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Calculus II - 35

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Stewart Calculus 13.1&2

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Calculus II - 35

  1. 1. Vector equation of a line: = +If = , , , = , , , = , , ,then , , = + , + , +Parametric equation: = + , = + , = +symmetric equation: = =
  2. 2. Vector equation of a plane: ·( )=If = , , , = , , , = , , ,then: , , · , , =Scalar equation: ( )+ ( )+ ( )=Linear equation: + + + =
  3. 3. Distance from a point ( , , ) to a plane + + + = is given by: | + + + | = + +Ex: Find the distance between ( , , ) and + = .Ex: Find the distance between the two planes + = , + = .
  4. 4. 13.1 Vector Functions and Space CurvesA 3D space curve can be represented by theparametric equations = ( ), = ( ), = ()It can be written as a vector function: ()= ( ), ( ), ( ) = ( ) + ( ) + ( )Ex: ()= , ( ),
  5. 5. Ex: ()= , ,
  6. 6. Ex: ()= ( + ) ,( + ) ,
  7. 7. Ex: ()= ( + . ) ,( + . ) , .
  8. 8. 13.2 Derivatives andIntegrals of Vec. Func.Definition of the derivative of a vector function: ( + ) () ()= ()= =If ()= ( ), ( ), ( ) = ( ) + ( ) + ( )then ()= ( ), ( ), ( ) = ( ) + () + ()
  9. 9. Differentiation Rules: ( ( ) + ( )) = ( )+ ( ) ( ( )) = () ( ( ) ( )) = ( ) ( ) + ( ) ( ) ( ( ) · ( )) = ( )· ( )+ ( )· ( ) ( () ( )) = () ( )+ ( ) () ( ( ( ))) = ( ) ( ( ))
  10. 10. Ex: Prove that the tangent line of a circle ata point is perpendicular to the connectingline of the point and the center.
  11. 11. Ex: Prove that the tangent line of a circle ata point is perpendicular to the connectingline of the point and the center.A circle can be represented as | ( )| =
  12. 12. Ex: Prove that the tangent line of a circle ata point is perpendicular to the connectingline of the point and the center.A circle can be represented as | ( )| =We want to show that () ()
  13. 13. Ex: Prove that the tangent line of a circle ata point is perpendicular to the connectingline of the point and the center.A circle can be represented as | ( )| =We want to show that () () ( ) · ( ) = | ( )| =
  14. 14. Ex: Prove that the tangent line of a circle ata point is perpendicular to the connectingline of the point and the center.A circle can be represented as | ( )| =We want to show that () () ( ) · ( ) = | ( )| = [ ( ) · ( )] =
  15. 15. Ex: Prove that the tangent line of a circle ata point is perpendicular to the connectingline of the point and the center.A circle can be represented as | ( )| =We want to show that () () ( ) · ( ) = | ( )| = [ ( ) · ( )] = ( )· ( )=
  16. 16. Ex: Prove that the tangent line of a circle ata point is perpendicular to the connectingline of the point and the center.A circle can be represented as | ( )| =We want to show that () () ( ) · ( ) = | ( )| = [ ( ) · ( )] = ( )· ( )= () ()
  17. 17. Definite integral: () = () , () , ()The Fundamental Theorem of Calculus:If ( ) = ( ), then () = () = ( ) ( )

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