The Path of an Object Moving in Xyz-Space Is Given $( x ( t ) , y ( t ) , z ( t ) ) = \left( 3 t ^ { 2 } , t + 1 , t ^ { 3 } \right)$

question 68

Essay

The path of an object moving in xyz-space is given by $( x ( t ) , y ( t ) , z ( t ) ) = \left( 3 t ^ { 2 } , t + 1 , t ^ { 3 } \right)$ .
The temperature at a point (x, y, z)in space is given by $f ( x , y , z ) = x ^ { 2 } y - 3 z$ Calculate $\left. \frac { d } { d t } f ( x ( t ) , y ( t ) , z ( t ) ) \right| _ { t = 2 }$ .

---

Definitions:

Monopolists

Entities or individuals that have exclusive control over the production and sale of a particular good or service, allowing them to manipulate market conditions.

Monopolist

A single seller in a market who has significant control over the price and supply of a product or service.

Perfectly Competitive Firm

A firm that operates in a market where there are many buyers and sellers, no barriers to entry or exit, and all firms sell identical products.

Natural Monopolies

A type of monopoly that arises due to high infrastructure costs and other barriers to entry which make it inefficient for more than one provider to operate (e.g., water supply).