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?

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 \( x=500 \).

A hotdog vendor must pay a monthly fee to operate a food cart in the city park. The cost in dollars of selling \( x \) hundred hotdogs in a month can be approximated by \( \mathrm{C}(\mathrm{x})=150+60 \mathrm{x} \) Complete parts (a) through ( d\( ) \) below. Interpret these results, choose the correct answer below. Take the sales level of 1000 hotdogs as an example. Select the correct choice below and fill in the answer box to complete your choice. A. At a sales level of 1000 hotdogs, the average cost is \( \$ 75 \) per 100 hotdogs. (c) Find the marginal average-cost function. The marginal average-cost function is \( \bar{C}^{\prime}(x)=\square \) (d) Find and interpret the marginal average cost at sales levels of 1000 hotdogs and 5500 hotdogs. The marginal average cost at sales levels of 1000 hotdogs is \( \$ \square / 100 \) hotdogs) \( { }^{2} \). (Round to the nearest cent as needed.)

Si la posición de un objeto está dada por la función \( s(t) = 5t^3 - 2t^2 + 3t \), ¿cuál es la expresión para su velocidad en términos de \( t \)?