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1. A group of students find out that the growth of the population increases at a rate that is proportional to the amount of people at any given time. They relate this increase on the population to the amount of people that will use a highway in the upcoming years in a city. A function that is used to model this increase is ex, and an infinite series (Maclaurin series) that is used to approximate e is given as: X x² ex=1+ + + + ... + 1! 2! 3! xn n! a) Prove that this Maclaurin series expansion is a special case of the Taylor series expansion with x₁ = 0 and h = x b) The highway will close on weekends, and there will be an exponential decay on pollution during those days in the area. The decay can be modeled by the function e*. Use the Taylor series to estimate f(x) = eat X₁+1=1 for x₁ = 0.2. Employ the zero-, first-, second-, and third-order versions and compute the || for each case.

1. A group of students find out that the growth of the population increases at a rate that is proportional to the amount of people at any given time. They relate this increase on the population to the amount of people that will use a highway in the upcoming years in a city. A function that is used to model this increase is ex, and an infinite series (Maclaurin series) that is used to approximate e is given as: X x² ex=1+ + + + ... + 1! 2! 3! xn n! a) Prove that this Maclaurin series expansion is a special case of the Taylor series expansion with x₁ = 0 and h = x b) The highway will close on weekends, and there will be an exponential decay on pollution during those days in the area. The decay can be modeled by the function e*. Use the Taylor series to estimate f(x) = eat X₁+1=1 for x₁ = 0.2. Employ the zero-, first-, second-, and third-order versions and compute the || for each case.

Advanced Engineering Mathematics
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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1. A group of students find out that the growth of the population increases at a rate that is proportional to the amount of people at any given time. They relate this increase on the population to the amount of people that will use a highway in the upcoming years in a city. A function that is used to model this increase is ex, and an infinite series (Maclaurin series) that is used to approximate e is given as: e = 1 + x² x³ + + + + X 1! 2! 3! x" n! a) Prove that this Maclaurin series expansion is a special case of the Taylor series expansion with x₁ = 0 and h = x b) The highway will close on weekends, and there will be an exponential decay on pollution during those days in the area. The decay can be modeled by the function e*. Use the Taylor series to estimate f(x) = eat X₁+1=1 for x = 0.2. Employ the zero-, first-, second-, and third-order versions and compute the || for each case.
Transcribed Image Text:1. A group of students find out that the growth of the population increases at a rate that is proportional to the amount of people at any given time. They relate this increase on the population to the amount of people that will use a highway in the upcoming years in a city. A function that is used to model this increase is ex, and an infinite series (Maclaurin series) that is used to approximate e is given as: e = 1 + x² x³ + + + + X 1! 2! 3! x" n! a) Prove that this Maclaurin series expansion is a special case of the Taylor series expansion with x₁ = 0 and h = x b) The highway will close on weekends, and there will be an exponential decay on pollution during those days in the area. The decay can be modeled by the function e*. Use the Taylor series to estimate f(x) = eat X₁+1=1 for x = 0.2. Employ the zero-, first-, second-, and third-order versions and compute the || for each case.
Expert Solution
Step 1

Given Data:

Let us consider the given data,

ex=1+x1!+x22!+x33!+....+xnn!

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