Mechanical Engineering Exam  >  Mechanical Engineering Questions  >  In the Taylor series expansion of exp(x) + si... Start Learning for Free
In the Taylor series expansion of exp(x) + sin(x) about the point x = π, the coefficient of (x – π)2 is  
  • a)
    exp (π )    
  • b)
    0.5 exp (π)  
  • c)
    exp (π ) + 1      
  • d)
    exp(π)  – 1 
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
In the Taylor series expansion of exp(x) + sin(x) about the point x = ...
View all questions of this test
Most Upvoted Answer
In the Taylor series expansion of exp(x) + sin(x) about the point x = ...
Taylor Series Expansion

The Taylor series expansion is a mathematical tool used to represent a function as a sum of infinite terms in a power series. It is used to approximate functions that are difficult to calculate by hand.

Given a function f(x), the Taylor series expansion of f(x) about a point x=a is given by:

f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...

where f'(a), f''(a), f'''(a), etc. are the first, second, third, etc. derivatives of f(x) evaluated at x=a.

Problem Statement

In the given problem, we are asked to find the coefficient of (x-a)^2 in the Taylor series expansion of f(x) = exp(x) sin(x) about the point x=a.

Solution

To find the Taylor series expansion of f(x), we first need to find its derivatives:

f(x) = exp(x) sin(x)
f'(x) = exp(x) sin(x) + exp(x) cos(x)
f''(x) = 2exp(x) cos(x)

Now, we can write the Taylor series expansion of f(x) about x=a as:

f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ...

where:

f(a) = exp(a) sin(a)
f'(a) = exp(a) sin(a) + exp(a) cos(a)
f''(a) = 2exp(a) cos(a)

Substituting these values in the Taylor series expansion, we get:

f(x) = exp(a) sin(a) + [exp(a) sin(a) + exp(a) cos(a)](x-a) + [2exp(a) cos(a)](x-a)^2/2! + ...

Simplifying this expression, we get:

f(x) = exp(a) sin(a) + exp(a)(sin(a) + cos(a))(x-a) + exp(a)cos(a)(x-a)^2/2! + ...

The coefficient of (x-a)^2 is given by the coefficient of exp(a)cos(a)/2! which is 0.5exp(a)cos(a).

Therefore, the correct answer is option B: 0.5exp(a)cos(a).
Attention Mechanical Engineering Students!
To make sure you are not studying endlessly, EduRev has designed Mechanical Engineering study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Mechanical Engineering.
Explore Courses for Mechanical Engineering exam

Top Courses for Mechanical Engineering

In the Taylor series expansion of exp(x) + sin(x) about the point x = π, the coefficient of (x – π)2 is a)exp (π ) b)0.5 exp (π) c)exp (π ) + 1 d)exp(π) – 1Correct answer is option 'B'. Can you explain this answer?
Question Description
In the Taylor series expansion of exp(x) + sin(x) about the point x = π, the coefficient of (x – π)2 is a)exp (π ) b)0.5 exp (π) c)exp (π ) + 1 d)exp(π) – 1Correct answer is option 'B'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about In the Taylor series expansion of exp(x) + sin(x) about the point x = π, the coefficient of (x – π)2 is a)exp (π ) b)0.5 exp (π) c)exp (π ) + 1 d)exp(π) – 1Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the Taylor series expansion of exp(x) + sin(x) about the point x = π, the coefficient of (x – π)2 is a)exp (π ) b)0.5 exp (π) c)exp (π ) + 1 d)exp(π) – 1Correct answer is option 'B'. Can you explain this answer?.
Solutions for In the Taylor series expansion of exp(x) + sin(x) about the point x = π, the coefficient of (x – π)2 is a)exp (π ) b)0.5 exp (π) c)exp (π ) + 1 d)exp(π) – 1Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mechanical Engineering. Download more important topics, notes, lectures and mock test series for Mechanical Engineering Exam by signing up for free.
Here you can find the meaning of In the Taylor series expansion of exp(x) + sin(x) about the point x = π, the coefficient of (x – π)2 is a)exp (π ) b)0.5 exp (π) c)exp (π ) + 1 d)exp(π) – 1Correct answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of In the Taylor series expansion of exp(x) + sin(x) about the point x = π, the coefficient of (x – π)2 is a)exp (π ) b)0.5 exp (π) c)exp (π ) + 1 d)exp(π) – 1Correct answer is option 'B'. Can you explain this answer?, a detailed solution for In the Taylor series expansion of exp(x) + sin(x) about the point x = π, the coefficient of (x – π)2 is a)exp (π ) b)0.5 exp (π) c)exp (π ) + 1 d)exp(π) – 1Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of In the Taylor series expansion of exp(x) + sin(x) about the point x = π, the coefficient of (x – π)2 is a)exp (π ) b)0.5 exp (π) c)exp (π ) + 1 d)exp(π) – 1Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an ample number of questions to practice In the Taylor series expansion of exp(x) + sin(x) about the point x = π, the coefficient of (x – π)2 is a)exp (π ) b)0.5 exp (π) c)exp (π ) + 1 d)exp(π) – 1Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice Mechanical Engineering tests.
Explore Courses for Mechanical Engineering exam

Top Courses for Mechanical Engineering

Explore Courses
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev