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Unformatted text preview: Math2011, Vector-Valued Functions 2.1 Curves Definition 2.1 A parametric representation of a curve (or a curve) is a func- tion r , in which a number is plugged in and a vector is returned (or a vector- valued function). Remark 2.2 Let r be a vector-valued function. The collection of points r ( t ) where t runs through all the values which the function is defined, is called the graph of r in the textbook. Example 2.3 Let r , v be vectors. Consider the function r ( t ) = r + tv for all numbers t . Then this is a curve. Such kind of curve is usually called a line. Example 2.4 Consider the function r ( t ) = (2 cos t/ 2 , 2 sin t/ 2 , 2) for all numbers t . Then, this is a curve. First of all, r ( t ) belongs to the plane with defining equation z = 2 for all t . Secondly, || r ( t )- (0 , , 2) || = || (2 cos t/ 2 , 2 sin t/ 2 , 0) || = q 4 cos 2 t/ 2 + 4 sin 2 t/ 2 = 2 Such curve is called a circle. Example 2.5 Consider the function r ( t ) = (2 cos t/ 2 , 2 sin t/ 2 , 2 t ) for all numbers t . This is a curve which is often called a circular helix. Remark 2.6 If r is a function in which a number is plugged in and a vector is returned, it can also be consider as the discription of the motion of a moving particle. Here r ( t ) is the location of the particle at time t . 1 Definition 2.7 If r is a function in which a number is plugged in and a vector is returned. The derivative of r at t is defined as r ( t ) = lim h → r ( t + h )- r ( t ) h . Remark 2.8 For a function r in which a number is plugged in and a vector is returned, it can be considered as a curve. In this case, r ( t ) is called the tangent vector of the curve at t . The tangent line of the curve at...
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