5 3 3. 9. 4. x, x. 4, f(x, ) :=x x + =4,x,.. 4 (, 3) (, 5) (3, 5), (4, 9) 95 9 (g) 4 6 8 (cm).9 3.8 6. 8. 9.9 Phsics 85 8 75 7 65 7 75 8 85 9 95 Mathematics = ax + b
6 3 (, 3) 3 ( a + b). f(a, b) ={3 (a + b)} + {5 (a + b)} + {5 (3a + b))} + {9 (4a + b)} 3.. 9 x, : f(x) (3.) f(x). x x R n f(x) f( x) f( x) (3.) x. x x R n f(x) f( x) f( x) (3.) x
3.. 7. f(x) =x +x +3 f(x) =x +x +3=(x +) + x = f(x) f( ) = x x O x. f(x) =x 3 x ( ) f(x) f 3 = 3 3 O 3 3 3 x / 3 x. x x x x f(x) >f( x) f x x x x x f(x) <f( x) f x f(x) f( x) 3.
8 3 3.: 3 ( ). 5 f(x, ) =x +x + f(x, ) =(x + ) (x, ) f(x, ) f(x, ) =(x + ) = x + = 5-4 - x 3.: x +x + f(, ) = f (, ) f (x, ) =(t, t) f f(t, t) =(t t) = (, ) f(, ) f (, ) 4-4 - 4 3. ( ) 3.. x x f f ( x) = a 6 ( ). x, f( x) =.
3.. 9. a a f f(a) =. f(x, ) =x 3 3x + 3 (x, ) =(, ) 3x 3 f(x, ) = 3 3x f 3.3 z - -.5.5.5 -.5 - -.5 - -.5 -.5 - -.5 x.5.5 -.5 -.5 -.5.5 3.3: z = x 3 3x + 3 (x,, z) =(,,f(, )) (,,f(, )) u =(x, ) (, ) f z = f(, ) + f x (, )(x ) + f (, )( ) (,,f(, )) z = f(, ) f x (, ) f(, ) = = f (, )
3 3.. () f(x, ) =x x + x () f(x, ) =x 3 3x + 3 6 (x, ) (x, ) = x + (a, b) (a, b) (x, ) f(x, ) f(a, b) (x, ) = (x, ), h(t) =f((a, b)+t(x, )) t>, (a, b)+t(x, ) (a, b), h(t) h(). t = h(t) h () =, h (t) = d dt {f(a + tx, b + t)} = f x(a + tx, b + t)x + f x (a + tx, b + t) h () = f x (a, b)x + f (a, b)x =. (x, ) (x, ) = (u, v) (x, ) =(, ) (x, ) =(, ) f(a, b) =. 3... 4. f(x, ) =x f(, ) =,. f(, ) (, ) (x, ) f f(, ) 8 6 4 - -4-6 -8 - - -5 x 5-3.4: x 6-5 5 3..3
3 - - -3-3 - - 3 3.. 3. 3.5: 6 f(x, ) =x 3 3x + 3 f (, ) (, ) 3.6 (, ) (, ) (, ) (, ) - - -.5 - -.5.5.5.5.5 -.5 - -.5 3.6: z = x 3 3x + 3 3.7: x 3 3x + 3 f x x, x f.. 7 ( ). (). x f( x) = f( x) (). f( x) = f( x) x
3 3 (3). f( x) x.. () () (3),., (), (x, ) f(x, ) f f(a, b) (x, ) (a, b) f xx (x, ),f x (x, ),f (x, ) f xx (a, b),f x (a, b),f (a, b) f(a, b) f(x, ) f (a, b) f(a, b) (a, b) f (a, b) f(x, ) f(x, ) 3.8: (3) f(a, b) f(x, ) < (a, b) (x, ) f (a, b)
3.3. 33 3.9: f(a, b) = f(a, b) f(a, b) = f(a, b) 3.9 (a, b) 5. f(x, ) =x 4 + 4 4x 3 f(x, ) =, f(x, ) = 4 3 x (, ) (, ) (x, ) f(x, ) (, ) (x, ) (, ) f(x, ) > (, ) 3.3 6. f(x, ) =x + x + 9x 9 +7.
34 3 6 x + 9 f(x, ) = x + 9 {x + 9= x + 9= (x, ) =(3, 3) 7. f(x, ) = (x, ) =(, ) f(, ) = f(, ), 3, 4 f(, ) f xx (, ) > f(, ) > 5 f(, ),?? (3, 3), f(3, 3) =.. 7. f(x, ) =x 3 3x + 3 [ 3x 3 f(x, ) =, 6x f(x, ) = 3 3x 3 ] 3 6. f(x, ) =, { 3x 3 = 3 3x =. (x, ) =(, ) (, ).
3.3. 35 x**+x*+**-9*x-9*+7 3 3 5 5 5 5 5 5-5 - -5 5 5 - -5 5 3.: x + x + 9x 9 +7. (x, ) =(3, 3) (, ) f(, ) <,.. (, ) f xx (, ) = 6 > f(, ) =7>,.,. (x, ) (, ) (, ) f(x, ) + f xx (x, ) + f(x, ) 3.3. (, ) (, ) 3.: x 3 3x + 3 (, ) (, )
36 3 f(a, b) = f(a, b) f(a, b) = f(a, b) 8. f(x, ) =x 3 + 3 6x 6 3x 6 f(x, ) =, f(x, ) = 3 6 6x 6x { 3x 6= 3 6= (x, ) =(±, ± ) (x, ) (, ) (, ) (±, ) f(x, ) + + f xx (x, ) + f(x, ) 8 8 (x, ) =(, ) 8 (x, ) =(, ) 8 8 8 3. 4 4 (, ) (, ) (±, ) (, ) (, )
3.4. 37 x**3+**3-6*x-6* 5 5 5 5-5 - -5-4 -3 - - 3 - - -3 4-4 3 4-5 - -5 3.: x 3 + 3 6x 6 3..?? 3.4 x 8. n f. f. a R n. f x R n f(x) f(a)+ f(a)(x a) f(a) = f(x) f(a) a f
38 3 = = 3.3:. () f(x, ) =x 3 3x + 3 () f(x, ) =3x +x + (3) f(x, ) =x 3 + 3 9x + 3.5 7. ( ) (a, b) (x, ) R h(t) = f ((a, b)+t(x, )) h(t) h() t (3.) t = h(t) =h() + h ()t + h ()t + o(t ) (3.3) h x () = f(a, b) = (3.) (3.3) h ()t + o(t ) t t o(t )/t h () = [ x ] x f(a, b) (3.4) h(t) (x, ) 3.4 (x, ) f(a, b) ( ) (a, b) f(a, b) λ.?? λ > x f ((a, b)+(x, )) = f(a, b)+ f(a, b) + ] x [ x f(a, b) + o ( (x, ) )
3.5. 7 39 o(δ ) lim = δ + δ ε <ε<λ/ (, ) (x, ) δ = (x, ) o ( (x, ) ) ε (x, ) { } x f ((a, b)+(x, )) f(a, b)+ f(a, b) + ] x [ x f(a, b) (a, b) f ((a, b)+(x, )) >f(a, b)+ ] x [ x f(a, b) ε (x, ) <ε (x, ) u = x (x, ) u =,?? t u f(a, b)u λ, x x f(a, b) λ (x, ). ε, ( ) f ((a, b)+(x, )) >f(a, b)+ λ ε (x, ) >f(a, b). (x, ) (, ), (a, b). ( ) (a, b) f(a, b) 7
4 3 f(x, ) Step Step f x (x, ) = f (x, ) = f(a, b) (x, ) =(a, b) f(a, b) = (a, b) f(a, b) = (a, b) f(x, ) f(x, ) f(a, b) = f(a, b) = f(a, b) f(a, b) = f(a, b)