8. THEOREM If the partial derivatives f x. and f y exist near (a, b) and are continuous at (a, b), then f is differentiable at (a, b).

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1 8. THEOREM I the partial derivatives and eist near (a b) and are continuous at (a b) then is dierentiable at (a b).

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3 For a dierentiable unction o two variables z= ( ) we deine the dierentials d and d to be independent variables; that is the can be given an values. Then the dierential dz also called the total dierential is deined b d ( ) d ( ) d z d z d

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5 For such unctions the linear approimation is and the linearization L ( z) is the right side o this epression. ) )( ( ) )( ( ) )( ( ) ( ) ( c z c b a b c b a a c b a c b a z z

6 I w= ( z) then the increment o w is w ( z z) ( z) The dierential dw is deined in terms o the dierentials d d and dz o the independent variables b dw w d w d w a dz

7 2. THE CHAIN RULE (CASE 1) Suppose that z= ( ) is a dierentiable unction o and where =g (t) and =h (t) and are both dierentiable unctions o t. Then z is a dierentiable unction o t and dz dt d dt d dt

8 dz dt z d dt z d dt

9 3. THE CHAIN RULE (CASE 2) Suppose that z= ( ) is a dierentiable unction o and where =g (s t) and =h (s t) are dierentiable unctions o s and t. Then dz d z d ds z d ds dz dt z d dt z d dt

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13 4. THE CHAIN RULE (GENERAL VERSION) Suppose that u is a dierentiable unction o the n variables 1 2 n and each j is a dierentiable unction o the m variables t 1 t 2 t m Then u is a unction o t 1 t 2 t m and u t 1 2 i u 1 d dt i u 2 dt i u n t n i or each i=12 m.

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15 F ( )=. Since both and are unctions o we obtain F d d F d d But d /d=1 so i F/ we solve or d/d and obtain d d F F F F

16 F ( z)= F d F d F z d d z But ( ) 1 and ( ) 1 so this equation becomes F F z z I F/ z we solve or z/ and obtain the irst ormula in Equations 7. The ormula or z/ is obtained in a similar manner. dz d F F z dz d F F z

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18 2. DEFINITION The directional derivative o at ( o o ) in the direction o a unit vector u=<a b> is i this limit eists. h hb ha D h u ) ( ) ( lim ) (

19 3. THEOREM I is a dierentiable unction o and then has a directional derivative in the direction o an unit vector u=<a b> and b a D u ) ( ) ( ) (

20 8. DEFINITION I is a unction o two variables and then the gradient o is the vector unction deined b j i ) ( ) ( ) (

21 D u ( ) ( ) u

22 1. DEFINITION The directional derivative o at ( z ) in the direction o a unit vector u=<a b c> is i this limit eists. h z hc z hb ha z D h u ) ( ) ( lim ) (

23 D u ( hu) ( ) ( ) lim h h

24 z i j z k

25 D u ( z) ( z) u

26 15. THEOREM Suppose is a dierentiable unction o two or three variables. The maimum value o the directional derivative D u () is () and it occurs when u has the same direction as the gradient vector ().

27 ) )( ( ) )( ( ) )( ( z z z F z F z F z The smmetric equations o the normal line to soot P are ) ( ) ( ) ( z F z z z F z F z The equation o this tangent plane as

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31 1. DEFINITION A unction o two variables has a local maimum at (a b) i ( ) (a b) when ( ) is near (a b). [This means that ( ) (a b) or all points ( ) in some disk with center (a b).] The number (a b) is called a local maimum value. I ( ) (a b) when ( ) is near (a b) then (a b) is a local minimum value.

32 2. THEOREM I has a local maimum or minimum at (a b) and the irst order partial derivatives o eist there then (a b)=1 and (a b)=.

33 A point (a b) is called a critical point (or stationar point) o i (a b)= and (a b)= or i one o these partial derivatives does not eist.

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35 3. SECOND DERIVATIVES TEST Suppose the second partial derivatives o are continuous on a disk with center (a b) and suppose that (a b) and (a b)= [that is (a b) is a critical point o ]. Let D D( a b) ( a b) ( a b) [ ( a b)] (a)i D> and (a b)> then (a b) is a local minimum. (b)i D> and (a b)< then (a b) is a local maimum. (c) I D< then (a b) is not a local maimum or minimum. 2

36 NOTE 1 In case (c) the point (a b) is called a saddle point o and the graph o crosses its tangent plane at (a b). NOTE 2 I D= the test gives no inormation: could have a local maimum or local minimum at (a b) or (a b) could be a saddle point o. NOTE 3 To remember the ormula or D it s helpul to write it as a determinant: D ( ) 2

37 z

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40 4. EXTREME VALUE THEOREM FOR FUNCTIONS OF TWO VARIABLES I is continuous on a closed bounded set D in R 2 then attains an absolute maimum value ( 1 1 ) and an absolute minimum value ( 2 2 ) at some points ( 1 1 ) and ( 2 2 ) in D.

41 5. To ind the absolute maimum and minimum values o a continuous unction on a closed bounded set D: 1. Find the values o at the critical points o in D. 2. Find the etreme values o on the boundar o D. 3. The largest o the values rom steps 1 and 2 is the absolute maimum value; the smallest o these values is the absolute minimum value.

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44 ( z ) g( z )

45 METHOD OF LAGRANGE MULTIPLIERS To ind the maimum and minimum values o ( z) subject to the constraint g ( z)=k [assuming that these etreme values eist and g on the surace g ( z)=k]: (a) Find all values o z and such that ( z) g( z) and g( z) k (b) Evaluate at all the points ( z) that result rom step (a). The largest o these values is the maimum value o ; the smallest is the minimum value o.

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