Problem 35
Question
Estimate the error if \(P_{3}(x)=x-\left(x^{3} / 6\right)\) is used to estimate the value of \(\sin x\) at \(x=0.1 .\)
Step-by-Step Solution
Verified Answer
The error is approximately 0.0000041667.
1Step 1: Understand the Problem
We are given a polynomial, \(P_3(x) = x - \frac{x^3}{6}\), which is a third-degree Taylor polynomial approximation of \(\sin(x)\). We need to estimate the error when using this polynomial to approximate \(\sin(x)\) at \(x = 0.1\).
2Step 2: Identify the Remainder Term
The error of a Taylor polynomial approximation is given by the remainder term \(R_n(x)\) of the Taylor series. For \(\sin(x)\), this is \(R_3(x) = \frac{f^{(4)}(c)}{4!}x^4\) for some \(c\) in the interval \([0, x]\). The fourth derivative of \(\sin(x)\) is \(\sin(x)\) again.
3Step 3: Bound the Derivative
The function \(f(x) = \sin(x)\) has a maximum absolute value of 1 for its fourth derivative over any real interval, including \([0, 0.1]\). This tells us \(|f^{(4)}(c)| \leq 1\).
4Step 4: Calculate the Error Bound
Substituting the bound and the value of \(x\) into the remainder term gives us \[R_3(0.1) = \frac{1}{4!} \cdot (0.1)^4.\] Calculating this gives an error: \[R_3(0.1) = \frac{1}{24} \cdot 0.0001 = \frac{0.0001}{24} \approx 0.0000041667.\]
5Step 5: Conclude the Error Estimate
From this calculation, the error of the approximation \(P_3(x)\) for \(x = 0.1\) is estimated to be at most \(0.0000041667\). This means our approximation is quite precise in the estimate of \(\sin(0.1)\).
Key Concepts
Taylor polynomialerror estimationremainder term
Taylor polynomial
A Taylor polynomial is a specific type of polynomial used to approximate complex functions. It is derived from a Taylor series, by which we take several terms to estimate the value of a function around a particular point. The core idea involves evaluating the function and its derivatives at this point to create a simpler polynomial that resembles the original function nearby.
The third-degree Taylor polynomial, like the given example of \(P_3(x) = x - \frac{x^3}{6}\), is constructed to approximate the function \(\sin(x)\). To form such a polynomial, we need data on the values and derivatives of the function at the center of approximation, often \(x = 0\). More terms and derivatives used generally mean the Taylor polynomial approaches the actual function more accurately away from this point.
Taylor polynomials are particularly useful because they allow us to estimate and work with complex functions using simple algebraic equations. They can be adapted for different levels of precision by changing the degree of the polynomial.
The third-degree Taylor polynomial, like the given example of \(P_3(x) = x - \frac{x^3}{6}\), is constructed to approximate the function \(\sin(x)\). To form such a polynomial, we need data on the values and derivatives of the function at the center of approximation, often \(x = 0\). More terms and derivatives used generally mean the Taylor polynomial approaches the actual function more accurately away from this point.
Taylor polynomials are particularly useful because they allow us to estimate and work with complex functions using simple algebraic equations. They can be adapted for different levels of precision by changing the degree of the polynomial.
error estimation
Estimating errors in Taylor series approximations is crucial to understand how close our polynomial approximates the actual function. This estimation often involves the remainder term of the Taylor series, which tells us the potential deviation from the true function value.
When using a Taylor polynomial to approximate a function, we cannot always achieve exact results, especially as we move further from the center of approximation. Therefore, understanding how well a Taylor polynomial performs hinges on its error estimation.
Error estimation helps in assessing how significant an approximation error we might encounter. It involves mathematically bounding this error using derivatives and considering the highest order term not included in the Taylor polynomial. This way, by knowing the bounds, we can assess the reliability of the approximation provided by the Taylor polynomial.
When using a Taylor polynomial to approximate a function, we cannot always achieve exact results, especially as we move further from the center of approximation. Therefore, understanding how well a Taylor polynomial performs hinges on its error estimation.
Error estimation helps in assessing how significant an approximation error we might encounter. It involves mathematically bounding this error using derivatives and considering the highest order term not included in the Taylor polynomial. This way, by knowing the bounds, we can assess the reliability of the approximation provided by the Taylor polynomial.
remainder term
In a Taylor series, the remainder term often represents the potential error present in a polynomial approximation. Understanding the remainder term, sometimes noted as \(R_n(x)\), is essential for knowing how accurately a Taylor polynomial represents a function.
For our given exercise, the remainder term is calculated using the formula \(R_3(x) = \frac{f^{(4)}(c)}{4!}x^4\), where \(f^{(4)}(c)\) is the fourth derivative of the function evaluated at some point \(c\) within the interval \([0, x]\). In the case of \(\sin(x)\), this derivative repeats as \(\sin(x)\). With this term, we understand what lies beyond the third degree polynomial.
By bounding the fourth derivative to ensure consistency, we apply this within our error estimation to find how tight this polynomial's fit is around \(x = 0\). In practice, by estimating \(R_3(0.1)\) to be very small, we know the approximation of \(\sin(0.1)\) by \(P_3(x)\) is precise, with only a minor difference from the actual function.
For our given exercise, the remainder term is calculated using the formula \(R_3(x) = \frac{f^{(4)}(c)}{4!}x^4\), where \(f^{(4)}(c)\) is the fourth derivative of the function evaluated at some point \(c\) within the interval \([0, x]\). In the case of \(\sin(x)\), this derivative repeats as \(\sin(x)\). With this term, we understand what lies beyond the third degree polynomial.
By bounding the fourth derivative to ensure consistency, we apply this within our error estimation to find how tight this polynomial's fit is around \(x = 0\). In practice, by estimating \(R_3(0.1)\) to be very small, we know the approximation of \(\sin(0.1)\) by \(P_3(x)\) is precise, with only a minor difference from the actual function.
Other exercises in this chapter
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