The Path of an Object Moving in Xyz-Space Is Given $( x ( t ) , y ( t ) , z ( t ) ) = \left( 4 t ^ { 2 } , 2 t + 1,2 t ^ { 3 } \right)$

question 51

Essay

The path of an object moving in xyz-space is given by $( x ( t ) , y ( t ) , z ( t ) ) = \left( 4 t ^ { 2 } , 2 t + 1,2 t ^ { 3 } \right)$ The temperature at a point (x,y,z)in space is given by $f ( x , y , z ) = x ^ { 2 } y - 2 z$ .
(a)At time $t = 1$ , what is the object's velocity $\vec { v }$ ? What is its speed?
(b)Calculate the directional derivative of f in the direction of $\vec { v }$ at the point (1,2,1), where $\vec { v }$ is the velocity vector you found in part (a).
(c)Calculate $\left. \frac { d } { d t } f ( x ( t ) , y ( t ) , z ( t ) ) \right| _ { t - 1 }$ (d)Explain briefly how your answers to part (a), (b)and (c)are related.Interpret them in terms of temperature.

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Definitions:

Sexual Drive

The basic biological and psychological urge or desire for sexual activity or expression.

Emotional Responses

The emotional reactions or feelings stimulated by an event, situation, or stimulus.

Ivan Pavlov

A Russian physiologist best known for his work in classical conditioning, demonstrating the learning process now known as Pavlovian conditioning.

Conditioned Responses

Learned reactions or behaviors in response to specific conditioned stimuli that have been associated with an unconditioned stimulus.