Consider a function f(t). It is known that it satisfies the following conditions: f"(t) = f(t), f(1) = 0, f'(1) = 1.
Consider a function f(t). It is known that it satisfies the following conditions: f"(t) = f(t), f(1) = 0, f'(1) = 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![4. Giving a precise mathematical description of natural phenomenon can be quite difficult. It
is often more practical for scientists and engineers to describe nature by looking for relations
among how it changes. In certain situations, Taylor polynomials can be used to approximate
functions without an explicit description.
Consider a function f(t). It is known that it satisfies the following conditions:
f"(t) = f(t), f(1) = 0, f'(1) = 1.
%3D
%3D
We also know |f(t)| and |f'(t)| are bounded above by 3t+1 over the interval [0, 2].
(a) Compute Taylor polynomial of degree 3 for f(t) centred at 1.
(b) Using information available, what is the lowest degree Taylor polynomial for f(x) centred
at 1 to guarantee an approximation of f(2) to an absolute error to within 0.001?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73102c60-1498-4636-80ef-4cdd7b0f71d1%2F0db42eab-c9eb-45ae-ad57-60c0f3272740%2Fhv6886t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Giving a precise mathematical description of natural phenomenon can be quite difficult. It
is often more practical for scientists and engineers to describe nature by looking for relations
among how it changes. In certain situations, Taylor polynomials can be used to approximate
functions without an explicit description.
Consider a function f(t). It is known that it satisfies the following conditions:
f"(t) = f(t), f(1) = 0, f'(1) = 1.
%3D
%3D
We also know |f(t)| and |f'(t)| are bounded above by 3t+1 over the interval [0, 2].
(a) Compute Taylor polynomial of degree 3 for f(t) centred at 1.
(b) Using information available, what is the lowest degree Taylor polynomial for f(x) centred
at 1 to guarantee an approximation of f(2) to an absolute error to within 0.001?
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