Consider the function f(x)=sec(2x)
The McLaurin series of a function f(x) is given by the expansion formula as follows:
f(x)≈f(0)+1!f′(0)x+2!f′′(0)x2+3!f′′′(0)x3+...+...
Now, find the first derivative of the function as,
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)=4sec(2x)tan2(2x)+4sec3(2x)
Find the third derivative of the function as,
f′′′(x)=4[sec(2x)dxdtan2(2x)+tan2(2x)dxdsec(2x)]+4[dxdsec3(2x)]
f′′′(x)=4[2tan(2x)sec3(2x)+2tan3(2x)sec(2x)]+4[6sec3(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))]
Next, find the function value and derivative value at x=0 as,
f(0)=1
f′(0)=0
f′′(0)=4
f′′′(0)=0
fiv(0)=80
So, the third order McLaurin expansion is given by,
f(x)≈1+10x+24x2+60x3+2480x4+...
f(x)≈1+2x2+0⋅x3+...
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