For the function \( f(x)=15+10 x+6 x^{2}+3 x^{3} \), find the following Taylor polynomials (all centered at \( a=0 \) ). \( T_{0}(x)=\square \) \( T_{1}(x)=\square \) \( T_{2}(x)=\square \) \( T_{3}(x)=\square \) \( T_{4}(x)=\square \) \( T_{118}(x)=\square \)

integral \( \int_{0}^{1} \int_{e^{y}}^{e} \frac{x}{\ln x} d x d y \) (by

Calcule \( \int \frac{x^{4} - 5x^{2} + 6}{x-2} \,dx \) aplicando o método de divisão longa.

8. If \( f(x)=x^{2}+10 \sin x \), show that there is a number \( c \) such that \( f(x)=1000 \).

By using Newton-Raphson Method, determine the root for \( x^{3}+2 x^{2}-1=0 \). Given \( x_{0}=0.5 \). Give the answer correct to four decimal places. Dengan mengomenan

\( \lim _ { x \rightarrow 0 } \frac { 3 ( x - \operatorname { cts } x ) } { x } \)