Use the Alternating Series Estimation Theorem to estimate the range of values of x for which the given approximation is accurate to within the stated error. (Round the answer to three decimal places.) Check your answer graphically.
Given:
sin x ~= x - x^3/6, \(|error| < 0.00000001\)
Solve:
? < x < ?
To use the Alternating Series Estimation Theorem, we need to find the interval of x values for which the error of the given approximation is within the desired tolerance.
The Alternating Series Estimation Theorem states that for an alternating series \( \sum (-1)^n a_n \), if the terms satisfy:
1. The terms are decreasing in absolute value, that is, \( a_{n+1} < a_n \)
2. The terms approach zero as \( n \) approaches infinity, that is, \( \lim_{n\to\infty} a_n = 0 \)
Then the error of the alternating series approximation is less than or equal to the absolute value of the next term not included in the sum.
In this case, we have an alternating series approximation for \( \sin x \), given by \( x - \frac{x^3}{6} \).
To find the interval of x values for which the approximation is accurate within the desired error, we need to find the next term not included in the sum and set the absolute value of that term less than the desired error.
Let's find the next term:
The given approximation is \( x - \frac{x^3}{6} \).
The next term would be \( \frac{x^5}{120} \).
To ensure the error is less than \( 0.00000001 \), we set the next term's absolute value less than this value:
\( \left| \frac{x^5}{120} \right| < 0.00000001 \)
Simplifying this inequality gives:
\( |x|^5 < 0.00000001 \times 120 \)
\( |x|^5 < 0.0000012 \)
To solve for the range of x values, we take the fifth root of both sides:
\( |x| < \sqrt[5]{0.0000012} \)
Taking the fifth root of 0.0000012 gives approximately 0.01113.
Therefore, the range of x values for which the given approximation is accurate to within the stated error is:
\( -0.01113 < x < 0.01113 \)
To check this graphically, plot the graph of the given approximation \( x - \frac{x^3}{6} \) along with the graph of \( \sin x \). The range of x values for which the two graphs are within 0.00000001 of each other is expected to be within the interval \( -0.01113 < x < 0.01113 \).