If T4(x) is the fourth Maclaurin polynomial for the function f(x) = e*, then |e²-T₁(2)| 4e² 15 ≤-
If T4(x) is the fourth Maclaurin polynomial for the function f(x) = e*, then |e²-T₁(2)| 4e² 15 ≤-
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
true or false
![**Understanding the Approximations of Exponential Functions with Maclaurin Polynomials**
In calculus, Maclaurin polynomials provide an important method to approximate functions. Let’s explore this concept with the exponential function \( f(x) = e^x \).
Consider \( T_4(x) \), the fourth Maclaurin polynomial for the function \( f(x) = e^x \). The Maclaurin series for \( e^x \) is an infinite series, but for approximation purposes, we often use a finite number of terms.
For \( T_4(x) \), the polynomial looks like this:
\[ T_4(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} \]
Given this fourth-degree polynomial, we can approximate \( e^2 \) as \( T_4(2) \).
The error bound in this polynomial approximation is given by the difference between the actual function value and the polynomial value at a specific point. Mathematically, this is expressed as:
\[ | e^2 - T_4(2) | \leq \frac{4e^2}{15} \]
This inequality indicates that the absolute error between the actual value \( e^2 \) and its approximation using the fourth Maclaurin polynomial \( T_4(2) \) is at most \( \frac{4e^2}{15} \).
Understanding this can help you estimate how accurate your polynomial approximation might be in practical applications. Maclaurin polynomials are widely used for numerical methods, analysis, and solving differential equations, providing a simpler way to work with complex functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff24651e9-0a43-4fcf-8170-c5e34dc6dcab%2F6c19921d-44b2-4b69-8a4c-4eb25a8e75ce%2Fvaekm8n_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding the Approximations of Exponential Functions with Maclaurin Polynomials**
In calculus, Maclaurin polynomials provide an important method to approximate functions. Let’s explore this concept with the exponential function \( f(x) = e^x \).
Consider \( T_4(x) \), the fourth Maclaurin polynomial for the function \( f(x) = e^x \). The Maclaurin series for \( e^x \) is an infinite series, but for approximation purposes, we often use a finite number of terms.
For \( T_4(x) \), the polynomial looks like this:
\[ T_4(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} \]
Given this fourth-degree polynomial, we can approximate \( e^2 \) as \( T_4(2) \).
The error bound in this polynomial approximation is given by the difference between the actual function value and the polynomial value at a specific point. Mathematically, this is expressed as:
\[ | e^2 - T_4(2) | \leq \frac{4e^2}{15} \]
This inequality indicates that the absolute error between the actual value \( e^2 \) and its approximation using the fourth Maclaurin polynomial \( T_4(2) \) is at most \( \frac{4e^2}{15} \).
Understanding this can help you estimate how accurate your polynomial approximation might be in practical applications. Maclaurin polynomials are widely used for numerical methods, analysis, and solving differential equations, providing a simpler way to work with complex functions.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
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,
