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

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ma 242 003
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
slide5

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

Corollary: The corresponding mixed second partial derivatives of polynomials are always equal.

slide7

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

slide8

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

slide9

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

slide10

Section 11.4:

Tangent planes and linear approximations

or

On the differentiability of multivariable functions

slide11

Section 11.4:

Tangent planes and linear approximations

or

On the differentiability of multivariable functions

Recall the following definition from Calculus I:

slide12

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.

slide13

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

slide14

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.

slide19

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)).

slide20

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)).

slide23

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.

slide24

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

slide25

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

slide26

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

slide27

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.

slide30

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

section 11 5

Section 11.5

THE CHAIN RULE