013 4 16 5 54 (03-5465-7040) nkiyono@mail.ecc.u-okyo.ac.jp hp://lecure.ecc.u-okyo.ac.jp/~nkiyono/inde.hml 1.. y f(, y) = a + by + cy + p + qy + r a, b, c 0 y b b 1 z = f(, y) z = a + by + cy z = p + qy + r (, y) z = p + qy + r 1 y = + + 1 y = y = + 1 6 + + 1 ( = + 1 ) + 7 4 16 y y y + = O O O y = y = + 1 y = + + 1 6: ac b 0 a + by + cy + p + qy + r = a( 0 ) + b( 0 )(y y 0 ) + c(y y 0 ) + r
0, y 0, r ( 0, y 0 ) 1 p = a 0 by 0 q = b 0 cy 0 ac b 0 r = r a 0 b 0 y 0 cy 0 ac b 0 f(, y) = a + by + cy ac b = 0 1 ac b = 0 z = f(, y) f(, y) 1 a, b, c 0 a 0 f(, y) = a ( ( + b ) ) a y ac b + a y 1 y = y = z = f(, y) a a z = ( + ba ) y + ac b a y z = ( + ba ) y ac b a y 3 y 1 X, Y X = + b a y 0, y 0, r 3 z = f(, y) y
3 Y ac b ac b b ac Y = a y Y = a y ac b = 0 Y = y X Y a > 0, ac b > 0 z = X + Y a > 0, ac b = 0 z = X a > 0, ac b < 0 z = X Y a < 0, ac b > 0 z = X Y a < 0, ac b = 0 z = X a < 0, ac b < 0 z = X + Y a 0 a = 0 c 0 y a = c = 0 b 0 by = b ( ( + y) ( y) ) X = + y, Y = y b z = X Y z = X + Y a 0 z = X Y 1 Y z = X z = X 7 ac b = 0 1 z = X z = X 7:
4 X Y 1 z = X + AX + BY + C z = X + AX + BY + C Y 1 X 7 zx z = X + AX + C z = X + AX + C X = A X = A Y Y z z = BY + C 1 z = X + Y XY (, y) = (0, 0) XY z = k X + Y = k k z = X + Y 1 z 1 Xz Xz Y = 0 z = X + Y z = X z z = X Y 8 X, Y, y z = X + Y z = X Y 8: y z = X Y z = X + Y X Y z = X Y XY X Y = k
5 Y z = 0 z = O X z = z = 3 z = 1 z = 3 z = 1 9: z = X Y k X k Y k = 0 Y = ±X 9 XY Xz Y = k z = X k X = 0 k z = k Y z Xz z = X Y z z = Y z = X Y? 10 z = X Y 10: z = X Y 4. (1) + 4y + 5y + + 3y + 1 () + 6y + 5y + + 3y + 1
6 1..3 f(, y) z = f(, y) z = y = = f(, y) 1 g() = + y z = f(, y) z = g() 0 z z = X + Y = X + Y 1 1 1 3 4 1 4 3 4 n 1 1 1 1.1 1 a f() f() f(a) lim a a (3) f() = a f() = a f (a) df d (a) d d f() =a
7 f() a f() a f (a) f() f (), df d (), d d f() = a a f() = f () = f(3) = 3 = 6 f f() f(3) 3 (3) = lim = lim 3 3 3 3 = lim ( 3)( + 3) 3 3 = lim 3 ( + 3) = 3 + 3 = 6 = 3 a f f() f(a) a (a) = lim = lim a a a a = lim ( a)( + a) a a = lim a ( + a) = a + a = a a ( ) = 5. 1 { < 0 f() = 3 0. (3) f() f(a) f(a) f() a
8 a f(a) a a 1 (a, f(a)) (, f()) 11 y f() f(a) f() f() f(a) a f(a) a 1 y = f() O a a + 1 11: a = a 1 y = f() (a, f(a)) 1.3 1 1 1 1
9 y f (a) f() f(a) a O a a + 1 1: f() = c = a 0 f() f(a) a = c c a = 0 0. 0 n n = a na n 1 n a n = ( a)( n 1 + n a + + a n + a n 1 ) n a n lim a a = lim a (n 1 + n a + + a n + a n 1 ) = a n 1 + a n a + + aa n + a n 1 = na n 1 n n n 1 (a 0 + a 1 + a + + a n n ) = a 0 (1) + a 1 () + a ( ) + + a n ( n ) = a 1 + a + 3a 3 + + na n n 1
10 d d sin = cos d. d cos = sin d d e = e an log ( f()g() ) = f ()g() + f()g () f()g() f(a)g(a) = f()g() f(a)g() + f(a)g() f(a)g(a) f()g() f(a)g(a) lim = lim a a = (f() f(a))g() + f(a)(g() g(a)) f() f(a) a a = f (a)g(a) + f(a)g (a) ( ) 1 = f () f() (f()) lim a 1 f() 1 f(a) a g() g(a) g() + lim f(a) a a f(a) f() 1 = lim a a f()f(a) = f 1 (a) f(a)f(a) an an = sin cos ( (an ) = sin 1 ) ( ) = (sin ) 1 1 cos cos + sin cos = cos 1 ) ( ( cos + sin (cos ) cos = 1 + sin sin ) cos = 1 + sin cos = 1 + an 1 cos an = 1 + an = cos cos + sin cos = 1 cos
11 (f(g()) = f (g())g () g() f() f(g()) = g () = 1 f (g()) log log e e log = (e ) = e (log ) = 1 e log = 1 > 0 < 0 (log( )) = 1 ( ) = 1 ( 1) = 1 (log ) = 1 r ( r ) = r r 1 > 0 r = e r log ( r ) = (e r log ) = e r log (r log ) = r r = rr 1 6. f() = e sin g() = 3 + + 1 h() = log(cos ) 3 n 1,,..., n f f n 1
1 n = n = n 3 3.1 (a, b) f(, y) y = b 1 4 = a φ (a) φ() := f(, b) f(, y) (a, b) = a y 1 y = b ψ (b) ψ(y) := f(a, y) f(, y) (a, b) y φ() 1. f (, y) (, y) = (a, b) f f(, b) f(a, b) lim a a f (a, b) (a, b) f (a, b) f(a, b) (a, b) f y f(a, y) f(a, b) (a, b) = lim y y b y b. 4 φ() := f(, b) φ() f(, b) := =:
13 1 d MS-IME f(, y) (a, b) f (a, b) (a, b) 1 (a, b) (, y) 1 1 1 f(, y) = y 3 f (, y) = y 3 y f y (, y) = 3 y. y 1 y f (a, b) (a, b) f (a, b) (a, b) f(, y) f (, y). f(, ) () f f(, ) () 1 φ() = f (, () ) f f (, () ) () df (, ) d φ () (, ) 1 7. f(, y) f(, y) = sin ( 3 y ) (, y), (, y) y
14 3. 1 1 1 y = f() = a f (a) y = a 1 z = f(, y) (, y) = (a, b) (, y) = (a, b) (a, b) (, y) = (a, b) z 1 y (a, b) y y f(, y)? y 1. 5 T V 6 T V U f(, y) = T y = V U = f(t, V ) f(, y) V 0 T 0 7 (T 0, V 0 ) (T 0, V 0 ) 5 6 7
15. f(, y) y f T T T V f U = f(t, V ) T V V 0 T T = T 0 T (T 0, V 0 ) U f f U f U T V U U = U(T, V ) 1. 0 (cm) f(, ) 13: f(, )
16 1 ( ) f (, ) := (, ) 1 f(, ) (, ) (, ) = C f (, ) C 8. > 0 f(, ) = 1 e f (, ) = 4 (, ) 13 1 5 3. g(, ) 14: g (, ) = c g (, ) (4) c 9. c f(, ) = sin( + c) + sin( c) (4) 14
17 1 1 f (a, b) φ() = f(, b) 1 φ = a f (a, b) φ() (a, φ(a)) φ() φ() f(, y) f(, y) yz z = f(, y) z = φ() φ() f(, b) z = φ() z = f(, y) y = b y = b yz y = b z b = 0 z b 0 z y y b φ() z = f(, y) ( a, b, f(a, b) ) z f (a, b) φ (a) z y = b z = φ() z = φ() = f(, b) b a 15: ( a, b, f(a, b) ) 15 f y (a, b)
18 z = f(, y) ( a, b, f(a, b) ) yz ( a, b, f(a, b) ) f (a, b) z = f(a, b) (a, b, f(a, b)) f y (a, b) y 16 = a f y (a, b) z y = b f (a, b) b a 16: y
19 4 1 a y b y c (1) a = 1, b =, c = 5 a = 1 > 0 ac b = 1 5 = 5 4 = 1 > 0 z = X + Y () a = 1, b = 3, c = 5 ac b = 1 5 3 = 5 9 = 4 < 0 z = X Y 5 < 0 f() f () = < 0 > 0 f() 3 f () = 3 > 0 = 0 f() 0 = 0 f() f(0) lim 0 0 f (0) f() 0 0 0 0 +0 f (0) f() f(0) lim = lim 0 0 0 = lim = 0 0 f() f(0) 3 lim = lim +0 0 +0 = lim +0 = 0
0 f (0) = 0 { < 0 f () = 3 0 0 f() = 0 = 0 0 f() = 3 +0 3 = 0 6 f () = (e ) sin + e (sin ) = e sin + e cos = e (sin + cos ) g () = ( ) ( 3 + + 1) ( 3 + + 1) ( 3 + + 1) = 4 + 4 + 3 4 ( 3 + + 1) = 4 + + ( 3 + + 1) h () = log (cos ) (cos ) = 1 ( sin ) = an cos 7 y 1 1 y g() = 3 y 1 f(, y) sin z z = g() sin z = cos z g () = ( 3) y = 3 y (, y) = ( sin g() ) g () = ( cos ( 3 y )) 3 y = 3 y cos ( 3 y )
1 h(y) = 3 y y 1 f(, y) sin z z = h(y) y (, y) = ( sin h(y) ) h (y) = ( cos ( 3 y )) 3 y = 3 y cos ( 3 y ) 8 1 f(, ) (, ) = e 1 f (, ) = ( e ) = e e = 4 e 1 f(, ) 1 (, ) = e f (, ) = 4 + 1 e e = 4 e e = = 4 (, ) 9 (, ) = cos( + c) + cos( c) ( ) f (, ) = (, ) = sin( + c) sin( c) = f(, ) f(, ) (, ) = c cos( + c) c cos( c) f (, ) = ( ) (, ) = c sin( + c) c sin( c) = c f(, ) f (, ) = c f(, ) = c ( f(, )) = c f (, ) (4)