Consider the function f: RR defined by f(x) = cos(x), where n is an even integer....

 

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Consider the function f: RR defined by f(x) = cos(x), where n is an even integer. Recall from lectures that one can express f(x) as a linear combination of terms of the form cos(kx), where k are non-negative integers. Note that the term where k = 0 is the constant term. (a) What is the constant term in the expansion of f(x) described above? 1/(2^n) * (n!)/((n/2)!)^2 Syntax advice: Remember to use correct Maple syntax. For example, ⚫ the factorial n! may be written as n! or factorial (n) ⚫the binomial coefficient (*) may be written as binomial (n,k) k ⚫the term a is written a^x (b) Now, consider the function g: R→ R defined by 9(x)=sin911(x) f(x). Briefly explain in words why g(x) may be written as a linear combination of terms of the form sin(kx), where k are non-negative integers. DX Equation Editor A-AI BI U S x, x*² Styles Font Size Words: 0 (c) In the above expansion of g(x), what is the coefficient of sin ((912 + n)x)? (d) Similarly, what is the coefficient of sin ((911+ n)x)? (e) What is the coefficient of sin ((909 + n)x)? Consider the function f: RR defined by f(x) = cos(x), where n is an even integer. Recall from lectures that one can express f(x) as a linear combination of terms of the form cos(kx), where k are non-negative integers. Note that the term where k = 0 is the constant term. (a) What is the constant term in the expansion of f(x) described above? 1/(2^n) * (n!)/((n/2)!)^2 Syntax advice: Remember to use correct Maple syntax. For example, ⚫ the factorial n! may be written as n! or factorial (n) ⚫the binomial coefficient (*) may be written as binomial (n,k) k ⚫the term a is written a^x (b) Now, consider the function g: R→ R defined by 9(x)=sin911(x) f(x). Briefly explain in words why g(x) may be written as a linear combination of terms of the form sin(kx), where k are non-negative integers. DX Equation Editor A-AI BI U S x, x*² Styles Font Size Words: 0 (c) In the above expansion of g(x), what is the coefficient of sin ((912 + n)x)? (d) Similarly, what is the coefficient of sin ((911+ n)x)? (e) What is the coefficient of sin ((909 + n)x)?

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a The constant term in the expansion of fx can be found by setting z 0 in fx cos z When z 0 cos0 1 s... View the full answer