Chapter 1 Topics in Analytic Geometry

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••••MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 66r 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.

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 67(b) If r(t) = 〈x(t),y(t),z(t)〉 = x(t)i+y(t)j+z(t)k, then〈〉limr(t) = limx(t),limy(t),limz(t)t→a t→a t→a t→a= limt→ax(t)i+limt→ay(t)j+limt→az(t)kprovided the limits of the component functions exist. Conversely, the limits of thecomponent functions exist provided r(t) approaches a limiting vector as t approachesa.Example 4.7 Find limt→0((t 2 +1)i+5costj+sintk ) .Solution .........Example 4.8 Find limt→0〈t 2 ,e t ,−2cosπt〉.Solution .........Motivated by the definition of continuity for real-valued functions, we define a vectorvaluedfunction r(t) to be continuous at t = a iflimr(t) = r(a) (4.8)t→aThat is, r(a) is defined, the limit of r(t) as t → a exists, and the two are equal. As in thecase for real-valued functions, we say that r(t) is continuous on an interval I if it iscontinuous at each point of I. It follows from Theorem 4.1 that a vector-valued function iscontinuous at t = a if and only if its component functions are continuous at t = a.DerivativesThe derivative of a vector-valued function is defined by a limit similar to that for thederivative of a real-values function.Definition 4.2 If r(t) is a vector-valued function, we define the derivative of r withrespect to t to be the vector-valued function r ′ given byr ′ (t) = limh→0r(t+h)−r(t)h(4.9)The domain of r consists of all values of t in the domain of r(t) for which the limit exists.The function r(t) is differentiable at t if the limit in (4.9) exists. All of the standardnotations for derivatives continue to apply. For example, the derivative of r(t) can beexpressed asddt [r(t)], drdt , r′ (t), or r ′

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