mupad for highschool section 8

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1 8 ( III) 8. ( ) ( ) ( )

2 8 ( III) ) a n n f () a f () a+ f () a it(a(n), n = infinity); it(f(), = a); it(f(), = a, Right); it(f(), = a, Left); f () diff(f(), ) n f (n) () diff(f(), $n); f ()d int(f(), ) b a f ()d int(f(), = a b) sin sin() cos cos() tan tan() a a e ep() E log a log(a, ) log ( ) ln() ( ) PI e( ) E ( ) infinity ) 8. ( ) 8.. MuPAD ( ) infinity, ( ) infinity n n n? it(( n )/( n ), n = infinity); >> n? it(( n )/( n ), n = infinity); >> in f inity n n n? it(( n )/( n ), n = infinity); >> in f inity n n ( n + ) n? it(( + /n) n, n = infinity); >> ep() n n ) ( ), ( ) ) diff differentiate( ) int integral( ) int(f(), = a b);.. M

3 MuPAD e ep() ep() = e = e. e ; e = n (n + n ) n 8.. S n S n n {a k } n sum(a(k), k =..n); ) sum, S = + + ( ) + + ( ) n + ( ) n ; S n = n ( ) k,s n k= sum((/) (k ), k =..n); >> (/) n it(%, n = infinity); >> ( ), n S n,,( ), S n = { ( ) n } S n = n ( ) n = 8.., a f () it(f(), = a); sin? it(sin()/, = ); >> ) sum(k,k=..) k= k = + + =

4 cos () sin? it(cos() /( sin()), = PI/); >> = t, t cos () sin = t cos ( t) sin ( t) = t sin t cos t = t cos t cos t = t ( + cos t) = e h? it((e h )/h, h = ); >> h h log( + t)? it(ln( + t)/t, t = ); >> t t ( + t) t? it(( + t) (/t), t = ); >> ep() t ( + )? it(( + /), = infinity); >> ep() ( + )? it(( + /), = infinity); >> ep() MuPAD e = ep(),ep()= e = e e h h h = t log( + t) t = t ( + t) t ( = e + ) = e ± 8.., f (), f () a a a+ a ( ) a a ( ) MuPAD f () it(f(), = a, Right), f () a+ a it(f(), = a, Left) ) f () = +? it(/( ), =, Right); in f inity? it(/( ), =, Left); in f inity? it(/( ), = ); unde f ined ) <, Left > <, Right > MuPAD

5 > >, < <, + =, = MuPAD y = f () a b plotfuncd(f(),=a..b) y c y d plotfuncd(f(),=a..b,y=c..d) 5) y =, y plotfuncd(/( ), =.., y =..); window, = = MuPAD 5) plotfuncd plot( )+function( )+d( ), 5

6 8. f () diff(f(),) 8.. (sin ) cos (cos ) sin (tan ) cos (e ) e (log ) ( α ) α α MuPAD MuPAD (e) E, e = ep(), log = ln()( ) ( ) (sin )? diff(sin(), ); >> cos() (cos )? diff(cos(), ); >> sin() (tan )? diff(tan(), ); >> tan() + (e )? diff(ep(), ); >> ep() (log )? diff(ln(), ); >> ( a )? diff( a, ); >> a a ( )? diff(sqrt(), ); >> (tan ) = cos = + tan, ( ) = ( ) = = 6) 8. n f (n) () diff(f(), $ n) 7) f () = f ()? diff(, $); 6 f () diff(, $); 6 ( ) =, ( ) = ( ) = 6, ( ) = (6) = 6 8. (u + v) = u + v (uv) ( ) = u v + uv u v = u v uv v f (g()) = f (g())g () 6) MuPAD diff(tan(),); rewrite(%,sincos); simplify(%); rewrite(f(),sincos) f () sin() cos() 7) $ k $k=..5; >>,,9,6,5 6

7 , diff () ( ) diff((7 6)/( + ), ); >> 7 + (7 6) ( + ) normal(%) normal(%); >> factor(%); >> ( + ) 8) () ( ) sin +sin diff(( + sin())/( sin()), ); >> cos() sin() + factor, factor(%); >> ( ) cos() (sin() + ) ( ) sin = ( sin ) ( + sin ) ( sin )( + sin ) + sin ( + sin ) cos ( + sin ) ( sin )cos = ( + sin ) = cos (sin + ) cos()( sin()) (sin() + ),MuPAD, 9) 8.5 f () int(f(),) MuPAD C e d d α d(α=\ ) e + C log + C α+ α+ + C sin d cos d cos d cos + C sin + C tan + C 8) factor(); normal(%); factor(%); factor(%); 9) 7

8 MuPAD e d? int(ep(), ); ep() d? int(/, ); >> ln() ln() = log(), ep() = e sin d? int(sin(), ); >> cos() cos d? int(cos(), ); >> sin() cos d? int(/cos(), ); >> sin() cos() +, d = tan tan cos rewrite(%, tan); >> { tan } ( tan() ( + tan() ) ) + + tan() normal normal(%); >> tan cos d? int(cos(), ); >> + sin() sin cos d? int(sin( ) cos(), ); >> cos() cos() 6 cos = +cos cos d = + cos d = + sin + C sin cos = (sin + sin ) sin cos d = (sin + sin )d = cos 6 cos + C, MuPAD 8

9 8.6 sin d? int(sin(), =..PI); >> cos d? int(cos(), =..PI/); >> e d? int(e, =..); >> ep() d? int(/, =..); >> ln() (e e )d? int(e E ( ), =..); >> ep() ep( )? simplify(%); >> d? int(/( 6 + 8), = 6..8); >> ln() ln() ln(6) ? combine(%, ln); >> ln sin d = [ cos ] = e d = [ e ] = e (e e )d = [ e +e ] = (e+e ) (e +e) = 8 ( d = cos d = [ sin ] = d = [ log ] = log ) d = [ log ] 8 [ log ] = (log log ) (log 6 log ) = log (log + log ) + log= log log = log 8 MuPAD e, log ep(),ln(), simplify ( ), combine(,ln) (log ), normal ( ), factor( ) float( ) MuPAD int e e sin( log ) d? int(sin(pi ln())/, =..E); >> log d? int( ln(), =..E); >> ep()

10 log = t, d = dt, e t e sin( log ) d = sin tdt = [ ] cos t = e log d = e ) ( ) [ log d = log ] e e e d = [ ] e 9 = e ( ) MuPAD abs() MuPAD, cos d? int(abs(cos()), =..PI); >> sin + cos d? int(abs(sin() + cos()), =.. PI); >> sin + cos d? int(abs( sin() + cos()), =.. PI); >> cos d = cos d + ( cos )d = [ sin ] [ sin ] sin + cos = sin ( ) +, + = t d = dt, t + = sin + cos d = + sin t dt sin t + sin t dt = sin t dt = sin tdt = [ cos t ] = (, ) ), ) MuPAD ) sin + cos = ( tan, tan )

11 8.8 a c g() f (), c b f () g() = a, = b, y = f (), y = g() S S = = = c a c a b a { f () g()}d + f () g() d + f () g() d b c b c {g() f ()}d f () g() d y = g() y = f () f () g() a c b C : y = sin ( ), C : y = sin ( ) S y y = sin O α y = sin,mupad S = sin sin d int(abs(sin( ) sin()), =..PI); >> 8 < < α ( <α< ) sin α = sin α sin α = sinα sin α sin α sinα = sin α (sin α )(sin α + ) = <α< sin α = α =

12 C, C = S = α (sin sin )d + α (sin sin )d = [ cos + cos ] α + [ cos + = ( ) cos α + cos α ( ) + + cos ] α ( cos α cos α = cos α + cosα = cos + cos = ) S = = 8 )? C : y = sin ( ), C : y = sin ( ) S y y = sin O α y = sin,mupad S = sin sin d int(abs(sin() sin( )), =..PI); >> < < α ( <α<) sin α = sinα sin α = sinα cos α sin α( cosα) = cos α = S = α ( sin sin )d + α (sin sin)d = [ cos + cos ] α + [ cos + cos ] α = ( cos α + cos α) + (cos α cos α) = cosα + cosα + = ( cos α ) + cosα + ) Maple + MuPAD

13 cos α = S = cos α + cosα + = + + = 7! MuPAD ) 8.9 MuPAD intlib(integral library) stdlib(standard library; ) computer hold(); hold() hold eval(); subs(); ) 5) f ()d = g(t) b a intlib :: changevar(hold(int)(f(), ), = g(t)); f ()d = g(t) intlib :: changevar(hold(int)(f(), = a..b), = g(t)); f ()g ()d, intlib :: byparts(hold(int)(f() g (), ), g ()); b f a ()g ()d, intlib :: byparts(hold(int)(f() g (), = a..b), g ()); 8.9. ( ) f ()d = g(t) intlib::changevar(hold(int)(f(),),=g(t)) hold eval subs(, ) ( ) 5 d = t intlib :: changevar(hold(int)( ( ) 5, ), = t); >> int(t 6 t 5, t); hold, t = t ( ) 5 d = (t 6 t 5 )dt eval eval(%); >> t7 7 t6 6 t,subs(, ) subs(%, t = ); >> ( )7 7 ( )6 6 ), Maple cos cos = arccos int(*sin(*) sin(),=..arccos(/))+ int(sin() *sin(*),=arccos(/))..pi) MuPAD int(*sin(*) sin(),=..a)+ int(sin() *sin(*),=a..pi) ) changevar(),bypart(),eval(),subs() change variable( ),integration by parts( ), evaluate( ),substitute( ) subs(f,=a) f =a 5) hold(); intlib :: changevar(hold(int(f(), )), = g(t));

14 8.9. ( ) () ( ) 5 d = t intlib :: changevar(hold(int)( ( ) 5, =..), = t); >> int(t 6 t 5, t =..); = t ( )5 d = (t6 t 5 )dt eval() eval(%); >> () + d = tan t intlib :: changevar(hold(int)(/( + ), =..), = tan(t)); >> int(, t =..PI/); d = dt eval + eval(%); = tan t, d = cos t dt, t, + = +tan t >> PI = cos t + d = cos t cos t dt = dt = [ t ] = 8.9. () cos d cos d = (sin ) d cos intlib :: byparts(hold(int)( cos(), ), cos()); >> sin() int(sin(), ) sin d = sin sin d eval eval(%); >> cos() + sin() cos d = (sin ) d = sin () sin d = sin sin d = sin + cos cos d = ( ) cos d intlib :: byparts(hold(int)( cos(), ), ); >> cos() ( ) int sin(),

15 () I = e sin d e sin d = (e ) sin d I := intlib :: byparts(hold(int)(ep() sin(), ), ep()); >> ep() sin() int(ep()cos(), ),I = e sin d= (e ) sin d= e sin e cos d eval() eval(%); >> sin()ep() cos()ep() 8.9. MuPAD,,,,,,, MuPAD, MuPAD, 5

A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π

4 4.1 4.1.1 A = f() = f() = a f (a) = f() (a, f(a)) = f() (a, f(a)) f(a) = f 0 (a)( a) 4.1 (4, ) = f() = f () = 1 = f (4) = 1 4 4 (4, ) = 1 ( 4) 4 = 1 4 + 1 17 18 4 4.1 A (1) = 4 A( 1, 4) 1 A 4 () = tan