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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please help.
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,
Step by step
Solved in 5 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
