The equation of a curve is \( y=\frac{75}{(2 x+3)^{2}} \). (i) A point \( P \) moves along the curve in such a way that the \( x \)-coordinate of \( P \) increases at a constant rate of 0.04 units per second. Find the \( x \)-coordinate of \( P \) at the instant the \( y \)-coordinate is decreasing at a rate of 0.2 units per second.

Question 1: \( (15 \) mark \( ) \) (1)- Derivative of \( X \) from first principles?

Determina las dimensiones de una caja \sin tapa con un volumen de 300 cm³ que minimizan el área superficial, utilizando relación entre largo, ancho y alto.

(c) Use Bisection method to solve \( x^{3}-9 x+1=0 \) for the root between \( x=2 \) and \( x=4 \). 5 marks) (d) Using \( \sin (0.1)=0.09983 \) and \( \sin (0.2)=0.198 \), find an appropriate value of \( \sin (0.15) \) by using Langranges interpolation.

Câu 37. Cho hàm số \( y=f(x) \) xác định trên R và có \( \lim _{x \rightarrow 3} \frac{x^{2}}{x-3}=5 \). Tỉnh \( \lim _{x \rightarrow 3} \frac{x^{2}+x-12}{x^{2}-x} \) ? Câu 38. Cho hàm số \( y=f(x) \) xác định trền R và có \( \lim _{x \rightarrow 1} \frac{f(x)-2}{x-1}=4 \). Tỉnh \( \lim _{x \rightarrow 1} \frac{\sqrt{x+8}+\sqrt{f(x)-1}-4}{x^{2}} \) ?

The revenue (in thousands of dollars) from producing \( x \) units of an item is modeled by \( R(x)=13 x-0.01 x^{2} \). a. Find the average rate of change in revenue as \( x \) changes from 1004 to 1005 . b. Find the marginal revenue at \( x=500 \).