(Solved): (a) Find the third Taylor polynomial \( P_{3}(x) \) for the function \( f(x)=\sqrt[3]{x+1} \) abou ...

(a) Find the third Taylor polynomial \( P_{3}(x) \) for the function \( f(x)=\sqrt[3]{x+1} \) about \( c=0 \). \[ P_{N}(x)=f(c)+f^{\prime}(c)(x-c)+\frac{f^{\prime \prime}(c)}{2 !}(x-c)^{2}+\cdots+\frac{f^{(N)}(c)}{N !}(x-c)^{N}=\sum_{n=0}^{N} \frac{f^{(n)}(c)}{n !}(x-c)^{n} \] (b) Approximate \( \sqrt[3]{0.5} \) and \( \sqrt[3]{1.5} \) using \( P_{3}(x) \). (c) Compare your result with actual cube roots (as given by a calculator). (d) Find the error estimates \( \left|R_{3}(0.5)\right| \) and \( \left|R_{3}(-0.5)\right| \) given by the Taylor's remainder. \[ R_{N}(x)=\frac{f^{(N+1)}(\xi)}{(N+1) !}(x-c)^{N+1}, \] where \( \xi \) is between \( x \) and \( c \). (e) Compare the actual error with the error estimate.