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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