When \( x=16 \), the rate at which \( x^{3 / 4} \) is :- increasing is \( k \) times the rate at :- which \( \sqrt{x} \) is increasing. What is the :- value of \( k \) ? (A) \( \frac{1}{8} \) (B) \( \frac{3}{8} \) (C) 2 (D) 3 (E) 8

(Q6; 2,6 балла) Найдите наименьшее значение выражения \( 16 x+\frac{9}{25 x}(x>0) \), используя \( \frac{a+b}{2} \geq \sqrt{a b} \).

(4) show that \( \lim _{z \rightarrow a} \) \( \frac{i z^{3}+i z-1}{2 z+3 i(z-i)} \)

Calculate the volume of the solid formed by rotating the area between \( y = \sqrt{x} \) and \( y = 0 \) from \( x = 0 \) to \( x = 1 \) about the line \( y = -1 \) utilizing the washer method.

If \( \frac{d^{n t}}{d x^{n}}\left(e^{a x}\right)=\quad \) where \( a, b \) are constar (ii) \( a^{n-1} e^{a x} \) (iii) \( n!a^{n} \) (iii) \( (n-1)!a^{n-1} \) (iv) \( a^{n} e^{a x} \)

Calculate the area bounded by the curves \( y = \sin(x) \) and \( y = \cos(x) \) over the interval \([0, \frac{\pi}{4}]\).