Hi Everyone :) I don't really understand Taylor Polynomias and I was just looking for some help with determining the third taylor polynomial at x=0 for the function? f(x)=#2*sqrt(5x+1)# Thank you guys so much this is a huge help

Question

#2*sqrt(5x+1)#

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Answer 1 · 2018-07-19T17:45:51

The answer is #=2-5x-25/4x^2+125/8x^3+...#

The function is

#f(x)=2xx sqrt(1+5x)#

The Taylor expansion is

#f(x)=sum_(n=0)^ oo f^n(0)/(x!)x^n#

#=f(0)+(f'(0))/(1!)x+(f''(0))/(2!)x^2+(f'''(0))/(3!)x^3+.......#

We shall first caculate for #f(x)=sqrt(1+5x)# and then multiply the result by #2#

#f(x)=sqrt(1+5x)#, #=>#, #f(0)=1#

#f'(x)=5/(2sqrt(1+5x))#, #=>#, #f'(0)=5/2#

#f''(x)=-25/(4(1+5x)^(3/2))#, #=>#, #f''(0)=-25/4#

#f'''(x)=375/(8(1+5x)^(5/2))#, #=>#, #f'''(0)=375/8#

Finally,

#sqrt(1+5x)=1+5/2x-25/8x^2+125/16x^3+.....#

And

#f(x)=2sqrt(1+5x)=2-5x-25/4x^2+125/8x^3+....#