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1 [] (,65 [] (,3 ( ) /=45 /= /=36 4/= /=38 /= /=364 9/= /=45 /= /=9 /= ( ) ( ) ( ) ( ) ( )

2 (mean value) {x, x, x 3,, x n } x = x + x + x x n n = n i= x i c n f c n f c k n k f k x = n c i n i = i= c i f i i= (median) ( ) (n ) n + (n ) n n + (mode) 3

3 3 3 (sample variance) s x = n (x i x) = n i= s x = n k (c i x) n i = x i x i= i= i= (unbiased variance) k (c i x) f i = n k c i n i x = i= k c i f i x i= u x = n i= (x i x) = n k (c i x) n i i= ( ) ((sample) standard deviation) s x = s x x i x s x + 5 (mean deviation) n 4 () n () n x i x i= (x i x) i= (x i x) = n i= (3) u x n x i x i= x i x i= 5 () () ()

4 4 4 (skewness) n ( ) 3 xi x = i= s x n s 3 x (x i x) 3 i= (kurtosis) n ( ) 4 xi x 3 = i= s x n s 4 x (x i x) 4 3 i= (coefficient of variation) c v = s x x 5 ( ) n (x, y ), (x, y ),, (x n, y n ) (sample covariance) s xy = n (x i x)(y i y) i= 6

5 5 s xy = n x i y i xy i= 7 (sample correlation coefficient) r xy = s xy s x s y 8 9 r xy < = n (x, y ), (x, y ),, (x n, y n ) y i = ax i + b (i =,, 3,, n, a > ) r xy a < = d d j d s c f f j f s f c i f i f ij f is f i c r f r f rj f rs f r f f j f s n 6 s xy = x i y i xy n i= x i y i x i = a + bu i, y i = c + dv i x i x y i y x i s x y i

6 6 s y (x i, y i ) s xy r xy u i u v i v u i s u v i s v (u i, v i ) s uv r uv

7 7 6 n (x, y ), (x, y ),, (x n, y n ) y x y x (y = ax + b) x y L a, b ( ) L = ê i = {y i (ax i + b)} = nb b i= i= (y i ax i ) + i= (y i ax i ) = nb nb(y ax) + n(s y + y ) an(s xy + x y) + na (s x + x ) = n [ ] {b (y ax)} + a s x as xy + s y [ ( = n {b (y ax)} + s x a s ) ] xy + s xs y s xy s x s x a = s xy s y = r s xy, b = y ax = y x s xy s y = y xr x s x s xy x s x i= 3 S e a, b L a = x i {y i (ax i + b)} = n [ (s xy + x y) a(s x + x ) bx ] = i= L b = {y i (ax i + b)} = n{y (ax + b)} = i= a, b a, b ê i = y i (ax i + b) S e = ê i = {y i (ax i + b)} R = i= i= n i= (ax i + b y) n i= (y i y) 4 S e R () S e = ns xy( r xy) () R = r xy 5 n 3 (x, y, z ), (x, y, z ),, (x n, y n, z n )

8 8 z x y

9 9 7 n 3 (x, y, z ), (x, y, z ),, (x n, y n, z n ) z x y z x y (z = ax + by + c) x, y z L a, b, c ( ) L = {z i (ax i + by i + c)} i= L a = x i {z i (ax i + by i + c)} = i= L b = y i {z i (ax i + by i + c)} = i= L c = {z i (ax i + by i + c)} = i= a = s zxs y s zy s xy s xs y s xy, b = s zys x s zx s xy s xs y s xy, c = z ax by ẑ i = ax i + by i + c z i ẑ i R r zx + rzy r zx r zy r xy R = rxy R = r zx + rzy r zx r zy r xy rxy 6 a, b, c 7 z i ẑ i R

10 3 n k p k = n C k ( 3 ) k ( ) n k 3 3 p n k p k = n C k p k q n k ( p + q = ) B(n, p) 8 EXCEL n =, p = 3, q = 3 p, p, p,, p µ = kp k k= σ = (k µ) p k k= σ = σ µ = np, σ = npq 9

11 Poisson n n np = λ λ λk p k = e k! (k =,,, ) e e = 788 λ Poisson P o(λ) Poisson Poisson µ = kp k σ = (k µ) p k µ = λ, σ = λ k= k= Poisson 4 % k B(, ) np = Poisson p + p 3 + p 4 + = p p = e e = 64 4 EXCEL Poisson 64 EXCEL λ = Poisson p, p, p,, p

12 3 X (, ) f(x) > =, f(x) dx = X (a, b) P (a < X < b) P (a < X < b) = b a f(x) dx f(x) (probability density function) P (X < x) = F (x) = µ = x f(t) dt (distribution function) xf(x) dx µ = (x µ) f(x) dx 3 (a, b) = {x a < x < b} f(x) = F (x) = for x < a b a for a < = x < = b for b < x for x < a x a b a for a < = x < = b for b < x (a, b) (uniform distribution) U(ab) y b a a b x 3 (a, b) U(ab) µ σ

13 3 3 3 f(x) = πσ exp { } (x µ) σ exp x = e x µ σ (normal distribution) N(µ, σ ) µ =, σ = (standard normal distribution) N(, ) F (x) = x } (t µ) exp { dt πσ σ y y = e x π x 4 e x dx = π exp { πσ } (x µ) dx = σ 5 N(µ, σ ) µ σ 6 EXCEL N(, )

14 4 33 Γ ( ) B( ) Γ (s) = B(p, q) = Γ B e x x s dx (s > ) x p ( x) q dx (p >, q > ) () Γ (s + ) = sγ (s) ( Γ () =, n Γ (n + ) = n! ) () Γ (s) = (3) B(p, q) = (4) Γ ( ) = π, e t t s dt Γ (p)γ (q) Γ (p + q) = (5) Γ (p) = p π Γ (p)γ π cos p θ sin q θ dθ ( e t dt = ( ) π Γ = ( p + ) ( ) ) n x + x + x x < n = r r n x + x + x x n = r r n n { }} { n V n (r) = dx dx dx n x +x +x 3 + +x n < = r (n ) { }} { n S n (r) = x +x +x 3 + +x n =r ds () S n (r) = π n r ( n n ) Γ () dv n(r) dr (3) V n (r) = = S n (r) r S n (t) dt = π n r ( n n ) nγ

15 3 5 7 n =,, 3, 4, 5 n

16 6 Γ B () Γ (s + ) = e x x s dx = { e x } x s dx = [ ( e x )x s] + e x (sx s ) dx = + sγ (s) Γ () = e x x dx = [ e x] = ( ) = Γ (n + ) = nγ (n) = n(n )Γ (n ) = = n!γ () = n! () Γ (s) = = (3) B(p, q) = e x x s dx (x = t, dx = t dt, x : t : ) e t t s (t) dt = x p ( x) q dx e t t s dt (x = cos θ, dx = cos θ sin θ dθ, x : θ : π ) = π Γ (p)γ (q) = 4 = 4 cos p θ sin q θ( cos θ sin θ) dθ = D D e x x p e y y q dxdy e (x +y ) x p y q dxdy π cos p θ sin q θ dθ (D = {(x, y) < = x, < = y}) { π } x = r cos θ, y = r sin θ, D = (r, θ) < = r, < = θ < = x y J = r r cos θ sin θ = x y r sin θ r cos θ = r θ θ dxdy = J drdθ = rdrdθ Γ (p)γ (q) = 4 e r r p+q cos p θ sin q θ drdθ = 4 D e r r p+q dr = Γ (p + q)b(p, q) B(p, q) = Γ (p)γ (q) Γ (p + q) (4) ( Γ ( )) = Γ ( + )B(, ) = π Γ ( ) > Γ ( ) = π π cos p θ sin q θ dθ

17 3 7 Γ (5) B(p, q) = ( ) = x p ( x) q dx e t dt e t dt = π B(p, q) = B(p, p) = x = t, x = t dt, dx = dt, x : t : t p ( t ) q tdt = x p ( x) p dx = t = x, x = t + B(p, p) = p t p ( t ) q dt {x( x)} p dx, dx = dt, x : t : ( t ) p dt = p = p B(, p) = p B(, p) B(p, p) = B(, p) p Γ (p) Γ (p) = Γ ( )Γ (p) p Γ(p + ) Γ (p) = p Γ ( ) Γ (p)γ (p + ) = p π Γ (p)γ (p + ) ( t ) p dt = p ( t ) p β α β α β α (x α) p (β x) q dx x = (β α)t + α, dx = (β α)dt, x : α β t : x α = (β α)t, (β x) = (β α)( t) (x α) p (β x) q dx = (β α) p+q t p ( t) q dt = (β α) p+q B(p, (x α) p (β x) q dx = (β α) p+q B(p, q)

18 8 n (3) V n (r) = π n r ( n n ) nγ n = ( )= V (r) = r, ( )= π r ( ) = r Γ n = k V k (a) = π k a ( k ) k kγ n = k + V k+ (r) = dx dx dx k dx k+ D k+ (r) D k+ (r) = {(x, x,, x k, x k+ ) x + x + + x k + x < k+ = r } ( ) V k+ (r) = = r r r r π k D k( r x k+) dx dx dx k ( r x k+ kγ ( k ) ) k dx k+ dx k+ = π k r ( ) k kγ ( k) r x dx r = π k kγ ( k ) r r (x + r) k (r x) k dx = π k k kγ ( k)(r) ++ k + B( k +, k + ) = k+ π k Γ ( k + ) kγ ( k) Γ (k + ) rk+ = k+ k π ( k ) Γ ( k ) kγ ( k) (k + )kγ (k) rk+ = k π k Γ ( k ) (k + )Γ (k) rk+ Γ (k) = k Γ ( k)γ ( ) k+ π = k π k Γ ( k) (k + ) k Γ ( k)γ ( ) rk+ = k+ π π k+ (k + )Γ ( k+ )r k+

19 ) f(x) = B(a, b) xa ( x) b ( < x < ) (x <, < x) (a >, b > para-meter a, b (beta distribution) Be(a, b) µ = a a + b, σ = ab (a + b) (a + b + ) 8 35 f(x) = { λe λx (x > ) (x < ) (λ > ) para-meter λ (exponential distribution) Ex(λ) µ = λ, σ = λ F (x) = x λe λt dt = [ e λt] x = e λx 9 36 f(x) = c ( x ) c { ( x c } exp a a a) (x > ) (x < ) (a >, c > ) para-meter (a, c) (Weibull distribution) W e(a, c) c = Ex( a ) F (x) = x c a ( t a ) c exp { ( ) c } t { ( x c } dt = exp a a) 3

20 37 f(x) = Γ (α)β α xα exp( x ) (x > ) β (x < ) (α >, β > ) Ga(α, β) µ = αβ, σ = αβ α = Ex( ) β 3 f(x) dx = 38 (χ n) X, X, X 3,, X n N(, ) X = X + X + X3 + + Xn = n χ n ( n ) x n e x (x > ) f n (x) = Γ n (x < ) k= n χ n Ga(α, β) α = n, β = χ n = Ga( n, ) X k F (x) F (x) = = ( π) n e x e x e x n dx dx dx n x +x + +x n < = x π π π x +x + +x n < = x exp{ (x + x + + x n)} dx dx dx n

21 3 x = rω, x = rω,, x n = rω n r = x + x + + x n, ω + ω + + ω n = dx dx dx n = r n drds ds n x F (x) = ( exp( r π) n ) rn drds = ( ds π) n S n S n x exp( r ) rn dr = n ( π ( π) n n ) Γ x exp( r ) rn dr f n (x) = F (x) = n ( π ( π) n n ) exp( ( x) )( x) n { x} = Γ ( n ) e x x n Γ n 39 t X, Y X N(, ) Y n χ n T = X Y/n n ( )t t n ( ) n + Γ ( ) n+ f(t) = ( n ) + t nπγ n n = f(t) = π( + t ) Cauchy 3 t Cauchy f(t) = Cauchy F (x) = x f(t) dt π( + t )

22 t F (t) F (t) = e x D t π Γ ( n ) n y n e y dxdy D t = {(x, y) < x <, < = y <, x y/n < = t } (x, y) (s, u) s = x u, u = y x = s y/n n, y = u D t = {(s, u) < s < = t, < = u < } x y u J = s s = n u u x y s u u nu = n, dxdy = n dsdu F (t) = e x ( D t π n ) y n e y dxdy Γ n = = πγ ( n ) n nπγ ( n ) n exp( Dt us n )u n e u u n dsdu t { } exp{ u s + n n } du ds u n f(t) = F (t) = v = u t + n n f(t) = nπγ ( n ) n u = nv t + n, du = nπγ ( n ) n u n ( n ) n+ t + n exp{ u t + n n } du n dv, u : v t + n v n e v du = nπγ ( n ) n ( ) n+ n Γ t + n ( ) n + = ( ) n + Γ ( n ) nπγ ( ) n+ + t n

23 3 BASIC ) download (Yahoo Google BASIC ) ) download BASIC734zip ( ) BASIC734setupexe (USB ) 3) BASIC BASICEXE BASIC ) END ) ( ) 3) ( ) $ 4) LET = (LET ) INPUT INPUT PROMPT " ": 5) + - * / ^ 5 ^5 () 6) PRINT ( ) PRINT, ( ) PRINT ; ( ) 7)

24 4 PRINT " " PRINT " " LET a= LET b=5 LET c$="university of Hyogo" LET d$="kobe University of Commerce" PRINT a+b,a-b,a*b,a/b PRINT c$ PRINT d$ END INPUT PROMPT "x=(x>)":x INPUT PROMPT "y=":y PRINT "x+y=";x+y,"x-y=";x-y,"x^y=";x^y END 3 ( ) (I) FOR NEXT FOR = TO [ STEP 3] NEXT 3 ( EXIT FOR) 3 FOR x= TO print x,"kobe" NEXT x END

25 BASIC 5 4( ) LET S= FOR x= TO LET S=S+x NEXT x PRINT S END INPUT n 4 () n () (3) n ( )n n (4) n! = 3 4 n (5) +! +! + 3! + + n! 3 a n+ = a n + a n a = a a 3,, a (II) DO WHILE LOOP ( ) DO WHILE LOOP ( EXIT DO)

26 6 () <> >< () >= => (3) <= =< (4) AND, OR (5) NOT 5( ) LET S= LET X= DO WHILE X<= LET S=S+X LET X=X+ LOOP PRINT S END (III) DO LOOP WHILE ( ) DO LOOP WHILE ( EXIT DO) (IV) DO LOOP UNTIL ( ) DO LOOP UNTIL ( )

27 BASIC 7 (V) DO UNTIL LOOP ( ) DO UNTIL LOOP ( ) 6( ) INPUT PROMPT " x=":x INPUT PROMPT " y=":y LET q= DO UNTIL x<y LET x=x-y LET q=q+ LOOP PRINT " =";q PRINT " =";x END DO UNTIL x<y DO WHILE x>=y 4 3%,,3, (DO LOOP WHILE ) () ABS(x) x x () SQR(x) x x (3) INT(x) x [x] (Gauss ) (4) sin x SIN(x) cos x COS(x) tan x TAN(x) sin x ASIN(x) cos x ACOS(x) tan x ATN(x) (5) e x EXP(x) log e x LOG(x) log x LOG(x) (6) RND < = RND < RANDOMIZE

28 8 5 3 (DO LOOP WHILE ) SQR(), an > a n ^(-) 4 (I) IF ( ) IF THEN IF THEN ELSE (II) IF THEN END IF ( ) IF THEN END IF END IF (III) IF THEN ELSE END IF ( ) IF THEN ELSE 3 END IF 3

29 BASIC 9 (IV) ELSEIF IF THEN ELSEIF THEN ELSE 3 END IF 3 7( ax = b ) INPUT PROMPT "a=":a INPUT PROMPT "b=":b IF a<> then LET x=b/a PRINT "x=";x ELSEIF b= THEN PRINT " ( )" ELSE PRINT " " END IF END 8( ) RANDOMIZE LET X=INT(RND*)+ DO INPUT Y LOOP WHILE X<>Y PRINT " " END

30 3 6 () () X > Y X < Y (3) (V) DO LOOP WHILE UNTIL DO LOOP WHILE UNTIL DO LOOP FOR NEXT EXIT DO EXIT FOR (VI) FOR NEXT DO LOOP 3 EXIT FOR EXIT DO GOTO GOTO (= ) GOTO 7 () 3% 5 () 3% 5 (6 ) (3) 3% (4) %

31 BASIC 3 5 (I) SET WINDOW,,, (II) PLOT POINTS: x, y SET POINT STYLE point style (III) PLOT LINES: x,y ; x,y (x, y ) (x, y ) SET LINE COLOR , SET LINE STYLE,, 3, 4 (IV) DRAW circle ( line color ) (x, y) r DRAW circle WITH SCALE(r)*SHIFT(x,y) (V) DRAW AXES (x y ) DRAW GRID (x y ) (VI) CLEAR (VII) FLOOD x,y ( (x, y) (x, y) area color ) PAINT x,y ( (x, y) line color area color )

32 3 SET AREA COLOR ( ) SET WINDOW -3,3,-3,3 PLOT LINES:-,-;,-;,;-,;-,- PLOT LINES:-,-;, PLOT LINES:-,;,- DRAW circle WITH SCALE() PLOT POINTS:,5 SET AREA COLOR 6 paint, END ( ) SET WINDOW -3,3,-3,3 PLOT LINES:-,-;,-;,;-,;-,- FOR x=- TO STEP 4 PLOT LINES:x,-;x, NEXT x FOR y=- TO STEP 4 PLOT LINES:-,y;,y NEXT y FOR x=-+ TO STEP 4 FOR y=-+ TO STEP 4 DRAW circle WITH SCALE()*SHIFT(x,y) NEXT y NEXT x END 8 () () ( 55) (3) (4) 6

33 BASIC 33 FOR y=-+ TO STEP 4 FOR y=-+ TO x STEP 4 (5) 55 (6) (5) 55 ( 55)

34 34 y = f(x) y = f(x) x p, y q y = f(x p) + q x a, y b y = bf( x a ) f(x, y) = x p, y q f(x p, y q) = x a, y b f ( x a, y b ) = y = f(x) 3(y = x ( 3 < = x < = 3) graph) SET WINDOW -3,3,-,5 DRAW AXES DEF f(x)=x^ LET h= FOR x=-5 TO 5-h STEP h PLOT LINES:x,f(x);x+h,f(x+h) NEXT x END BASIC DEF ( )= PLOT LINES PLOT LINES:x,y; ( ) PLOT LINES PLOT LINES:x,y ( ) PLOT LINES ( )

35 35 4(y = ( 5 < x = x < = 5) graph) SET WINDOW -5,5,-5,5 DRAW AXES DEF f(x)=/x LET h= FOR x=-5 TO 5-h STEP h PLOT LINES:x,f(x);x+h,f(x+h) NEXT x END 4 x = 4-(y = ( 5 < x = x < = 5) graph) SET WINDOW -5,5,-5,5 DRAW AXES DEF f(x)=/x LET h= FOR x=-5 TO 5-h STEP h WHEN EXCEPTION IN PLOT LINES:x,f(x);x+h,f(x+h) USE PLOT LINES END WHEN NEXT x END BASIC WHEN EXCEPTION IN USE END WHEN

36 36 9 (x ) y = f(x) y = g(x) y = h(x) () y = f(x) = x 3 x, y = g(x) = x 4 x ( < = x < = ) () y = f(x) = x x ( < = x < = ) (3) y = f(x) = x + x, y = g(x) = x x ( 7 < = x < = 7) (4) y = f(x) = x ( < = x < = 3) ( x = SQR(x)) (5) y = f(x) = x +, y = g(x) = x ( 5 < = x < = 5) (6) y = f(x) = sin x, y = g(x) = cos x ( 7 < = x < = 7) (7) y = f(x) = cos x + sin 4x ( 5 < = x < = 5) (8) y = f(x) = x cos x ( < = x < = ) (9) y = f(x) = sin x ( 7 < = x < = 7) (sin x = SIN(x), () y = f(x) = f(x) = sin x + sin x ( < = x < = ) cos x = COS(x)) () y = f(x) = tan x, y = g(x) = tan x ( 7 < = x < = 7) (tan x = TAN(x), tan x = ATN(x)) () y = f(x) = x, y = g(x) = x, y = h(x) = log x ( 7 < = x < = 7) (log a x = LOG(X)/LOG(a) log x = LOG(x), log x = LOG(x) (3) y = f(x) = cosh x = ex + e x, y = g(x) = sinh x = ex e x y = h(x) = tanh x = sinh x cosh x = ex e x e x + e x ( 5 < = x < = 5) (e x = EXP(x), cosh x = COSH(x), sinh x = SINH(x), tanh x = TANH(x)) (4) y = f(x) = [x], y = g(x) = x [x] ( 5 < = x < = 5) ([x] = INT(x))

37 37 { x = f(t) t : α β y = g(t) t α β (f(t), g(t)) P xy para-meter t (para-meter) (f(α), g(α)) (f(β), g(β)) (cycloid) a P (cycloid) { x = aθ a sin θ y = a a cos θ θ : β 5(cycloid) SET WINDOW -,7,-,7 DRAW AXES DEF f(t)=t-sin(t) DEF g(t)=-cos(t) LET h= FOR t= TO 8-h STEP h PLOT LINES:f(t),g(t);f(t+h),g(t+h) NEXT t END (epicycloid) b a P (epicycloid) x = (a + b) cos θ b cos a + b θ b y = (a + b) sin θ b sin a + b θ b θ : β

38 38 3(hypocycloid) b a P (hypocycloid) ( b < a) x = (a b) cos θ + b cos a b θ b y = (a b) sin θ b sin a b θ : β θ b 4( (Lissajous) ) { x = a cos(ω t + α ) y = a cos(ω t + α ) t : β { x = A cos(at) y = B sin(bt + δ) t : β 5( ) x a + y b = { x = a cos θ y = b sin θ x a y b = { x = a cos θ y = b tan θ θ : π θ : π x = a cosh t = a et + e t t : y = b sinh t = b et e t ( ) (trochoid) (epitrochoid) (hypotrochoid)

39 39 x = a cos θ, y = a sin θ (a, ) 3 P xy- O O OX OP = r, XOP = θ P (r, θ) θ r OP = r, XOP = θ P P O r θ f(r, θ) = OX x O y 6( ) { x = r cos θ y = r sin θ r = x + y tan y x tan y θ = + π x π (x > ) (x < ) (x =, y > ) π (x =, y < ) θ = α (α : ) (r, α) A OA r cos(θ α) = r 7( ) r r = r (r : ) (r, α) r r + r rr cos(θ α) r = 8( ( )) r = aθ (a > )

40 4 9( ) r = a θ (a > ) ( ) r = a sin nθ (a >, n ) (Pascal (limaçon= )) r = a cos θ + b (a >, b > ) a = b (cardioid) 8( ) 9( ) 3 ( ) n =,, 3, 4, 5 4 (Pascal ) a > b, a = b, b = a r = f(θ) { x = f(θ) cos θ y = f(θ) sin θ 5 4 6

41 f(x) = α ( ) a = a a + b = a a + b = b = b b x [a, b] f(x) f(x) [a, b] f(x) f(a) f(b) f(a)f(b) < f(x) = [a, b] α f(a) <, f(b) > ( ) () a = a, b = b () a b a + b f(x) f( a + b ) = α = a + b f( a + b ) < a = a + b, b = b f( a + b ) > a = a, b = a + b

42 4 () a b a + b α a b f(x) f( a + b ) = α = a + b f( a + b ) < a = a + b, b = b f( a + b ) > a = a, b = a + b α a b (k) a k b k a k + b k f(x) f( a k + b k ) = α = a k + b k f( a k + b k ) < a k = a k + b k, b k = b k f( a k + b k ) > a k = a k, b k = a k + b k α a k b k b = k a k ( ) a k + b k E E < = b k a k = (b a) k n n f(a) <, f(b) > ( ) () a = a, b = b () [a, b ] n f(x) a b f(x) () [a, b ] n f(x) a b f(x)

43 3 43 (k) [a k, b k ] n f(x) a k b k f(x) b = k a k ( ) a k + b k E E < = b k a k = (b a) nk Newton x x x x [a, b] f(x) f(x) [a, b] f (x) f(x) f(a) f(b) f(a)f(b) < a b α f (x) >, f(a) <, f(b) > ( ) () x = b () x y = f(x) y = f (x )(x x ) + f(x ) x x x = x f(x ) f (x ) () x y = f(x) y = f (x )(x x ) + f(x )

44 44 x x x = x f(x ) f (x ) (k) x k y = f(x) y = f (x k )(x x k ) + f(x k ) x x k x k = x k f(x k ) f (x k ) x k = x k f(x k ) f (x k ) x = b ( x = a) x a b f(a) f(b) f (x) f(x) x = x k Taylor (n = ) f(α) = f(x k ) + f (x k )(α x k ) + f (c)(α x k ) E k = x k α = x k α f(x k ) f(α) f (x k ) = f (c) f (x k ) x k α m = min f (x), M = max f (x) x [a,b] x [a,b] E k < = M m x k α = M m E k n E k < = ce k ( ) Newton ( ) (x k, f(x k )), (x k, f(x k )) y = f(x k), f(x k ) x k x k (x x k ) + f(x k ) x x = x k (x k x k )f(x k ) f(x k ) f(x k ) x k+ Newton x k+ = x k (x k x k )f(x k ) f(x k ) f(x k ) x, x

45 3 45 x 3 x x x x Newton x, x f(x) f (x) Newton 5 f(x) = x = Newton 3 b a f(x) dx ( ) [a, b] a = x < x < x < < x n < x n = b n [x k, x k ](k =,, 3,, n) ξ k Riemann S(, {ξ k }) = f(ξ k )(x k x k ) k= lim S(, {ξ k }) {ξ k } b a f(x) dx n {ξ k } x k

46 46 x k = a + (b a) k, ξ k = x k = a + n Riemann S n () S () n = k= f ( a + (b a) (k ) n ) b a n (b a) (k ) n {ξ k } x k Riemann S n () S () n = k= f ( a + b a ) (b a) b a k n n f(x) dx n S n () S n () S () n S () n x x x x n x n x x x x x n x n x x x x x n x n x

47 3 47 ( ) (x k, y k ) (x k, y k ) y = f(x) (y + y ) b a n + (y + y ) b a n + + (y n + y n ) b a n = b a n {y + y n + (y + y + + y n )} ( ) S () n S () n S() n + S n () E E < = (b a) 3 M n M = max f (x) xı[a,b] Simpson x x x x n x n x n x [a, b] n x k = a + b a k (k =,,,, n) n y y k = f(x k ) n [x k, x k ] 3 (x k, y k ), (x k, y k ), (x k, y k ) f(x) g(x) b a g(x) dx = b a 6 ( g(a) + 4g ( ) a + b ) + g(b)

48 48 (a = x k, b = x k ) xk x k g(x) dx = b a 6n (y k + y k + 4y k ) b a f(x) dx = x x f(x) dx + x4 x f(x) dx + + xn x n f(x) dx = = x x g(x) dx + x4 x g(x) dx + + xn x n g(x) dx b a 6n (y + y n + 4(y + y y n )+ +(y + y y n )) Simpson Simpson E E < = (b a) 5 88n 4 M 4, M 4 = max x [a,b] f (4) (x) n order Simpson n order ( Simpson ) n4 4 6 dx + x Simpson

49 4 ( ) 49 4 ( ) 4 ( ) 7 ( ) a n P n P n+ P n 8 (Fibonacci ) a 8 a Fibonacci a n+ = a n+ + a n, a =, a = 9 n 3 n a n a n+ a n ( ) C p a( p/c)p ( p C ) (a ) n n p n p n+ p n ( ) N x x n+ x n = kx n (N x n ) ( ) a b c % d % n n R n F n

50 5, R n+ F n+ R n, F n [ ] [ ] Rn+ Rn = A F n+ F n 3( R F ( + A( R/C))R DRF QDRF n n R n F n, R n+ F n+ R n, F n ( A: C: D: Q: ) ( ) n n Pn P n P n 3 Pn 3 ( ) 5( 4 ) A B r a b (n + ) x n+ a, b, x n b 6(Markov chain ) 3 7 5

51 4 ( ) n x n y n z n x n+, y n+, z n+ x n, y n, z n 7 n k f(n, k) f(n +, k) f(n, k) f(n, k ) ( ) f(n, ) =, f(n, n) = f(5, 3) 8 n p(n) n k p(n, k) p(n) = p(n, k) k= p(n, k) x + x + + x k = n (x x x k ) (x, x,, x k ) p(n, k) p(n k, k) p(n, k ) ( x k = x k ) p() 9(Mandelbrot Set) z n+ = zn + α, z = n z n α Mandelbrot Set Mandelbrot Set (Julia Set) z n+ = zn + β, z = α n z n α Julia Set Julia Set β

Appendix A BASIC BASIC Beginner s All-purpose Symbolic Instruction Code FORTRAN COBOL C JAVA PASCAL (NEC N88-BASIC Windows BASIC (1) (2) ( ) BASIC BAS

Appendix A BASIC BASIC Beginner s All-purpose Symbolic Instruction Code FORTRAN COBOL C JAVA PASCAL (NEC N88-BASIC Windows BASIC (1 (2 ( BASIC BASIC download TUTORIAL.PDF http://hp.vector.co.jp/authors/va008683/