(b) By using the Maclaurin series for \( \cos x \) and the result from part (a), show that the Maclaurin series for \( \sec x \) up to and including the term in \( x^{4} \) is \( 1+\frac{x^{2}}{2}+\frac{5 x^{4}}{24} \)

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

Suppose \( f \) is continuous on \( [1,5] \) and the only solutions of the equation \( f(x)= \) 6 are \( x=1 \) and \( x=4 \). If \( f(2)=8 \), explain why \( f(3)>6 \). Show that \( F(x)=|x| \) is continuous evervwhere.

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 \( f(x)=1000 \).

Show that \( F(x)=|x| \) is continuous everywhere.