1. (MTH 250) Lecture 24 Calculus

2. Previous Lecture’s Summary Multivariable functions Limits along smooth curves Limits of multivariable functions Continuity of multivariable functions Partial differentiation

3. Today’s Lecture Recalls Differentiability of multivariable functions Differentials & linear approximations Chain rules for partial differentiation Implicit differentiation Extrema of multivariable functions

4. Recalls

5. Recalls

6. Recalls

7. Definition (formal): Recalls

8. Theorem: Def: Recalls

9. Recalls

10. Recalls

11. Tangent plane

13. Example: Sol.

14. Definition: Differentiability of multivariable functions

17. Theorem: If a function is differentiable at a point, then it is continuous at that point.

18. Differentiability of multivariable functions

19. Definition (Total differential): Differentials & linear approximations

22. Cont: Differentials & linear approximations

26. Local linear approximation

27. Differentials & linear approximations Solution:

28. Chain rules for partial differentiation Definition (chain rules):

29. Chain rules for partial differentiation

30. Definition (chain rules) for partial derivatives:

31. Chain rules for partial differentiation

35. Extrema of multivariable functions Definitions:

36. Extrema of multivariable functions

38. Theorem:

39. Extrema of multivariable functions

42. Lecture Summary Recalls Differentiability of multivariable functions Differentials & linear approximations Chain rules for partial differentiation Implicit differentiation Extrema of multivariable functions