09 II 09/12/ (3D ) f(, y) = 2 + y 2 3D- 1 f(0, 0) = 2 f(1, 0) = 3 f(0, 1) = 4 f(1, 1) = 5 f( 1, 2) = 6 f(0, 1) = z y (3D ) f(, y) = 2 + y

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1 09 II 09/12/ I 2D II 3D f() = f(, y) = y + y 2 6 4y f(1) = = 2 y = f() f(3, 2) = = 8 z = f(, y) y 2 1 z y 1 ( y ) 1 (0, 0) 2 (1, 0) 3 (0, 1) 4 (1, 1) 5 ( 1, 2) 6 (0, 1) y y

2 09 II 09/12/ (3D ) f(, y) = 2 + y 2 3D- 1 f(0, 0) = 2 f(1, 0) = 3 f(0, 1) = 4 f(1, 1) = 5 f( 1, 2) = 6 f(0, 1) = z y (3D ) f(, y) = 2 + y 2 3D hiver.html

3 09 II 09/12/ I II I II 1 f() = log f(, y) = y + y 2 6 4y 2 f () = 2 { f (, y) = I II 3 Fermat f () = 0 = a 3 Fermat { f (, y) = 0 0 { =? y =? 4 f () 4 = a 5 Hesse f (a) + 5 Hesse 1 ( 5.4 ) f() = log 3 3 = log 3 + log 3. ({}}{ ) ({}} {) ( {}}{ f () = log 3 + log 3 {}}{ ) 1 = 3 = 1 3 Fermat f () = = 0 = 1 3 f () = ( 1 ) ( ) 3 = 1 3 = 1 2 = 1 2 = 1 3 Hesse f ( 1 3 ) = 1 ( 1 = 9 < 0 = 1 ( 3 )2 3 f(1 3 ) = log 3 1 ) 3 1 = log 1 1 = 1 ( ) 3 3 //

4 09 II 09/12/ y = f() = y = f () y = dy d { y y = dy d = 2. y = u 3 + u dy du = 3. z = d = 4. z = y 3 + y dy = 5. z = d =

5 09 II 09/12/ z = 2y 3 + 4y dy = 7. z = d = 8. z = y y dy = 9. z = 3 a + 2 b + 2 d = 10. z = ay 3 + by dy = 11. z = 3 a b 2 + c 4 d = 12. z = a 3 y 3 + b 2 y 2 + c 4 dy =

6 09 II 09/12/ * z = 3 y y 2 + y 4 d = y 14. * z = 3 y y 2 + y 4 dy = y * ( ) = = ( {}}{{}}{ c f()) = c f (), ( ) {}}{ f() c = f {}}{ () c, ( ) {}}{{}}{ {}}{{}}{ a f() b = a f () b ( 3 ) = 0 ( c ) = 0 ( 3 2) = 3 2 = 6 ( a 3 ) = a ( 3 4 ) = 3a 4 ( 2 3 ) = 2 3 = 6 ( 3 a 2) = 3 4 a 2( = 3a 2 4) ( ) ( = = 24 a3 3 b 2) = a 3 ( 3 4 )b 2 = 3a 3 b 2 4 =

7 09 II 09/12/ d 2006 IBM z f (, y) z f z = f(, y) = 2 y z y f y (, y) }{{} z y f y }{{} y 1 1. z f z f z y f y z f z y f y 2. Chain Rule d d the partial derivative of z with respect to 3. f f y f f y f(, y, z) f f y f z

8 09 II 09/12/ f(, y) = 4 y {}}{ y 3 y {}}{ 4 3y 2 = 3 4 y 2 y 2. f(, y) = 3 2 y 4 y y 3. f(, y) = 4 3 y y y

9 09 II 09/12/ f(, y) = y + y y = 6 + 2y + 4 y y + 0 = 2 + 2y y 5. f(, y) = y + y 2 6 4y y y 6. f(, y) = 2 + 4y + 9y y + 2 y y

10 09 II 09/12/ f(, y) = 3 y y y y 8. * f(, y) = α y β ( α) = α α 1 α β y y 9. f(, y) = 1 2 y 1 2 y y II

11 09 II 09/12/ f(, y) = 1 3 y 2 3 y y 11. f(, y) = e + log y y y 12. * f(, y) = α log + (1 α) log y y y

12 09 II 09/12/ f(, y) = e y log y y 14. f(, y) = y + y y y 15. f(, y) = 2y + y y y ( ) f = f g f g g g 2

13 09 II 09/12/

14 09 II 09/12/

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