Let a planar curve consist of two consecutive parametric segments defined by…

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Let a planar curve consist of two consecutive parametric segments defined by (x1(t),y1(t)),  a≤t≤band(x2(s),y2(s)),  c≤s≤d,\bigl(x_1(t),y_1(t)\bigr),\; a\le t\le b \quad\text{and}\quad \bigl(x_2(s),y_2(s)\bigr),\; c\le s\le d, where (x1(b),y1(b))=(x2(c),y2(c))(x_1(b),y_1(b))=(x_2(c),y_2(c)). (1) Explain mathematically why the total length of the entire curve over the combined interval equals the sum of the lengths of the individual segments. (2) State any conditions on (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2) and on the sub-intervals [a,b][a,b] and [c,d][c,d] that must hold for the additivity you described in part (a) to be valid.

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