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Understanding Vector-Valued Functions: Definition, Tracing, and Smoothness

Learn the definition and practical tracing of vector-valued functions, as well as differentiation, integration, and smoothness concepts. Find intervals where curves are smooth.

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Understanding Vector-Valued Functions: Definition, Tracing, and Smoothness

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  1. Section 11.1 Vector Valued Functions

  2. Definition of Vector-Valued Function

  3. How are vector valued functions traced out?

  4. In practice it is often easier to rewrite the function. Sketch the curve represented by the vector-valued function and give the orientation of the curve. #26 r(t)= #34 r(t)=

  5. Definition of the Limit of a Vector-Valued Function

  6. Definition of Continuity of a Vector-Valued Function

  7. Section 11.2 • Differentiation and Integration of Vector-Valued Functions.

  8. Definition of the Derivative of a Vector-Valued Function

  9. Theorem 12.1 Differentiation of Vector-Valued Functions

  10. Theorem 12.2 Properties of the Derivative

  11. Definition of Integration of Vector-Valued Functions

  12. Smooth Functions • A vector valued function, r, is smooth on an open interval I if the derivatives of the components are continuous on I and r’ 0 for any value of t in the interval I. #30 Find the open interval(s) on which the curve is smooth.

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