MA 242.003

1. MA 242.003 • Day 24- February 8, 2013 • Section 11.3: Clairaut’s Theorem • Section 11.4: Differentiability of f(x,y,z) • Section 11.5: The Chain Rule

4. Note that the partial derivatives of polynomials are again polynomials.

5. Note that the partial derivatives of a polynomials are again polynomials. Corollary: The corresponding mixed second partial derivatives of polynomials are always equal.

7. If a rational function is continuous at a point, then its first and second partial derivatives will also be continuous at that point.

8. If a rational function is continuous at a point, then its first and second partial derivatives will also be continuous at that point.

9. Corollary: The corresponding mixed second partial derivatives of a rational function f are equal at each point of the domain of f.

10. Section 11.4: Tangent planes and linear approximations or On the differentiability of multivariable functions

11. Section 11.4: Tangent planes and linear approximations or On the differentiability of multivariable functions Recall the following definition from Calculus I:

12. Section 11.4: Tangent planes and linear approximations or On the differentiability of multivariable functions Recall the following definition from Calculus I: DEF: A function f(x) is differentiable at x = a if f’(a) exists.

13. Section 11.4: Tangent planes and linear approximations or On the differentiability of multivariable functions Recall the following definition from Calculus I: DEF: A function f(x) is differentiable at x = a if f’(a) exists. Example: f(x) = exp(sin(x)) and x = Pi/4

14. Section 11.4: Tangent planes and linear approximations or On the differentiability of multivariable functions Recall the following definition from Calculus I: DEF: A function f(x) is differentiable at x = a if f’(a) exists. Example: f(x) = exp(sin(x)) and x = Pi/4 We need a generalization of the above definition to multivariable functions.

15. Linear Approximations

16. Linear Approximations

17. The generalization of tangent line to a curve

18. The generalization of tangent line to a curve Is tangent plane to a surface

19. So we will say that a function f(x,y) is differentiable at a point (a,b) if its graph has a tangent plane at (a,b,f(a,b)).

20. So we will say that a function f(x,y) is differentiable at a point (a,b) if its graph has a tangent plane at (a,b,f(a,b)). We are going to show that if f(x,y) has continuous first partial derivatives at (a,b) then we can write down an equation for the tangent plane at (a,b,f(a,b)).

22. (continuation of example)

23. DEF: Let f(x,y) have continuous first partial derivatives at (a,b). The tangent plane to z = f(x,y) is the plane that contains the two tangent lines to the curves of intersection of the graph and the planes x = a and y = b.

24. Theorem: When f(x,y) has continuous partial derivatives at (a,b) then the equation for the tangent plane to the graph z = f(x,y) is

25. Def: When f(x,y) has continuous partial derivatives at (a,b) then the linear approximation of f(x,y) near (a,b) is:

26. Def: When f(x,y) has continuous partial derivatives at (a,b) then the linear approximation of f(x,y) near (a,b) is:

27. Def: When f(x,y) has continuous partial derivatives at (a,b) then the linear approximation of f(x,y) near (a,b) is: Let us now formulate the definition of differentiability for f(x,y) based on the linear approximation idea.

30. We will need this definition to justify the chain rule formulas in the next section of the textbook.

36. Section 11.5 THE CHAIN RULE

44. Proof: