Chapter 1 Topics in Analytic Geometry

••••MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 65r 0t(r 1 −r 0 )rOr 1r = (1−t)r 0 +tr 1It is common to call (4.5) the two-point vector form of a line. It is understoodin (4.5) that t varies from −∞ to +∞. However, if we restrict t to vary over the interval0 ≤ t ≤ 1, then r will vary from r 0 to r 1 . Thus, the equationr = (1−t)r 0 +tr 1 (0 ≤ t ≤ 1) (4.6)represents the line segment in 2-space or 3-space that is traced from r 0 to r 1 .4.2 Calculus of Vector-Valued FunctionsIn this section we will define limits, derivative, and integral of vector-valued functions.Limits and ContinuityOur first goal in this section is to develop a notion of what it means for a vector-valuedfunction r(t) in 2-space or 3-space to approach a limiting vector ̷L ar t approaches a numbera. That is, we want to definelimr(t) = ̷L (4.7)t→aDefinition 4.1 Let r(t) be a vector-valued function that is defined for all t in some openinterval containing the number a, except that r(t) need not be defined at a. We will writelimr(t) = ̷Lt→aif and only ifTheorem 4.1lim‖r(t)− ̷L‖ = 0t→a(a) If r(t) = 〈x(t),y(t)〉 = x(t)i+y(t)j, thenlimr(t) =t→a〈limt→ax(t),limt→ay(t)〉= limt→ax(t)i+limt→ay(t)jprovided the limits of the component functions exist. Conversely, the limits of thecomponent functions exist provided r(t) approaches a limiting vector as t approachesa.