Question

How would I find the fourth-order Taylor polynomial for the function?

Answer 1

7889/6114

Explanation:

`e^x \approx 1+x+(x^2)/(2!)+(x^3)/(3!)+(x^4)/(4!) \\ \\ =1+x+(x^2)/(2)+(x^3)/(6)+(x^4)/(24) \\ \\ \implies e^(3x) \approx 1+3x+((3x)^2)/(2)+((3x)^3)/(6)+((3x)^4)/(24) \\ \\ =1+3x+(9)/(2)x^2+(9)/(2)x^3+(27)/(8)x^4 \\ \\ \implies e^(1/4) \approx 1+3(1/12)+(9)/(2)(1/12)^2+(9)/(2)(1/12)^3+(27)/(8)(1/12)^4 \\ \\ =(7889)/(6144)`