QuestionLet .
Find the MacLaurin polynomial of degree 7 for .
Use this polynomial to estimate the value of .
Studdy Solution
STEP 1
1. We are given the function .
2. We need to find the Maclaurin polynomial of degree 7 for .
3. We will use the Maclaurin polynomial to estimate the value of .
STEP 2
1. Find the derivatives of up to the 7th derivative.
2. Evaluate each derivative at to find the coefficients of the Maclaurin series.
3. Construct the Maclaurin polynomial of degree 7.
4. Use the polynomial to estimate the integral from 0 to 0.73.
STEP 3
First, find the first derivative of using the Fundamental Theorem of Calculus:
STEP 4
Find the second derivative:
STEP 5
Find the third derivative:
STEP 6
Continue finding derivatives up to the 7th derivative. The pattern involves using the product rule and chain rule repeatedly.
STEP 7
Evaluate each derivative at :
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- Continue this process for higher derivatives.
STEP 8
Construct the Maclaurin polynomial using the evaluated derivatives:
Continue constructing the polynomial up to the 7th degree.
STEP 9
Use the Maclaurin polynomial to estimate the integral:
Calculate the value of the polynomial at .
The estimated value of the integral is:
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