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- うまじ ひがき
- 5 years ago
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Transcription
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3 (1) g(x) (2) 49
4 6.1 t
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6 8 OK
7 1 point y y = f(x) O a b x x = a, x = a + h 2 y y = f(x) f(a+h) f(a) O a a+h x
8 2 y f(a + h) f(a). h f(a+h) f(a+h) f(a) f(a) O a h a+h x h a + h a x = a!! y lim h 0 f(a + h) f(a). h f(a+h) f(a) O a a+h x f (a) y = f(x) x = a (differential coefficient)
9 . y = f(x) x = a x = a y = f(x) (differential coefficient) f f(a + h) f(a) (a) = lim. h 0 h. y = x x = 1. f(x) = x f f(1 + h) f(1) (1) = lim. h 0 h f f(1 + h) f(1) (1) = lim. h 0 h (1 + h) 1 = lim h 0 h h = lim h 0 h = 1.. y = x 2 x = 1. f(x) = x 2 f f( 1 + h) f( 1) ( 1) = lim. h 0 h
10 f (h 1) 2 ( 1) 2 ( 1) = lim. h 0 h (h 2 2h + 1) 1 = lim h 0 h h 2 2h = lim h 0 h h(h 2) = lim h 0 h = lim (h 2) = 2. h 0 a x x y = f(x) y = f(x) (derivative) f (x), dy dx. y = f(x) x y = f(x) y = f(x) (derivative) dy dx = f f(x + h) f(x) (x) = lim. h 0 h y = f(x) y = f(x) f (x) x ( x = 1) y = f(x). y = x 2 x = 1. f(x) = x 2 f f(x + h) f(x) (x) = lim. h 0 h
11 f (x + h) 2 x 2 (x) = lim. h 0 h (x 2 + 2hx + h 2 ) x 2 = lim h 0 h h(2x + h) = lim h 0 h = lim (2x + h) = 2x. h 0 f (x) = 2x. x = 1 f ( 1) = 2...!! 1.1 (1) y = f(x) x = a f (a) (2) y = f(x) f (x) (3) y = C (4) 1 y = ax + b (a, b ) (5) y = x 3 x = 1 (6) ( ) { af(x) + bg(x) } = af (x) + bg (x)
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13 2 point y = x n (1 n ) y = a x ( ) y = sin x, y = cos x ( ) y = log x ( ) 2.1
14 (1) [ ] sin π π 6 = 30 3 (2) 0 (1) sin π 6 = 1 2 [ ] cos 3π 4 sin π 4 = 45 O 3π 4 = 135 cos (2) cos x sin y cos 3π 4 = sin 0 = 0, sin π = 1, sin π = 0 2
15 cos 0 = 1, cos π = 0, cos π = 1 2 tan 0 = 0, tan π, tan π = sin 2 α + cos 2 α = 1 tan α = sin α cos α 1 + tan 2 α = 1 cos 2 α cos 2 α = 1 sin 2 α, sin 2 α = 1 cos 2 α (sin 2 cos 2 cos 2 sin 2 ) 2.1.2, sin (α ± β) = sin α cos β ± cos α sin β }{{}}{{} cos (α ± β) = cos α cos β sin α sin β }{{}}{{}
16 2.1.3 sin 2α = 2 sin α cos α. cos 2α = 1 2 sin 2 α = 2 cos 2 α 1. sin sin 2α = sin (α + α) (α + α ) = sin α cos α + cos α sin α ( ) = 2 sin α cos α sin 2 α 2 = 1 cos2 α. 2 cos 2 α 2 = 1 + cos2 α. 2 sin 2 α = 1 cos2 2α. 2 sin cos cos 2 α = 1 + cos2 2α. 2 cos 2α = 1 2 sin 2 α
17 sin 2 α 2 sin 2 α = 1 cos 2α sin 2 α = 1 cos 2α a m a n = a m+n (a m ) n = a mn a m = 1 a m a 1 m ( ) = m a (m m ) a 0 = 1 ( 0 1) 2.2.2
18 3 2 1 y 7 y = a x (a > 1) 6 5 y = a x (0 < a < 1) O x O y x y x = log a y y = a x O x a p = M }{{} a p M p = log a M }{{} a M? p
19 [ ] log 2 8 = 3 (2 8 3 ) 1 [ ] log 3 9 = 2 ( ) log a 1 = 0 (a 1 0 ) log a a = 1 (a a 1 ) log a a b = b (a a b b ) log a b c = c log a b ( ) log a M + log a N = log a MN log a M log a N = log a M N ( ) ( ) y y = log a x (a > 1) O x 2.3 (1) tan 2 α = 1 sin 2 α. (2) α x = sin α cos α x sin 2α, tan α x
20 (3) sin α cos β = 1 { sin (α + β) + sin (α β) } 2 cos α cos β = 1 { cos (α + β) + cos (α β) } 2 sin α sin β = 1 { cos (α + β) cos (α β) } 2 (4) (a > 0) ( a x a x ) 2 ( a x + a x ) =. 2 2 (5) (6) b x = a x log a b. log a b = log c b log c a
21 3 point!! 3.1 y = f(x) f (x) dy dx f (x) = dy dx = lim h 0 f(x + h) f(x). h
22 3.1.1 x n y = x n f(x) = x n f f(x + h) f(x) (x) = lim h 0 h (x + h) n x n = lim h 0 h = lim h 0 x n + n C 1 x n 1 h + n C 2 x n 2 h n C n 1 xh n 1 + h n x n h ( (x + h) n ) nc 1 x n 1 h + n C 2 x n 2 h n C n 1 xh n 1 + h n = lim h 0 h [ = lim nc 1 x n 1 + h( n C 2 x n n C n 1 xh n 3 + h n 2 ) ] h 0 = nx n 1. (x n ) = nx n 1 x n (!!) (x n ) = nx n y = cos x, y = sin x (tan ) f(x) = sin x
23 f f(x + h) f(x) (x) = lim h 0 h sin (x + h) sin x = lim h 0 h 2 cos 2x+h 2 sin h 2 = lim h 0 h ( ) ( ) 2 cos x + h 2 sin h 2 = lim h 0 h = lim sin h 2 h 0 h 2 }{{} 1 cos ( x + h 2 ) = cos x sin x cos x sin x sin x cos x (!!) (sin x) = cos x (cos x) = sin x y = a x (a > 0) f(x) = a x f f(x + h) f(x) (x) = lim h 0 h a x+h a x = lim h 0 h = lim a x ah 1. h 0 }{{ h } ah 1 h 1 (h 0 ) a a =
24 e. a h 1 lim h 0 h = 1 a e (Napier s constant) e = e. lim (1 + h) 1 h = e. h 0 e h 1 lim h 0 h = 1. h lim h 0 (eh 1) = lim h. h 0 1 lim h 0 eh = lim (1 + h). h 0 1 h e = lim h 0 (1 + h) 1 h. e y = e x f e x+h e x (x) = lim h 0 h = lim e x eh 1 h 0 }{{ h } 1 (e ) e x e x = e x.
25 e x (!!) (e x ) = e x e y = e x y = log e x e y = log x *1 y = log x f(x) = log x f log x + h log x (x) = lim h 0 h 1 = lim h 0 h log x + h x 1 = lim h 0 x x h log ( 1 + h x ) k = h x h 0 k 0 f 1 1 (x) = lim log (1 + k) k 0 x k = 1 x lim k 0 log (1 + k) 1 k }{{} e = 1 x log e = 1 x. log x (!!) (log x) = 1 x *1 e
26 (x n ) = nx n 1 (sin x) = cos x (cos x) = sin x (e x ) = e x (log x) = 1 x (i) y = x 3 + x 2 + x + 1 (ii) y = 2x2 + x x 3x + 5 (iii) y = (iv) y = x 2 ( ) 2 x ( ) 2 x (v) y = 3 sin x + 2 cos x (vi) y = log x e x (vii) y = 5e x log x (viii) y = 1 2 sin x + 2ex + x 2
27 4 point (formula) (rule) 5
28 4.2. y = xe x. *1 y = }{{} x }{{} ex x e x!! y = (x) }{{} (ex ) }{{} = 1 ex = e x. *1
29 f(x)g(x) }{{} f(x) g(x) *2 = f (x) g(x) + f(x) g (x). }{{}}{{}}{{}}{{} { xe x } = (x) }{{} e x }{{} = e x + xe x = e x (x + 1). + x }{{} (e x ) }{{}. y = x 2 cos x. y = }{{} x 2 cos }{{ x } y = (x 2 ) cos x + x 2 (cos x) = 2x cos x + x 2 ( sin x) = 2x cos x x 2 sin x = x(2 cos x x sin x). *2
30 . y = e x sin x. y = e x }{{} sin x }{{}. y = (e x ) sin x + e x (sin x) = e x sin x + e x cos x = e x (sin x + cos x). 4.3 y = f(x) g(x) y = f(x) g(x). f(x) g(x)!! y = f (x) g (x)!!
31 f(x) g(x) }{{} f(x) g(x) = {}}{{}}{{}}{{}}{ f (x) g(x) f(x) g (x) { g(x) } 2 }{{} ( ) 2 2 ( ). y = 5x2 + x 3x y = {}}{ 5x 2 + x 3x }{{} y = (5x2 + x) (3x 3 + 1) (5x 2 + x)(3x 3 + 1) (3x 3 + 1) 2 = (10x + 1)(3x3 + 1) 9x 2 (5x 2 + x) (3x 3 + 1) 2 = 30x4 + 10x + 3x x 4 9x 3 (3x 3 + 1) 2 = 15x4 6x x + 1 (3x 3 + 1) 2.
32 . (tan ) y = tan x tan x = sin x cos x y = {}}{ sin x cos x }{{} *3 y = (sin x) cos x sin x(cos x) cos 2 x cos x cos x sin x( sin x) = cos 2 x = cos2 x + sin 2 x cos 2 x = 1 cos 2 x ( 1 ) y = 1 cos 2 x. *3 tan x sin cos sin cos
33 . y = log x x 2 y = {}}{ log x x 2 }{{} y = (log x) x 2 log x(x 2 ) (x 2 ) 2 = = 1 x x2 2x log x x 4 x 2x log x x 4 x(1 2 log x) = x 4 = 1 2 log x x 3.. y = log x x y = {}}{ log x x }{{}
34 y = (log x) x log x( x) ( x) 2 1 x x log x 1 2 x = x = 1 1 x log x 2 x x = 2 log x 2x x ( ) (x n ) = nx n 1 (e x ) = e x (sin x) = cos x (cos x) = sin x (log x) = 1 x f(x)g(x) } = f (x)g(x) + f(x)g (x) { } f(x) g(x) = f (x)g(x) f(x)g (x) { g(x) } (1) y = (x 3 + 2x)(x 2 + 1) (2) y = x log x (3) y = e x cos x (4) y = x 2 e x
35 (5) y = sin 2x ( sin 2x 2 ) 2. ( 3 ver) { f(x)g(x)h(x) } = f (x)g(x)h(x) + f(x)g (x)h(x) + f(x)g(x)h (x).
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37 5 (1) point 2 ( )
38 5.1 f(x) = sin x, g(x) = 2x + 1 f(x) x g(x) f(g(x)) f(g(x)) = sin (2x + 1) x f(x), g(x) f(g(x)) x f, g. f(g(x)) (1) { f(x) g(x) = e x = 3x (2) { f(x) g(x) = log x = x 2 + x (3) { f(x) g(x) = sin x = e x
39 (1) f(x) x g(x) f(g(x)) = e 3x. (2) (3) f(g(x)) = log (x 2 + x). f(g(x)) = sin e x. f(x) g(x) f(g(x)) f(g(x)) f(x), g(x). f(x), g(x) (1) f(g(x)) = e 3x (2) f(g(x)) = log (x 2 + x) (3) f(g(x)) = sin e x (1) f(g(x)) f(x) = e x, g(x) = 3x. f(g(x)) = e 3x. (2) f(g(x)) f(x) = log x, g(x) = x 2 + x. f(g(x)) = log (x 2 + x). (3) f(x) = sin x, g(x) = e x. f(g(x)) f(g(x)) = sin e x.
40 f(x), g(x) 5.2 x x y = f(x) x y dy dx. t y = g(t) t dy dt dy dx, dy dt y x y t *1 5.3 *1
41 y = f(g(x)) t = g(x) y = f(t) dy dx = dy dt dt dx. dy dt dt dx y t x dy dx.
42 . x y = e 3x+1 f(g(x)) f(x) = e x, g(x) = 3x + 1 f(g(x)) = e 3x+1 dy }{{} dx y x (1) t = g(x) t = 3x + 1 y t y = f(t) = e t (2) dy dx dy dx = dy dt dt dx = e t 3 = 3e t = 3e 3x+1.
43 . x y = (5x + 3) 5 f(x) = x 5, g(x) = 5x + 3 f(g(x)) = (5x + 3) 5 (1) t = 5x + 3 }{{} g(x) y = t 5 }{{} f(t). (2) dy dx = dy dt dt dx = 5t 4 5 = 25(5x + 3) 4.. x y = sin (x 2 + x + 1) f(x) = sin x, g(x) = x 2 + x + 1 f(g(x)) = sin (x 2 + x + 1)
44 (1) t = x 2 + x + 1 }{{} g(x) y = sin t. }{{} f(t) (2) dy dx = dy dt dt dx = cos t (2x + 1) = (2x + 1) cos (x 2 + x + 1). x y = e x2 f(x) = e x, g(x) = x 2 f(g(x)) = e x2 (1) t = }{{} x 2 g(x) y = }{{} e t f(t). (2) dy dx = dy dt dt dx = e t 2x = 2xe x g(x) g(x) 1
45 . x (1) y = (3x + 7) 5 (2) y = e 2x+1 (3) y = sin (5x 1) (4) y = log (10x + 2) (1) f(x) = x 5, g(x) = 3x + 7 y = f(g(x)) t = 3x + 7 y = t 5 dy dx = dy dt dt dx = 5t4 3 = 3 5(3x + 7) 4. (2) f(x) = e x, g(x) = 2x + 1 y = f(g(x)) t = 2x + 1 y = e t dy dx = dy dt dt dx = et 2 = 2 e 2x+1. (3) f(x) = sin x, g(x) = 5x 1 y = f(g(x)) t = 5x 1 y = sin t dy dx = dy dt dt dx = cos t 5 = 5 cos (5x 1). (4) f(x) = log x, g(x) = 10x + 2 y = f(g(x)) t = 10x + 2 y = log t dy dx = dy dt dt dx = 1 t 10 = x + 2. { (3x + 7) 5 } = 3 5 (3x + 7) 4. { e 2x + 1 } = 2 e 2x + 1. { sin (5x 1) { log (10x + 2) } = 5 cos (5x 1). } = 10 1 (10x + 2).
46 y = cos (2x + 1) (1) 2x + 1 cos (2x + 1) }{{} cos (2) x sin (2x + 1) }{{} sin y = 2 sin (2x + 1).!! g(x) t f(ax + b) { f(ax + b) } = af (ax + b).!! (1) y = (x 2 + x + 1) 9 1 (2) y = x2 + 1 (3) y = e x3 (4) y = 2e x2 + sin (3x + 1)
47 (5) y = log (x 2 1) (6) y = 3 sin 5x + 2 cos x 2 (7) y = sin (cos x) (8) y = e sin x (9) y = (e x + 1) 3 (10) y = log (e x sin x) 2. ( { f(ax + b) } = af (ax + b) ) (1) y = (4x 9) 100 (2) y = 3 sin 2x (3) y = e x (4) y = e 3x + 3 cos 5x (5) y = x + 2 (6) y = log (2x 2)
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49 6 (2) point
50 6.1 t. y = sin (x 2 + x) t = x 2 + x y = sin t. dy dx = dy dt }{{} dt dx }{{} t t = cos }{{} t (2x + 1) = (2x + 1) cos (x 2 + x). }{{} t t t t t y = sin (x 2 + x) }{{} = cos (x 2 + x) }{{} x 2 + x }{{} = cos (x 2 + x) (2x + 1) = (2x + 1) cos (x 2 + x). t. y = e x2 +1
51 dy dx = e x2 + 1 }{{} x = 2xe x2 +1. }{{}. y = (x 3 + x) 10 dy ( ) 9 dx = 10 x 3 + x x 3 + x = 10(3x 2 + 1)(x 3 + x) 9. }{{}}{{}. y = log (x 2 + x + 1) dy dx = 1 x 2 + x + 1 x 2 + x + 1 }{{}}{{} = 2x + 1 x 2 + x + 1. t t. y = 1 x2 + a 2 (a )
52 { } y = (x 2 + a 2 ) 1 2 = 1 2 (x2 + a 2 ) 3 2 2x = x(x 2 + a 2 ) 3 2 x = (x 2 + a 2 ) x 2 + a y = e x sin 3x e x sin 3x g(x) x y = (e x ) ( sin 3x + e x sin 3x ) ( ) = e x sin 3x + 3e x cos 3x ( x ) = e x (3 cos 3x sin 3x).. y = (x 2 + 2) 2 (3x + 1) 3
53 t y = { (x 2 + 2) 2 } (3x + 1) 3 + (x 2 + 2) 2 { (3x + 1) 3 } ( ) = 4x(x 2 + 2)(3x + 1) 3 + 9(x 2 + 2) 2 (3x + 1) 2. ( ) = (x 2 + 2)(3x + 1) 2 { 4x(3x + 1) + 9(x 2 + x) } = x(x 2 + 2)(3x + 1) 2 (21x + 13). ( ) t... *1. y = 1 (3 sin x 2 2) 3 2 y = 1 (3 sin x 2 2) 3 = (3 sin x 2 2) 3 y = { (3 sin x 2 2) 3 } = 3(3 sin x 2 2) 4 (3 sin x 2 2) ( ) = 3(3 sin x 2 2) 4 3 cos x 2 2x (!!) = 18x(3 sin x 2 2) 4 cos x 2 = 18x cos x2 (3 sin x 2 2) 4 *1 dy dx = dy dt dt dx.
54 . sin x2 y = e 2 { } y = e sin x2 = e sin x2 { sin x 2 } = e sin x2 cos x 2 2x = 2x cos x 2 sin x2 e. y = e tan x y = e { } tan x tan x = e { tan x (tan x) 1 2 } = e tan x 1 2 (tan x) 1 2 (tan x) = e tan x 1 2 = 1 tan x e tan x 2 cos 2 x tan x. 1 cos 2 x 6.2
55 . y = x x (1) dy dx y = x x. (2) x x x log y = x log x (3) x y = x x (y x!!) log y 1 y }{{} y dy. }{{} dx y x dy dx!! x 1 dy y dx = log x + 1 ( ) (4) dy dx y dy dx = y(log x + 1). y dy dx = xx (log x + 1).!!
56 . y = (cos x) ex dy dx (1) log y = e x log (cos x). (2) x 1 y dy dx = ex log (cos x) + e x 1 ( sin x) } cos x {{}!! (3) y { ( dy dx = y e x log (cos x) sin x ) }. cos x y dy dx x = (cos x)e { e x (log (cos x) tan x) } y = log x dy dx = 1 x. 3
57 (1) y = log x e y = x. (2) x (3) e y e y dy dx = 1. dy dx = 1 e y = 1 x (ey = x ) y = sin 1 x. sin 1 x sin x x y = sin 1 x sin y = x. cos y dy dx = 1 dy dx = 1 cos y. cos 2 y = 1 sin 2 y = 1 x 2 cos y = 1 x 2 dy dx = 1 1 x 2.
58 . y = tan 1 x. x y = tan 1 x tan y = x 1 cos 2 y dy dx = 1 dy dx = cos2 y. 1 + tan 2 y = 1 cos 2 y cos2 y = 1 1+tan 2 y = 1 1+x 2 dy dx = x 2. cos 1 x (sin 1 x) = 1 1 x 2 (cos 1 x) 1 = 1 x 2 (tan 1 x) = x
59 dy dx = 1 dx dy 6.4!! 6.5 (1) (i) y = (4x 2 + x) 2 (ii) y = x 2 (x 2 + x) 3 (iii) y = log (cos x) (iv) y = eax a 2 +b (a sin bx b cos bx) 2 ( ) (v) y = 1 2 log 1 cos x 1+cos x ( (vi) y = 1 2 x x2 + a 2 + a 2 log (x + x 2 + a 2 ) ) (2) y = (sin x) cos x (3) y = cos 1 x
60 (4) x = y 2 + 4y 1 dy dx (5) x 0 y = f(x) = xe x g(x) g (e)
61 7 point ( ) 7.1 y = f(x) x = a *1 *1
62 y y = f(x) O a x y = mx + n x = a y = f(a) (a, f(a)) x = a f (a) f(a) = f (a) }{{}}{{}}{{} a +n. y m x n n = f(a) af (a) y = f (a)x + (f(a) af (a)). y f(a) = f (a)(x a). y = f(x) x = a y = f(x) x = a y f(a) = f (a)(x a). (a, b) m y b = m(x a) (a, f(a)), m = f (a) x
63 . x = 1 y = x 2 1 x = 1 y = 1 (1, 1) f (x) = 2x x = 1 f (1) = 2 y = mx + n 1 = n n n = 1 y = 2x 1. 2 a = 1, f(x) = x 2, f (x) = 2x y f(a) = f (a)(x a) y 1 = 2(x 1). y = 2x 1 1
64 . x = 1 y = x 3 2x 1 x = 1 y = 1 ( 1, 1) f (x) = 3x 2 2 x = 1 f ( 1) = 1 y = mx + n 1 = 1 ( 1) + n n n = 2 y = x a = 1, f(x) = x 3 2x, f (x) = 3x 2 2 y 1 = 1(x ( 1)) y = x y = x 3 2x x = 1
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66 !! x 1 x 3 2x x + 2 }{{}}{{} x=1 x = a y = f(x) x = a y = f(x) x a f(x) f (a)(x a) + f(a).
67 y = f(x) x = a θ (θ 0 ) sin θ θ. y = sin x x = 0 y = x
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87 f(x)g(x) }{{} f(x) g(x) = f (x) g(x) + f(x) g (x). }{{}}{{}}{{}}{{} { f(x)g(x) } = lim h 0 f(x + h)g(x + h) f(x)g(x) h f(x + h)g(x + h) f(x + h)g(x) + f(x + h)g(x) f(x)g(x) = lim h 0 h = lim h 0 g(x + h) g(x) h } {{ } g (x) = g (x)f(x) + f (x)g(x). f(x + h) + }{{} f(x) f(x + h) f(x) h } {{ } f (x) g(x)
88 17.2 f(x) g(x) }{{} f(x) g(x) = {}}{{}}{{}}{{}}{ f (x) g(x) f(x) g (x) { g(x) } 2 }{{} 2 { f(x) g(x) } = lim h 0 = lim h 0 f(x+h) g(x+h) f(x) g(x) h f(x+h)g(x) f(x)g(x+h) g(x+h)g(x) h = lim h 0 f(x + h)g(x) f(x)g(x + h) h 1 g(x + h)g(x) f(x + h)g(x) f(x)g(x) + f(x)g(x) f(x)g(x + h) 1 = lim h 0 h g(x + h)g(x) f(x + h)g(x) f(x)g(x) { f(x)g(x + h) f(x)g(x) } 1 = lim h 0 h g(x + h)g(x) = lim h 0 f(x + h) f(x) h } {{ } f (x) = f (x)g(x) f(x)g (x) { g(x) } 2. g(x + h) g(x) g(x) f(x) 1 }{{ h } g(x + h) g(x) }{{} g (x) g(x)
89 { f(x) g(x) } = [f(x) { g(x) } 1] = f (x) { g(x) } 1 + f(x) ( 1) { g(x) } 2 g (x) }{{} [{ g(x) } 1 ] = f 1 (x) g(x) f(x) g (x) { g(x) } 2 = f (x)g(x) f(x)g (x) { g(x) } 2 ( )
f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a
3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 (
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18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C
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Chap11.dvi
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II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R
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5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)
, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x
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III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2 lim. (x,y) (1,0) x 2 + y 2 lim (x,y) (0,0) lim (x,y) (0,0) lim (x,y) (0,0) 5x 2 y x 2 + y 2. xy x2 + y
III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2. (x,y) (1,0) x 2 + y 2 5x 2 y x 2 + y 2. xy x2 + y 2. 2x + y 3 x 2 + y 2 + 5. sin(x 2 + y 2 ). x 2 + y 2 sin(x 2 y + xy 2 ). xy (i) (ii) (iii) 2xy x 2 +
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(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e 0 1 15 ) e OE z 1 1 e E xy 5 1 1 5 e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0 Q y P y k 2 M N M( 1 0 0) N(1 0 0) 4 P Q M N C EP
chap1.dvi
1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt
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1 1 3 ABCD ABD AC BD E E BD 1 : 2 (1) AB = AD =, AB AD = (2) AE = AB + (3) A F AD AE 2 = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD 1 1
ABCD ABD AC BD E E BD : () AB = AD =, AB AD = () AE = AB + () A F AD AE = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD AB + AD AB + 7 9 AD AB + AD AB + 9 7 4 9 AD () AB sin π = AB = ABD AD
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2
θ i ) AB θ ) A = B = sin θ = sin θ A B sin θ) ) < = θ < = Ax Bx = θ = sin θ ) abc θ sin 5θ = sin θ fsin θ) fx) = ax bx c ) cos 5 i sin 5 ) 5 ) αβ α iβ) 5 α 4 β α β β 5 ) a = b = c = ) fx) = 0 x x = x =
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( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED)
rational number p, p, (q ) q ratio 3.14 = 3 + 1 10 + 4 100 ( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED) ( a) ( b) a > b > 0 a < nb n A A B B A A, B B A =