Answer to Question #108830 in Calculus for harry

Question #108830
Find a third degree McLaurin expansion of f (x) =sec2x
1
Expert's answer
2020-05-21T12:56:20-0400

Consider the function f(x)=sec(2x)f(x)=sec(2x)

The McLaurin series of a function f(x)f(x) is given by the expansion formula as follows:


f(x)f(0)+f(0)1!x+f(0)2!x2+f(0)3!x3+...+...f(x)\approx f(0)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+...+...


Now, find the first derivative of the function as,


f(x)=2sec(2x)tan(2x)f'(x)=2sec(2x)tan(2x)

Find the second derivative of the function as,


f(x)=2[2sec(2x)tan(2x)tan(2x)+sec(2x)(2sec2(2x))]f''(x)=2[2sec(2x)tan(2x)tan(2x)+sec(2x)(2sec^2(2x))]

f(x)=4sec(2x)tan2(2x)+4sec3(2x)f''(x)=4sec(2x)tan^2(2x)+4sec^3(2x)

Find the third derivative of the function as,


f(x)=4[sec(2x)ddxtan2(2x)+tan2(2x)ddxsec(2x)]+4[ddxsec3(2x)]f'''(x)=4[sec(2x)\frac{d}{dx}tan^2(2x)+tan^2(2x)\frac{d}{dx}sec(2x)]+4[\frac{d}{dx}sec^3(2x)]


f(x)=4[2tan(2x)sec3(2x)+2tan3(2x)sec(2x)]+4[6sec3(2x)tan(2x)]f'''(x)=4[2tan(2x)sec^3(2x)+2tan^3(2x)sec(2x)]+4[6sec^3(2x)tan(2x)]

Find the fourth derivative of the function as,

fiv(x)=2[(4(2sec(2x)+4sec3(2x))+24(6sec3(2x)tan2(2x)+2sec5(2x))]f^{iv}(x)=2[(-4\left(-2\sec \left(2x\right)+4\sec ^3\left(2x\right)\right)+24\left(6\sec ^3\left(2x\right)\tan ^2\left(2x\right)+2\sec ^5\left(2x\right)\right)]


Next, find the function value and derivative value at x=0x=0 as,


f(0)=1f(0)=1


f(0)=0f'(0)=0


f(0)=4f''(0)=4


f(0)=0f'''(0)=0


fiv(0)=80f^{iv}(0)=80


So, the third order McLaurin expansion is given by,


f(x)1+01x+42x2+06x3+8024x4+...f(x)\approx 1+\frac{0}{1}x+\frac{4}{2}x^2+\frac{0}{6}x^3+\frac{80}{24}x^4+...


f(x)1+2x2+0x3+...f(x)\approx 1+2x^2+0\cdot x^3+...


Therefore, the third degree McLaurin expansion of the function f(x)=sec(2x)f(x)=sec(2x) is f(x)1+2x2+0x3f(x)\approx 1+2x^2+0\cdot x^3

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