QuestionAssignment 4: Problem 11
(1 point)
Find , the degree 5 Taylor polynomial of the function at .
Find all values of for which this approximation is within 0.004794 of the right answer. Assume for simplicity that we limit ourselves to .
Note: You can earn partial credit on this problem.
Studdy Solution
STEP 1
1. We are finding the degree 5 Taylor polynomial of at .
2. We need to find the values of for which this approximation is within 0.004794 of the true value.
3. We are limited to .
STEP 2
1. Derive the Taylor series expansion for at .
2. Identify the terms up to degree 5 to form .
3. Calculate the error term for the Taylor approximation.
4. Determine the range of for which the error is within the specified bound.
STEP 3
The Taylor series for at is given by:
STEP 4
Identify the terms up to degree 5:
STEP 5
Write out the polynomial:
STEP 6
The error term for the Taylor polynomial is given by the next term in the series:
STEP 7
We want the error to satisfy:
STEP 8
Solve the inequality:
Take the sixth root:
STEP 9
Calculate the sixth root:
The degree 5 Taylor polynomial is:
The values of for which the approximation is within 0.004794 are:
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