \( |R_{n+1}(x)| \leq \frac{K|x|^{n+1}}{(n+1)!} \), the error term \( R_{n+1}(x) \) represents the difference between the actual function and the Taylor polynomial of degree \( n \). This is pivotal for determining how much error or deviation one can expect when approximating.The goal is to keep this error small enough to meet a specific tolerance level, usually denoted by a small number like \( 10^{-2} \) or 0.01. Here, \( K \) is the largest value of \( |f^{(n+1)}(t)| \) within the interval from 0 to \( x \). The factor \( (n+1)! \) in the denominator renormalizes the error bound, making the error smaller as \( n \) increases. For any function, knowing \( K \) helps set a boundary to ensure the polynomial accurately represents the function within desired limits.

Cosine Function

The cosine function, \( f(x) = \cos x \), is a periodic function known for its wave-like pattern. It is highly significant in both mathematics and physics for modeling periodic phenomena. When approximating \( \, \cos x \, \) with a Taylor polynomial, understanding its properties and derivatives is crucial.The cosine function has symmetrical properties: it repeats every \( 2\pi \) radians, displaying peaks and troughs at consistent intervals. This cyclic pattern means that its derivatives also repeat. For example:

• The first derivative, \( f'(x) = -\sin x \), flips the function to a sine wave.
• The second derivative, \( f''(x) = -\cos x \), mirrors the original function.
• This rotation continues with the third and fourth derivatives returning to \( \sin x \) and \( \cos x \) respectively.

Such periodicity simplifies calculations in Taylor polynomial approximations since the pattern is predictable, and for \( \, 0 \leq x \leq 1 \, \), the maximum magnitude of cosine or sine remains 1.

Derivatives

Derivatives provide insight into the behavior of a function by showing how it changes over different points. For functions like \( f(x) = \cos x \), derivatives help us construct a Taylor polynomial, offering a polynomial approximation that mirrors the function's behavior close to a specific point.When working with Taylor polynomials, the derivative's order increases the function's representation accuracy. Calculating these derivatives helps determine the polynomial's degree needed to achieve a specific error threshold. For the cosine function, derivatives are cyclical:

• With the first derivative, \( -\sin x \), the function's rate of change is determined.
• The second derivative, \( -\cos x \), indicates the original function's concavity.
• Higher-order derivatives continue this pattern, switching between sine and cosine.

By knowing that the maximum absolute value of these derivatives is 1 in the specified interval, one can effectively use this information to calculate \( K \) and thereby set limits on the approximation error. This understanding enables the determination of how many derivatives (or polynomial degree) are necessary to achieve the desired precision.