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The coefficient of (x - 2) ^ 3 in the Taylor expansion is of ex around x = 2is
(a) 1/(3!)
(b) (2 ^ 3)/(3!)
(c) (e ^ - 2)/(3!)
(d) (e ^ 2)/(3!)?
(a) 1/(3!)
(b) (2 ^ 3)/(3!)
(c) (e ^ - 2)/(3!)
(d) (e ^ 2)/(3!)?
Most Upvoted Answer
The coefficient of (x - 2) ^ 3 in the Taylor expansion is of ex around...
Understanding the Taylor Expansion
The Taylor expansion of a function f(x) around a point a is given by:
f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)²/2! + f'''(a)(x - a)³/3! + ...
In this case, we want the coefficient of (x - 2)³ in the Taylor series of e^x around x = 2.
Finding the Derivative
To find the coefficient of (x - 2)³, we need to calculate the third derivative of e^x:
- f(x) = e^x
- f'(x) = e^x
- f''(x) = e^x
- f'''(x) = e^x
Since the derivative of e^x is e^x itself, we can evaluate it at x = 2:
- f'''(2) = e²
Calculating the Coefficient
The coefficient of (x - 2)³ in the Taylor expansion is given by:
f'''(2) / 3! = e² / 3!
This means the coefficient we are looking for is e² divided by 6.
Conclusion
Now, comparing with the options given:
- (a) 1/(3!)
- (b) (2³)/(3!)
- (c) (e^(-2))/(3!)
- (d) (e²)/(3!)
The correct answer is (d) (e²)/(3!).
This matches our calculation and confirms that the coefficient of (x - 2)³ in the Taylor expansion of e^x around x = 2 is indeed (e²)/(3!).
The Taylor expansion of a function f(x) around a point a is given by:
f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)²/2! + f'''(a)(x - a)³/3! + ...
In this case, we want the coefficient of (x - 2)³ in the Taylor series of e^x around x = 2.
Finding the Derivative
To find the coefficient of (x - 2)³, we need to calculate the third derivative of e^x:
- f(x) = e^x
- f'(x) = e^x
- f''(x) = e^x
- f'''(x) = e^x
Since the derivative of e^x is e^x itself, we can evaluate it at x = 2:
- f'''(2) = e²
Calculating the Coefficient
The coefficient of (x - 2)³ in the Taylor expansion is given by:
f'''(2) / 3! = e² / 3!
This means the coefficient we are looking for is e² divided by 6.
Conclusion
Now, comparing with the options given:
- (a) 1/(3!)
- (b) (2³)/(3!)
- (c) (e^(-2))/(3!)
- (d) (e²)/(3!)
The correct answer is (d) (e²)/(3!).
This matches our calculation and confirms that the coefficient of (x - 2)³ in the Taylor expansion of e^x around x = 2 is indeed (e²)/(3!).
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The coefficient of (x - 2) ^ 3 in the Taylor expansion is of ex around x = 2is(a) 1/(3!)(b) (2 ^ 3)/(3!)(c) (e ^ - 2)/(3!)(d) (e ^ 2)/(3!)?
Question Description
The coefficient of (x - 2) ^ 3 in the Taylor expansion is of ex around x = 2is(a) 1/(3!)(b) (2 ^ 3)/(3!)(c) (e ^ - 2)/(3!)(d) (e ^ 2)/(3!)? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about The coefficient of (x - 2) ^ 3 in the Taylor expansion is of ex around x = 2is(a) 1/(3!)(b) (2 ^ 3)/(3!)(c) (e ^ - 2)/(3!)(d) (e ^ 2)/(3!)? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The coefficient of (x - 2) ^ 3 in the Taylor expansion is of ex around x = 2is(a) 1/(3!)(b) (2 ^ 3)/(3!)(c) (e ^ - 2)/(3!)(d) (e ^ 2)/(3!)?.
The coefficient of (x - 2) ^ 3 in the Taylor expansion is of ex around x = 2is(a) 1/(3!)(b) (2 ^ 3)/(3!)(c) (e ^ - 2)/(3!)(d) (e ^ 2)/(3!)? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about The coefficient of (x - 2) ^ 3 in the Taylor expansion is of ex around x = 2is(a) 1/(3!)(b) (2 ^ 3)/(3!)(c) (e ^ - 2)/(3!)(d) (e ^ 2)/(3!)? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The coefficient of (x - 2) ^ 3 in the Taylor expansion is of ex around x = 2is(a) 1/(3!)(b) (2 ^ 3)/(3!)(c) (e ^ - 2)/(3!)(d) (e ^ 2)/(3!)?.
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