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Understanding Vector Functions: Definitions, Examples, and Applications

Learn the definition of vector functions, see examples of graphing curves, understand parametric curves in 2D and 3D, explore limits, continuity, derivatives, geometric interpretations, tangents, chain rules, integrals, and calculating curve lengths.

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Understanding Vector Functions: Definitions, Examples, and Applications

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  1. 9.1 Vector Funcions • Def of vector function • Curve in xy-plane and xyz-plane • Graph r(t) • Curve of Intersection • Limit • Continuity • Derivatives • Smooth curve • Geometic representation of r’(t) • Tangent • Chain Rule • Integral of r(t) • Length of curve

  2. Vector Funcion Definition: A vector function is a function that takes one variable and returns a vector Examples: 3 ways to represent vector function

  3. Graph of Vector Functions Example1 2D Graph the curve traced by the vector funcion 0.3000 0.2955 1.2432 0.6000 0.5646 -0.4544 0.9000 0.7833 -1.8081 1.2000 0.9320 -1.7935 1.5000 0.9975 -0.4216 1.8000 0.9738 1.2694 2.1000 0.8632 1.9997 2.4000 0.6755 1.2167 2.7000 0.4274 -0.4871 t x y x=sin(t); y=2*cos(3*t); t=0.3:.3:2.7;

  4. Parametric Curve Example1 2D Graph the curve t=0.3:.01:2.7; % establish the t vector, with dot operation x=sin(t); y=2*cos(3*t); plot(x,y), grid % 2D curve drawing [t',x',y'] display(' t x y')

  5. Parametric Curve Example1 3D vector function Circular Helix Representation:

  6. Parametric Curve Example1 3D vector function Circular Helix 0 2.0000 0 0 1.5000 0.1415 1.9950 1.5000 3.0000 -1.9800 0.2822 3.0000 4.5000 -0.4216 -1.9551 4.5000 6.0000 1.9203 -0.5588 6.0000 7.5000 0.6933 1.8760 7.5000 9.0000 -1.8223 0.8242 9.0000 10.5000 -0.9511 -1.7594 10.5000 12.0000 1.6877 -1.0731 12.0000 13.5000 1.1898 1.6076 13.5000 15.0000 -1.5194 1.3006 15.0000 16.5000 -1.4048 -1.4236 16.5000 18.0000 1.3206 -1.5020 18.0000 t x y z t=0:.1:6*pi; % establish the t vector x=2*cos(t); y=2*sin(t); z=t; plot3(x,y,z), grid % 3D curve drawing ezplot3('2*cos(t)','2*sin(t)','t',[0,9*pi/2])

  7. Parametric Curve Example2 Circle in a plane Graph the curve traced by the vector function ezplot3('2*cos(t)','2*sin(t)','3',[0,2*pi])

  8. Curve of Intersection Example3 Curve of Intersection Find the vector function that describes the curve C of the intersection of the plane and the paraboloid

  9. Limit of a Vector Funcion Definition Limit of a Vector Funcion Theorem Properties of Limits

  10. Continuity of a Vector Funcion Definition Continuity of a Vector Funcion A vector funcion r is said to be continuous at if

  11. Derivative of Vector Function Differentiation of Components Derivative of Vector Function Example1 Find

  12. Smooth Curve Smooth Vector Function Smooth Curve continuous Smooth vector function The curve traced by r(t) is called a smooth curve Example1is it smooth vector function? Smooth curve? Circular Helix

  13. Geometric Interpretation Tangent Line If the vector at a point P The tangent line is defined to be the line through the point P and parallel to

  14. Rules of Differentiation Theorem Rules of Differentiation

  15. Rules of Differentiation Integrals of Vector Functions

  16. Integrals of Vector Functions Integrals of Vector Functions Example1 Find

  17. Length of a curve Length of a curve the length of the smooth curve traced by r is given by

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