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. 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?
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. 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?
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter2: Functions
Section: Chapter Questions
Problem 30P: In this problem you are asked to find a function that models in real life situation and then use the...
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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.
%3|
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%2Ff2b9fbfb-992e-4207-8e5e-c107bb952d65%2Fc2fe414a-d46f-4c6c-98e1-efb8acf5968e%2F2i7y4k_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.
%3|
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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