3. By the end of Week :
You would learn how to solve many problems involving limits, derivatives and integrals of vector-valued functions and questions regarding parametrization of curves.
These are just few examples:
Find parametric equations of the line tangent to the graph at a given point.
Determine if the parametrization is smooth.
Find the arc length parametrization given some parametrization.

4. Vector-Valued functions (VVF):
a number
a vector
component functions of r(t)

5. COLLABORATE:

6. The Limit of a VVF:

7. The Limit of VVF via Component Functions:
Continuity of VVF via Component Functions:
r(t) is continuous

8. The Derivative of VVF:
Geometric interpretation of the Derivative:
The Derivative via Component Functions:

9. The Integral of VVF via Component Functions:

10. Basic properties of Differentiation and Integration of Vector-Valued Functions:

11. Vector-Valued Version of the Fundamental Theorem of Calculus:

12. Tangent lines to Graphs of Vector-Valued Functions:
Derivatives of Dot and Cross Products:

13. Exercises:

16. Change of Parameter; Arc Length
A Parametrization of C:
A smooth parametrization of C:
Smooth parametrizations:
“Continuously turning” tangent (no cusps)

17. Example:

18. Example:

20. Arc Length for Smooth Parametric Curves:

21. A Change of Parameter:
Question: For what g is smooth?

22. Smooth Change of Parameter:
Why?
By the Chain Rule:

23. Arc Length as a Parameter:
The “arc length parameter“ s is the signed length of arc measured along the curve from some fixed reference point P and with orientation (s>0 in the “+” direction and s<0 in the “-” direction).
The arc length parametrization of C is a parametrization r(s) such that the length of the arc between P and r(s) is |s|.
Finding Arc Length Parametrizations: