Find the open interval on which the curve given by the vector-valued function is smooth.

Question

Find the open interval on which the curve given by the vector-valued function  is smooth.

Quizplus · Accepted Answer

For the given vector-valued function to be smooth, its derivative must exist and be continuous. Thus, we need to calculate the derivative of the given function:

r'(t) = <2et, 3sec^2(t), 5tan^4(t)>

The derivative is defined for all values of t, except for those where the secant is zero (i.e. t = π/2 + nπ), because the tangent is undefined at those points. Therefore, the curve is smooth on the open interval (-∞, π/2) U (π/2, ∞), which is option D.

Find the Open Interval on Which the Curve Given by the Vector-Valued