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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?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 64E
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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. We also know |f(t)| and |f'(t)| are bounded above by 3+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?
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. We also know |f(t)| and |f'(t)| are bounded above by 3+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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