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Which of the following is the coefficient of x^4 in the Maclaurin series generated by cos (3x)? 
a) 27/8
b) 9
c) 1/24
d) 0
e) -27/8
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Isaac K. | Reliable and Knowledgeable in Math, Science, and Test Prep SkillsReliable and Knowledgeable in Math, Scie...
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You know that f(x) can be expanded as a series as follows. 
f(x) = f(0) + f'(0)*x + (f''(0)/2)*x2 + (f'''(0)/6)*x3 + (f''''(0)/24)*x4 + ...
In case of cos(x), the coefficient of any odd power is zero, which matches well with the fact that cos(x) and even power are even functions.
cos(x) = 1 -1/2*x2 + 1/24*x4 + ...
When we apply the chain rule in computing derivatives of cos(3x), the Maclaurin series for cos(3x) is computed as follows,
cos(3x) = 1 -(32/2)*x2 + (34/24)*x4 + ....
So, the coefficient of x4 is 34/24, which is 27/8. 
The answer: a) 27/8