60. \( f(x)=\left\{\begin{array}{c}2 x^{2}-1, \text { if } x \leq 1 \\ 4 x-1, \text { if } x>1\end{array}\right. \), then the derivative of \( f \) at \( x=4 \) is A. 4 B. 3 C 0 D. does not exist 61. What is the maximum possible area of a rectangle in square units with diagonal of length 16 units ? A. 48 B. 64 C. 128 D. 256 62. The maximum profit that a company can make if the profit function company is given by \( P(x)=36+ \) \( 72 x-18 x^{2} \) is: A. 2,232 B. 108 C. 124 D. 31

A área transversal de um solide varia conforme é exato \( A(x) = 2x + 1 \). Encontre o volume desse sólido sobre o intervalo \([0, 4]\).

The network latency \( L(x)=-2 x^{2}+12 x-10 \) for data packets is studied for input size ' \( x \) '. If the latency is qual at \( x=1 \) and \( x=5 \), use Rolle's theorem to find a value ' \( c \) ' in the interval \( (1,5) \) where the latency's rate of hange is zero. In a network flow system, the following equations represent the flow through three nodes \( \mathrm{A}, \mathrm{B} \), and \( \mathrm{C}: 2 A+ \) \( +C=100 \) (total inflow); \( A+3 B+2 C=200 \) (flow through node B ); \( 3 A+B+4 C=300 \) (outflow from ode \( C \) )

Find the area of the region bounded by the curves \( y = \frac{1}{x} \) from \( x = 1 \) to \( x = 2 \) and the line \( y = 1 \).

Deduce that \( \left(x_{n}\right) \) is a Cauchy sequence and thus convergent.

Find the value of \( x \) at which the series converg and give the sum : \( \begin{array}{ll}\text { (a) } \sum_{n=1}^{\infty} n^{-3 x} & \text { (b) } \sum_{n=1}^{\infty}(\ln x)^{n}\end{array} \)