\( f(x)=\left\{\begin{array}{cc}a-x^{2} & \text { si, } \quad x \leq 0 \\ 2 & \text { si, } 0<x<1 \\ x^{2}-2 x+b & \text { si, } x \geq 1\end{array}\right. \)

Determine the following limits. (a) \( \lim _{x \rightarrow-4} f(x) \) (b) \( \lim _{x \rightarrow-1} f(x) \) (c) \( \lim _{x \rightarrow 1} f(x) \) (d) \( \lim _{x \rightarrow 3} f(x) \) (e) \( \lim _{x \rightarrow 5} f(x) \)

\( \int _ { - 2 } ^ { 2 } \int _ { \sqrt { 4 - x ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { \sqrt { x ^ { 2 } + y ^ { 2 } } } ^ { \sqrt { 8 - x ^ { 2 } - y ^ { 2 } } } ( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } ) d z d y d x \)

The total revenue from the sale of a popular book is approximated by the rational function \( R(x)=\frac{1300 x^{2}}{x^{2}+4} \), where \( x \) is the number of years since publication and \( R(x) \) is the total revenue in millions of dollars. Use this function to complete parts a through \( d \). (a) Find the total revenue at the end of the first year. \( \$ 260 \) million (b) Find the total revenue at the end of the second year. \( \$ 650 \) million (c) Find the revenue during the second year only. \( \$ \square \) million

\( \int _ { - 2 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { \sqrt { x ^ { 2 } + y ^ { 2 } } } ^ { \sqrt { 8 - x ^ { 2 } - y ^ { 2 } } } ( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } ) d z d y d x \)

If \( y=\frac{\left(e^{x}+1\right)^{3}}{1+e^{x}} \), then the gradient of the curve at the point \( (0,4) \) is \( -\frac{1}{4} \) 2