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  13.2 Derivatives and Integrals of Vector Functions
  
  # 13.2 Derivatives and Integrals of Vector Functions - Math...
  
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  Math 224 – Calculus III 1 13.2 Derivatives and Integrals of Vector Functions Recommended Homework: #1-39 Derivatives of Vector Functions Recall: From Calc I, 0 ( ) ( ) ( ) lim h f x h f x fx h slope of the tangent line to f at x What does () d t dt r r mean? Picture: x y z
  
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  Math 224 – Calculus III 2 So, we can take derivatives component-wise. If ( ) ( ), ( ), ( ) t f t g t h t r then: ( ) ( ), ( ), ( ) d t f t g t h t dt r r Geometrically (and analogously to calc I derivatives) ( ) ( ), ( ), ( ) t f t g t h t r represents the tangent vector to the space curve ( ) ( ), ( ), ( ) t f t g t h t r . To find the tangent line to ( ) ( ), ( ), ( ) t f t g t h t r , we would find the equation of the line through the tangent point and parallel to the tangent vector () t r We denote the unit tangent vector by t r T r Example: Given the vector function 2 ( ) 1,5 2 t t t r and the value 1 t Sketch the curve for 03 t Find t r Sketch the position vector (1) r and the tangent vector r on the graph above.
  Math 224 – Calculus III 3 Examples: Find the derivatives of the following vector functions: 12 ( ) tan ln( 1) t t t te t r i j k ( ) 2 tt r i j k ( )
  
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explanation: The provided text is an excerpt from a Calculus III course, specifically covering the topic of derivatives and integrals of vector functions. The text includes definitions, geometric interpretations, and examples of how to compute derivatives of vector functions. It also contains a few example problems, but no explicit questions or answers are presented in a question-and-answer format.
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distilabel_metadata.content: {[ promptMessage ]}
  
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  13.2 Derivatives and Integrals of Vector Functions
  
  # 13.2 Derivatives and Integrals of Vector Functions - Math...
  
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  Math 224 – Calculus III 1 13.2 Derivatives and Integrals of Vector Functions Recommended Homework: #1-39 Derivatives of Vector Functions Recall: From Calc I, 0 ( ) ( ) ( ) lim h f x h f x fx h slope of the tangent line to f at x What does () d t dt r r mean? Picture: x y z
  
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  Math 224 – Calculus III 2 So, we can take derivatives component-wise. If ( ) ( ), ( ), ( ) t f t g t h t r then: ( ) ( ), ( ), ( ) d t f t g t h t dt r r Geometrically (and analogously to calc I derivatives) ( ) ( ), ( ), ( ) t f t g t h t r represents the tangent vector to the space curve ( ) ( ), ( ), ( ) t f t g t h t r . To find the tangent line to ( ) ( ), ( ), ( ) t f t g t h t r , we would find the equation of the line through the tangent point and parallel to the tangent vector () t r We denote the unit tangent vector by t r T r Example: Given the vector function 2 ( ) 1,5 2 t t t r and the value 1 t Sketch the curve for 03 t Find t r Sketch the position vector (1) r and the tangent vector r on the graph above.
  Math 224 – Calculus III 3 Examples: Find the derivatives of the following vector functions: 12 ( ) tan ln( 1) t t t te t r i j k ( ) 2 tt r i j k ( )
  
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distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is an excerpt from a Calculus III course, specifically covering the topic of derivatives and integrals of vector functions. The text includes definitions, geometric interpretations, and examples of how to compute derivatives of vector functions. It also contains a few example problems, but no explicit questions or answers are presented in a question-and-answer format.
  
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