Problem 46 - Calculus: One and Several Variables

Question

Let \(P_{s}\) be the \(n\) th Taylor polynomial for the function $$f(x)=\sin x$$ Find the least integer \(n\) for which: (a) \(P_{n}(1)\) approximates sin 1 within 0.001; (b) \(P_{n}(2)\) approximates sin 2 within 0.001. (c) \(P_{n}(3)\) approximate \(\sin 3\) within 0.001.

Step-by-Step Solution

Answer

In conclusion, the least integer \(n\) for which the \(n\)th Taylor polynomial approximates \(\sin(1)\) within 0.001 is 6, approximates \(\sin(2)\) within 0.001 is 10, and approximates \(\sin(3)\) within 0.001 is 12.

Step 1: 1. Recall the Taylor polynomial formula for the sine function

The Taylor polynomial formula for the sine function is given by: \(P_n(x) = \sum_{k=0}^n \frac{(-1)^k}{(2k + 1)!}x^{2k+1}\) To find the least integer n for which the Taylor polynomial approximates sin(x) within 0.001, we will use the Lagrange error bound formula:

Step 2: 2. Recall the Lagrange error bound formula

The error, E, in approximating a function using Taylor polynomials is bounded by the Lagrange error bound formula: \(E = \frac{M|((x - a)^{n+1})|} {(n + 1)!}\) where M is an upper bound for the absolute value of the (n+1)th derivative of the function in the interval containing a and x. For the sine function, the derivatives repeat every four steps. Specifically, the upper bound for the absolute value of each derivative will be 1.

Step 3: 3. Solving for the least integer n

Now, we need to find the least integer n for which: (a) \(E = \frac{1 \cdot |(1-0)^{n+1}|}{(n + 1)!} \leq 0.001\) (b) \(E = \frac{1\cdot|(2-0)^{n+1}|}{(n + 1)!} \leq 0.001\) (c) \(E = \frac{1\cdot|(3-0)^{n+1}|}{(n + 1)!} \leq 0.001\) (a) and (b) can be simplified by removing the absolute value as x and a are both positive. (a) To find n for \(P_{n}(1)\): \(E = \frac{1^{n+1}}{(n + 1)!} \leq 0.001\) Try n=2: \(E = \frac{1^{3}}{(3)!} = \frac{1}{6} = 0.166666... > 0.001\) Try n=4: \(E = \frac{1^{5}}{(5)!} = \frac{1}{120} = 0.008333... > 0.001\) Try n=6: \(E = \frac{1^{7}}{(7)!} = \frac{1}{5040} = 0.0001984... \leq 0.001\) So, for \(P_{n}(1)\), n=6. (b) To find n for \(P_{n}(2)\): \(E = \frac{2^{n+1}}{(n + 1)!} \leq 0.001\) Try n=6: \(E = \frac{2^{7}}{(7)!} = \frac{128}{5040} = 0.025396... > 0.001\) Try n=8: \(E = \frac{2^{9}}{(9)!} = \frac{512}{362880} = 0.001410... > 0.001\) Try n=10: \(E = \frac{2^{11}}{(11)!} = \frac{2048}{3991680920} = 0.000816\) So, for \(P_{n}(2)\), n=10. (c) To find n for \(P_{n}(3)\): \(E = \frac{3^{n+1}}{(n + 1)!} \leq 0.001\) Try n=10: \(E = \frac{3^{11}}{(11)!} = \frac{177147}{39916800} = 0.004433... > 0.001\) Try n=12: \(E = \frac{3^{13}}{(13)!} = \frac{1594323}{6227020800} = 0.000256\) So, for \(P_{n}(3)\), n=12. In conclusion, the least integer n for which the nth Taylor polynomial approximates sin(1) within 0.001 is 6, approximates sin(2) within 0.001 is 10, and approximates sin(3) within 0.001 is 12.

Key Concepts

Lagrange Error Bound

The Lagrange Error Bound is a crucial tool for estimating the accuracy of a Taylor polynomial approximation. When we use a Taylor polynomial to approximate a function, the error bound helps us understand how close our approximation is to the actual value of the function. This is vital when determining how many terms of the polynomial we need to include to achieve a specific degree of accuracy.

The formula for the Lagrange Error Bound is:

• \(E = \frac{M|((x - a)^{n+1})|}{(n + 1)!}\)

Here, \(M\) represents an upper bound on the absolute value of the (\(n+1\))th derivative of the function over the interval from \(a\) to \(x\). This ensures that we're accounting for the largest possible error introduced by excluding higher-order terms.

For the sine function, whose derivatives are cyclic, the maximum value \(|f^{(n+1)}(x)|\) can attain is 1. Thus, the error bound simplifies our calculations significantly. By finding \(E\) less than or equal to the desired accuracy (e.g., 0.001 in our case), we strategically choose \(n\) to meet our precision requirements.

Sine Function

The sine function, typically denoted as \(f(x) = \sin(x)\), is one of the principal trigonometric functions, which you will frequently encounter in mathematics and physics. This function is periodic with a period of \(2\pi\), meaning its values repeat after every \(2\pi\) interval.

The derivatives of the sine function form a cycle:

• \(f(x) = \sin(x)\)
• \(f'(x) = \cos(x)\)
• \(f''(x) = -\sin(x)\)
• and it continues in this pattern.

This repeating nature significantly helps anticipate the behavior of derivatives for higher-order Taylor polynomials. When approximating \( \sin(x) \), the pattern ensures that the derivatives do not grow unbounded, facilitating a straightforward application of the Taylor series and Lagrange Error Bound. Knowing these characteristics is essential to manipulate the function and its approximations effectively.

Polynomial Approximation

Polynomial approximation is a fundamental concept in calculus that allows us to estimate complex functions using simpler polynomial expressions. This approach is particularly helpful when dealing with functions like \(\sin(x)\), which can be cumbersome to calculate precisely by hand or using basic calculators.

The Taylor polynomial is a special kind of polynomial approximation that uses derivatives of the function at a specific point to create an approximation that gets increasingly accurate as more terms are added. For the function \(f(x) = \sin(x)\), its Taylor series around \(x = 0\) is:

• \(P_n(x) = \sum_{k=0}^n \frac{(-1)^k}{(2k + 1)!}x^{2k+1}\)

Each term in this polynomial represents an effort to correct the approximation using derivatives of the sine function. The more terms we include, the closer the polynomial comes to representing the actual sine function. However, each additional term also means more complex calculations, so concepts like the Lagrange Error Bound become invaluable for deciding when having enough terms balances practical computation with the desired level of accuracy.