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The coefficient of (x – 1)2 in the Taylor series expansion of f(x) = xex (x ∈ R) about the point x = 1 is
- a)e/2
- b)2e
- c)3e/2
- d)3e
Correct answer is option 'C'. Can you explain this answer?
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The coefficient of (x – 1)2 in the Taylor series expansion of f(...
To find the coefficient of (x - 1)^2 in the Taylor series expansion of f(x) = x * e^x, we can use the formula for the nth derivative of f(x) evaluated at x = 1 divided by n factorial.
Let's start by finding the derivatives of f(x):
f(x) = x * e^x
f'(x) = (1 * e^x) + (x * e^x) = (1 + x) * e^x
f''(x) = [(1 + x) * e^x]' = (1 * e^x) + (1 + x) * e^x = (2 + x) * e^x
f'''(x) = [(2 + x) * e^x]' = (1 * e^x) + (2 + x) * e^x = (3 + x) * e^x
We can observe a pattern here:
f^(n)(x) = (n + 1 + x) * e^x
Now let's evaluate these derivatives at x = 1:
f(1) = 1 * e^1 = e
f'(1) = (1 + 1) * e^1 = 2e
f''(1) = (2 + 1) * e^1 = 3e
f'''(1) = (3 + 1) * e^1 = 4e
From this pattern, we can see that the nth derivative evaluated at x = 1 is (n + 1) * e.
Now, let's find the coefficient of (x - 1)^2 in the Taylor series expansion:
The coefficient of (x - 1)^2 is given by the second derivative evaluated at x = 1 divided by 2 factorial:
Coefficient = f''(1) / 2!
= (3e) / 2
= 3e/2
Therefore, the correct answer is option C: 3e/2.
Let's start by finding the derivatives of f(x):
f(x) = x * e^x
f'(x) = (1 * e^x) + (x * e^x) = (1 + x) * e^x
f''(x) = [(1 + x) * e^x]' = (1 * e^x) + (1 + x) * e^x = (2 + x) * e^x
f'''(x) = [(2 + x) * e^x]' = (1 * e^x) + (2 + x) * e^x = (3 + x) * e^x
We can observe a pattern here:
f^(n)(x) = (n + 1 + x) * e^x
Now let's evaluate these derivatives at x = 1:
f(1) = 1 * e^1 = e
f'(1) = (1 + 1) * e^1 = 2e
f''(1) = (2 + 1) * e^1 = 3e
f'''(1) = (3 + 1) * e^1 = 4e
From this pattern, we can see that the nth derivative evaluated at x = 1 is (n + 1) * e.
Now, let's find the coefficient of (x - 1)^2 in the Taylor series expansion:
The coefficient of (x - 1)^2 is given by the second derivative evaluated at x = 1 divided by 2 factorial:
Coefficient = f''(1) / 2!
= (3e) / 2
= 3e/2
Therefore, the correct answer is option C: 3e/2.
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The coefficient of (x – 1)2 in the Taylor series expansion of f(x) = xex (x ∈R) about the point x = 1 isa)e/2b)2ec)3e/2d)3eCorrect answer is option 'C'. Can you explain this answer?
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The coefficient of (x – 1)2 in the Taylor series expansion of f(x) = xex (x ∈R) about the point x = 1 isa)e/2b)2ec)3e/2d)3eCorrect answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about The coefficient of (x – 1)2 in the Taylor series expansion of f(x) = xex (x ∈R) about the point x = 1 isa)e/2b)2ec)3e/2d)3eCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The coefficient of (x – 1)2 in the Taylor series expansion of f(x) = xex (x ∈R) about the point x = 1 isa)e/2b)2ec)3e/2d)3eCorrect answer is option 'C'. Can you explain this answer?.
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