265Lesson05 - LESSON 5 Curves in 3-Space and...

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Unformatted text preview: LESSON 5 Curves in 3-Space and Differentiation READ: Sections 14.1, 14.2 NOTES: In addition to surfaces, a second type of object in 3-space we will be working with is a curve . When you hear the word curve you ought to picture something that looks like a piece of string floating in 3-space. Or, if you more comfortable with physics than string-ology , you can think of a space curve as the path traced out by a moving particle. A convenient way to describe a space curve is to use parametric equations that specify the location in space of a particle in terms of a parameter t , which we would normally think of as representing time. Thus we would write x = x ( t ) y = y ( t ) t I z = z ( t ) where I is the parameter interval that specifies the domain for t . If it has been a while since youve thought about parametric equations, it would be a good idea to do a quick review of section 12.1 before continuing in chapter 14. A particular example of a curve is a straight line, as we saw a few lessons back. For example, x = 2 + 3 t y = 1- t t [2 , 5] z = 2 + t could be thought of as the path traced out by a particle starting at time t = 2 at the point (8 ,- 1 , 4) in space, moving in a straight line, and stopping at time t = 5 at the point (17 ,- 4 , 7), the direction of motion indicated by way the path is traced out as t increases from 2 to 5. Just as in the case of the equation of the line in space, the parametric equations for any curve can be rewritten in the form of a vector equation, and, in fact, thats the way we will normally write equations of curves since the vector form is very handy for doing calculus, allowing us to write complicated formulas very compactly. The curve C given by parametric equations x = x ( t ) y = y ( t ) t I z = z ( t ) would be written in vector form as- r ( t ) = x ( t )- + y ( t )- + z ( t )- k , t I....
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265Lesson05 - LESSON 5 Curves in 3-Space and...

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