Find the first 4 non-zero terms of the Taylor polynomial centered at x=0 for f(x)=cos 6x.

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Find the first 4 non-zero terms of the Taylor polynomial centered at  x=0 for  f(x)=cos 6x.

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Find f(0): To find the Taylor polynomial of f(x) = cos(6x) centered at x = 0, we need to calculate the derivatives of f(x) at x = 0 and use the Taylor series formula: $$ P(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \ldots $$ First, we find f(0): $$ f(x) = \cos(6x) $$ $$ f(0) = \cos(6 \cdot 0) $$ $$ f(0) = \cos(0) $$ $$ f(0) = 1 $$First Derivative: Next, we find the first derivative of f(x) and evaluate it at x = 0: $$ f'(x) = -6\sin(6x) $$ $$ f'(0) = -6\sin(6 \cdot 0) $$ $$ f'(0) = -6\sin(0) $$ $$ f'(0) = 0 $$ Since the first derivative at x = 0 is zero, it does not contribute to the non-zero terms of the Taylor polynomial.Second Derivative: Now, we find the second derivative of f(x) and evaluate it at x = 0: $$ f''(x) = -36\cos(6x) $$ $$ f''(0) = -36\cos(6 \cdot 0) $$ $$ f''(0) = -36\cos(0) $$ $$ f''(0) = -36 $$ The second derivative at x = 0 is non-zero and contributes to the Taylor polynomial.Third Derivative: Next, we find the third derivative of f(x) and evaluate it at x = 0: $$ f'''(x) = 216\sin(6x) $$ $$ f'''(0) = 216\sin(6 \cdot 0) $$ $$ f'''(0) = 216\sin(0) $$ $$ f'''(0) = 0 $$ Since the third derivative at x = 0 is zero, it does not contribute to the non-zero terms of the Taylor polynomial.Fourth Derivative: Finally, we find the fourth derivative of f(x) and evaluate it at x = 0: $$ f''''(x) = 1296\cos(6x) $$ $$ f''''(0) = 1296\cos(6 \cdot 0) $$ $$ f''''(0) = 1296\cos(0) $$ $$ f''''(0) = 1296 $$ The fourth derivative at x = 0 is non-zero and contributes to the Taylor polynomial.Write Taylor Polynomial: Now we can write the first 4 non-zero terms of the Taylor polynomial for f(x) = cos(6x) centered at x = 0: $$ P(x) = f(0) + \frac{f''(0)}{2!}x^2 + \frac{f''''(0)}{4!}x^4 $$ $$ P(x) = 1 - \frac{36}{2!}x^2 + \frac{1296}{4!}x^4 $$ $$ P(x) = 1 - 18x^2 + \frac{1296}{24}x^4 $$ $$ P(x) = 1 - 18x^2 + 54x^4 $$

Find the first $4$ non-zero terms of the Taylor polynomial centered at $x=0$ for $f(x)=\cos 6x$ .

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Find the first $4$ non-zero terms of the Taylor polynomial centered at $x=0$ for $f(x)=\cos 6x$ .