## Thomas' Calculus 13th Edition

a) $f(x) =\dfrac{x^2}{2}-\dfrac{x^{4}}{12}+\dfrac{x^{6}}{30}- ....$ $| Error| \lt 0.00052$ (b) $f(x) \approx\dfrac{x^2}{2}-\dfrac{x^{4}}{12}+\dfrac{x^{6}}{30}+ ....+(-1)^5\dfrac{x^{32}}{31 \cdot 22}$ and $| Error| \lt 0.00089$
a) Integrate the integral with respect to $x$ as follows: $f(x)=\int_0^{x} \tan^{-1} t dt=\int_0^{x} [t-\dfrac{t^3}{3}+\dfrac{t^{5}}{5}-...] dt \\ =\dfrac{x^2}{2}-\dfrac{x^{4}}{12}+\dfrac{x^{6}}{30}- ....$ Now, $| Error| \lt \dfrac{(0.5)^6}{30} \approx 0.00052 \implies | Error| \lt 0.00052$ b) Now, $f(x)=\int_0^{x} \tan^{-1} t dt=\int_0^{x} [t-\dfrac{t^3}{3}+\dfrac{t^{5}}{5}-...] \space dt \\=\dfrac{x^2}{2}-\dfrac{x^{4}}{12}+\dfrac{x^{6}}{30}- ....$ and $f(x) \approx\dfrac{x^2}{2}-\dfrac{x^{4}}{12}+\dfrac{x^{6}}{30}+ ....+(-1)^5\dfrac{x^{32}}{31 \cdot 22}$ Now, $| Error| \lt \dfrac{1}{33 \cdot 34} \approx 0.00089$ or, $| Error| \lt 0.00089$