23. Taylor Series

d.1. Taylor Polynomial Approximation

$\underline{\sin x \quad \text{about } x=0}$
$\underline{\ln x \quad \text{about } x=1}$
$\underline{x^{1/3} \quad \text{about } x=8}$

c. $x^{1/3}$ about $x=8$

The Taylor series for $x^{1/3}$ about $x=8$ is $\begin{aligned} x^{1/3}=&2+\dfrac{1}{12}(x-8)-\dfrac{1}{288}(x-8)^2+\dfrac{5}{20\,736}(x-8)^3 \\ &-\dfrac{5}{248\,832}(x-8)^4+\dfrac{11}{5\,971\,968}(x-8)^5+\cdots \end{aligned}$ Here are the graphs of $x^{1/3}$ (in BLUE) with the $0^\text{th}$ through $8^\text{th}$ degree Taylor polynomial approximations (in RED):

23. Taylor Series

d.1. Taylor Polynomial Approximation

c. x1/3x^{1/3}x1/3 about x=8x=8x=8

c. $x^{1/3}$ about $x=8$