Calculus 2-Error Estimation Using Taylor Polynomials Questions, guaranteed and verified 100%

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Error Estimation Using Taylor Polynomials- HW Problems


Approximate the function 𝑓(𝑥) with a Taylor polynomial of the given
degree around the given point 𝑎. Estimate the accuracy of the Taylor
polynomial on the given interval.
1. 𝑓(𝑥 ) = √𝑥 , 𝑎 = 9, 𝑛 = 2, 9 ≤ 𝑥 ≤ 9.3
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2. 𝑓(𝑥 ) = , 𝑎 = 1, 𝑛 = 2, 0.9 ≤ 𝑥 ≤ 1.1
𝑥3
𝜋 𝜋 𝜋
3. 𝑓(𝑥 ) = cos(𝑥 ) , 𝑎= , 𝑛 = 4, ≤𝑥≤
4 6 3


4. How many terms of the Maclaurin series for 𝑓(𝑥 ) = 𝑒 𝑥 are needed
so that the absolute value of the error in 𝑒 0.2 is within 0.0001?


𝑥2 𝑥4
5. For what values of 𝑥 is cos(𝑥 ) ≈ 1 −
2
+ 24
with the absolute
value of the error less than 0.01?


𝑥2 𝑥3
6. For what values of 𝑥 is ln(1 + 𝑥 ) ≈ 𝑥 −
2
+ 3
with the absolute
value of the error less than 0.0001?


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7. Approximate ∫0 cos(𝑥 2 ) 𝑑𝑥 using the first 3 non-zero terms of
the Maclaurin series for 𝑓(𝑥 ) = cos(𝑥 2 ). How accurate is this
approximation?