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Use the definition of a Taylor series to find the first four nonzero terms...

A) 0, 0, 0, 0
B) 1, 1, 1, 1
C) x, x², x³, x⁴
D) Non-zero terms not specified

1 Answer

4 votes

Final answer:

Option C. The question lacks details regarding a specific function for a Taylor series expansion, so the first four non-zero terms are represented generally by C1x, C2x², C3x³, C4x⁴, where the coefficients depend on the derivatives of the function at x=0.

Step-by-step explanation:

The student's question doesn't provide a specific function for which to determine the Taylor series, leaving the options A) 0, 0, 0, 0, B) 1, 1, 1, 1, C) x, x², x³, x⁴, and D) Non-zero terms not specified as incomplete without more context. However, if we assume the student is requesting the first four non-zero terms of a Taylor series of a generic function around x=0, then one would use the formula for the Taylor series expansion:

f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ...

Each term involves the n-th derivative of the function at 0, multiplied by x to the power of n, divided by n! (factorial). Since the first four non-zero terms are requested, assuming that the derivatives of the function at x=0 are non-zero, they would look like C1x, C2x², C3x³, C4x⁴, where C1, C2, C3, C4 are the values of the respective derivatives of the function f(x) at x=0 divided by n!. Without knowing the specific function, we can't provide the exact numeric coefficients.

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