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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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The provided text is a section from a Calculus III course, specifically discussing derivatives and integrals of vector functions. It includes definitions, geometric interpretations, and examples of how to compute derivatives of vector functions.
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the text does not contain any explicit mathematical questions or problems for the reader to solve. it is primarily expository, explaining concepts and providing examples.
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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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13.2 Derivatives and Integrals of Vector Functions - Math...

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The provided text is a section from a Calculus III course, specifically discussing derivatives and integrals of vector functions. It includes definitions, geometric interpretations, and examples of how to compute derivatives of vector functions.

TYPE: NO_QUESTION

The text does not contain any explicit mathematical questions or problems for the reader to solve. It is primarily expository, explaining concepts and providing examples.