To calculate the limit of a vector-valued function, calculate the limits of the component functions separately To calculate the derivative of a vector-valued function, calculate the derivatives of the component functions, then put them back into a new vector-valued function

The derivative of a vector-valued function r ( t ) is also a tangent vector to the curve. A vector-valued function is a function of the form r ( t ) = f ( t ) i + g ( t ) j or r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k , r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k , where the component functions f, g, and…

vector-valued function — a function of the form r ( t ) = f ( t ) i + g ( t ) j or r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k , r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k , where the component…. component functions — the component functions of the vector-valued function r ( t ) = f ( t ) i + g ( t ) j are f ( t ) and g ( t ) , and the component functions of the vector-valued function r ( t )…. derivative of a vector-valued function — the derivative of a vector-valued function r ( t ) is r ′ ( t ) = lim Δ t → 0 r ( t + Δ t ) - r ( t ) Δ t , r ′ ( t ) = lim Δ t → 0 r ( t + Δ t ) - r ( t ) Δ t , provided the…. plane curve — the set of ordered pairs ( f ( t ) , g ( t ) together with their defining parametric equations x = f ( t ) and y = g ( t ).