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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

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Note that the partial derivatives of polynomials are again polynomials.

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

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

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

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

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

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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.

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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

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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.

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Linear Approximations

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The generalization of tangent line to a curve

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Is tangent plane to a surface

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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)).

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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)).

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(continuation of example)

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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.

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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

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

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Let us now formulate the definition of differentiability for f(x,y) based on the linear approximation idea.

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

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Section 11.5 THE CHAIN RULE

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Proof:

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