Consider the Curve $\vec { r } ( t ) = ( t + 1 ) \vec { i } + ( 2 - t ) \vec { j } + \left( 2 t ^ { 2 } - t + 3 \right) \vec { k } , 0 \leq t \leq 1$

question 52

Essay

Consider the curve $\vec { r } ( t ) = ( t + 1 ) \vec { i } + ( 2 - t ) \vec { j } + \left( 2 t ^ { 2 } - t + 3 \right) \vec { k } , 0 \leq t \leq 1$ .
(a)Find a unit vector tangent to the curve at the point (1,2,3).
(b)Show that the curve lies on the surface $z = x ^ { 2 } + y ^ { 2 } - y$ .

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Definitions:

Modem

Telecommunications device that translates digital signals to analog for transmission.

Digital Signals

Electronic signals that represent data as sequences of discrete values, typically represented by binary code.

Analog Signals

Continuous signals that vary over time and can represent information such as temperature, sound waves, or electrical voltage.

Input Functions

The tasks or operations a system is designed to accept and process, often referring to software or electronic device inputs.