Applied Calculus Chapter 2 vector valued function
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This document discusses vector-valued functions and some key concepts related to them. It provides examples of parametric curves and vector functions, how to sketch their graphs, and how to find derivatives and integrals of vector functions. It also introduces concepts like the unit tangent vector, unit normal vector, binormal vector, curvature, and radius of curvature for curves defined by vector functions.
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Applied Calculus Chapter 2 vector valued function
- 1. VECTOR-VALUED FUNCTION [email protected] Prepared by :MISS RAHIMAH JUSOH @ AWANG
- 2. ectorposition v aasexpressedbecanD-3inequationcurveorlineA kjir zyx :equationparametricin the )(tfx )(tgy )(thz kjir )()()((t) thtgtf )(),(),((t) thtgtfr
- 3. DOMAIN Example 1 Determine the domain of the fo11owing function cos ,1n 4 , 1 So1ution: The first component is defined for a11 's. The second component is on1y defined for 4. The third component is on1y defined for t t t t t t r 1. Putting a11 of these together gives the fo11owing domain. 1,4 This is the 1argest possib1e interva1 for which a11 three components are defined. t
- 4. [email protected] kjir )4(3)(ofgraphSketch the(b) line.the sketchThen(2,3,-1)?and(1,2,2)pointsthepasses thatlinestraightaofequationlinetheisWhat(a) 2 ttt Example 2
- 9. [email protected] functionfollowingtheofeachofgraphSketch the 1,)( tt r(a) (b) ttt sin3,cos6)( r Solution : The first thing that we need to do is plug in a few values of t and get some position vectors. Here are a few, (a) Example 3
- 10. [email protected] The sketch of the curve is given as follows (red line).
- 11. [email protected] Solution : (b) As in Question (a), we plug in some values of t.
- 12. [email protected] The sketch of the curve is given as follows (red line).
- 13. functionfollowingtheofeachofgraphSketch the kjir cttatat sincos)( Example 4 CIRCULAR HELIX
- 23. [email protected] INTEGRATION OF VECTOR FUNCTIONS ))(())(())(()( thenb],[a,onofand,, functionsintegrablesomefrom)()()()(If ise.componentwdonealsoisfunctionsvectorofnIntegratio b a kjiF kjiF b a b a b a dtthdttgdttfdtt thgf thtgtft
- 24. [email protected] Example 7 Solution : kji kji kjiF kjiF F 802-42 ])5()2[( 4)52()43()( 4t)52()43()( if)(Find 3 1 4223 3 1 3 3 1 3 1 3 1 2 32 3 1 ttttt dttdttdtttdtt tttt dtt
- 25. [email protected] b a dt dy dt dx L 22 b a dt dz dt dy dt dx L 222 In 2-space In 3-space In general, b a b a tL dt d L )('or r r
- 26. Notes: Smooth Curve The graph of the vector function defined by r(t) is smooth on any interval of t where is continuous and . The graph is piecewise smooth on an interval that can be subdivided into a finite number of subintervals on which r is smooth. r 0t r
- 27. [email protected] Find the arc length of the parametric curve 4 3 0;2,sin,cos)( 33 tztytxa 10;2,,)( ttzeyexb tt Find the arc length of the graph of r(t) 42;6 2 1 )()( 23 ttttta kjir 20;2sin3cos3)()( tttttb kjir 4 3 : LAns 1 : eeLAns 58: LAns 132: LAns Example 8 Example 9
- 29. If r(t) is a vector function that defines a smooth graph, then at each point a unit tangent vector is t t t r T r UNIT TANGENT VECTOR 3 a) Find the derivative of 1 sin 2 b) Find the unit tangent vector at the point where 0. t t t te t t r i j k Example 10
- 30. curve.thetont vectorunit tangetheiswhere asbydenoted),(curvethevector to normalunitprinciplethedefinewe,0If T Nr T t dtd [email protected] )(' )(' T T t t dtd dtd T T N UNIT NORMAL VECTOR
- 31. ).(curvethetoly,respectivevector unitprincipaltheandnt vectorunit tangetheareandwhere asdefinediscurveaofvectorbinormalThe tr NT B [email protected] BINORMAL VECTOR NTB
- 32. Find the unit normal and binormal vectors for the circular helix cos sint t t t r i j k Example 11
- 33. curve.thetont vectorunit tangetheiswhere asdefinedis)(curvesmoothaofcurvatureThe T r t [email protected] )(' )(' t t dtd dtd r T r T CURVATURE 3 r r r
- 34. Curvature is the measure of how “sharply” a curve r(t) in 2-space or 3-space “bends”.
- 35. Find the curvature of the helix traced out by 2sin ,2cos ,4t t t tr Example 12
- 36. Radius of Curvature asdefinediscurvatureofradiusitsthen ),(curvesmooththeofcurvaturetheisIf tr 1
- 37. thentime,iswhere),r(ectorposition v bygivencurvethealongmovesparticleaIf tt [email protected] dt d t r v )(velocity 2 2 )(onaccelerati dt d dt d t rv a dt ds t )(speed v
- 38. [email protected] Example 13 .2when particletheofonacceleratiandspeedvelocity,theFind sincos)( bygivenisafter timeparticleaofectorposition vThe 3 t tttt t jir Solution : kji kjiv kji r v 1242.09.0 )2(3)2(cos)2(sin 2when )3()(cos)(sin velocityobtain thewe,w.r.tatingDifferenti 2 2 t ttt dt d t
- 41. [email protected] C Ce cc C t te c t ctce tdtdttdtet dtd t t t i kjir kji kji kji kjir rv 2 )0(2sin )0( 3 1 )0( cCwhere 2 2sin 3 1 ) 2 2sin () 3 1 ()( 2cos)( havewe,Since 30 321 3 32 3 1 2 Solution :
- 43. Find the position vector R(t), given the velocity V(t) and the initial position R(0) for 2 2 ; 0 4t t t e t V i j k R i j k Example 15
- 44. [email protected] [email protected] “All our dreams can come true, if we have the courage to pursue them”