Given the system of differential equations: \[ \frac{d \mathbf{x}}{d t}=A \mathbf{x} \] where \( A=\left(\begin{array}{ll}4 & 0 \\ 0 & 3\end{array}\right) \), what is the general solution for \( \mathbf{x}(t) \) ?

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.

USing first principle prove that \( y=\sin x \) then \( \frac{d y}{d x}=\cos x \)

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

11. Find the values of \( a \) and \( b \) that make \( f \) continuous everywhere. \[ f(x)=\left\{\begin{array}{ll}\frac{x^{2}-4}{x-2} & \text { if } x<2 \\ a x^{2}-b x+3 & \text { if } 2 \leq x<3 \\ 2 x-a+b & \text { if } x \geq 3\end{array}\right. \]

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