Web viewHighSchool-Calculus (Larson et al.) We selected the textbook "Calculus with Analytic Geometry 6th edition" written by . Roland E. Larson . Robert P. Hostetler

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HighSchool-Calculus (Larson et al.)

We selected the textbook

"Calculus with Analytic Geometry 6th edition" written by Roland E. Larson Robert P. HostetlerBruce H. Edwards

printed by Houghton Mifflin Company

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1


Chapter P Preparation for Calculus

■ Page 24 Basic types of transformations (c>0)

original graph y= f (x )

#> double f(x) = sqrt(|x-1|)*sin(x);#> plot.x(-2*pi,2*pi) ( f(x) );

Horizontal shift c units to the right y=f (x−c)#> plot.x(-2*pi,2*pi) ( f(x) );#> plot+.x(-2*pi,2*pi) ( f(x) ) .xmove(2);

Horizontal shift c units to the right y=f (x−c)#> plot.x(-2*pi,2*pi) ( f(x) );#> plot+.x(-2*pi,2*pi) ( f(x) ) .xmove(-2);

Vertical shift c units downward y=f ( x )−c#> plot.x(-2*pi,2*pi) ( f(x) );

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#> plot+.x(-2*pi,2*pi) ( f(x) ) .ymove(-2);

Vertical shift c units upward y=f ( x )+c#> plot.x(-2*pi,2*pi) ( f(x) );#> plot+.x(-2*pi,2*pi) ( f(x) ) .ymove(2);

reflection about the x-axis y=−f (x )#> plot.x(-2*pi,2*pi) ( f(x) );#> plot+.x(-2*pi,2*pi) ( f(x) ) .ysym;

reflection about the y-axis y=f (−x )#> plot.x(-2*pi,2*pi) ( f(x) );#> plot+.x(-2*pi,2*pi) ( f(x) ) .xsym;

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reflection about the y-axis y=−f (−x)#> plot.x(-2*pi,2*pi) ( f(x) );#> plot+.x(-2*pi,2*pi) ( f(x) ) .osym;

■ Page 32 Example 2 Fitting a quadratic model

#> t = [ 0, 0.02, 0.04, 0.06, 0.08, 0.099996, 0.119996, 0.139992, 0.159988, 0.179988, 0.199984, 0.219984, 0.23998, 0.25993, 0.27998, 0.299976, 0.319972, 0.339961, 0.359961, 0.379951, 0.399941, 0.419941, 0.439941 ]; t = [ 0 0.02 0.04 0.06 0.08 0.099996 0.119996 0.139992 0.159988 0.179988 0.199984 0.219984 0.23998 0.25993 0.27998 0.299976 0.319972 0.339961 0.359961 0.379951 0.399941 0.419941 0.439941 ]

#> h = [ 5.23594, 5.20353, 5.16031, 5.0991, 5.02707, 4.95146, 4.85062, 4.74979, 4.63096, 4.50132, 4.35728, 4.19523, 4.02958, 3.84593, 3.65507, 3.44981, 3.23375, 3.01048, 2.76921, 2.52074, 2.25786, 1.98058, 1.63488 ]; h = [ 5.23594 5.20353 5.16031 5.0991 5.02707 4.95146 4.85062 4.74979 4.63096 4.50132 4.35728 4.19523 4.02958 3.84593 3.65507 3.44981 3.23375 3.01048 2.76921 2.52074 2.25786 1.98058 1.63488 ]

#> p = .regress(t,h, 2); p = poly( 5.23405 -1.30177 -15.4492 ) = -15.4492x^2 -1.30177x +5.23405

#> p.plot(0,0.49941); Figure 3 : (null), (null), 0 0

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#> p.solve; ans = [ 0.54145 ] [ -0.625711 ]

Chapter 3 Application of Differentiation

■ Page 156 Example 1 Relative extrema

#> double f(x) = 9*(x^2-3)/x^3 ;

#> solve.x {1} ( f'(x) = 0 ); ans = 2.9999999

■ Page 158 Example 4

#> double f(x) = 2*sin(x)-cos(2*x) ;

#> xp = solve.x ( f'(x) = 0 ) .span(0,2*pi); xp = [ 1.5708 0 ] [ 3.66519 -1.16317e-009 ] [ 4.71239 0 ] [ 5.75959 2.87406e-009 ]

#> xp = xp.col(1).tr; xp = [ 1.5708 3.66519 4.71239 5.75959 ]

#> ( xp / pi ).ratio; (1/6) x [ 3 7 9 11 ]

#> f++(xp); // function upgrade by postfix ++ ans = [ 3 -1.5 -1 -1.5 ]

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■ Page 182 Example 3#> double f(x) = x^4-4*x^3;

#> xp = solve.x ( f''(x) = 0 ) .span(0,10); xp = [ 0 2e-010 ] [ 2 0 ]

#> xp = xp.col(1).tr; xp = [ 0 2 ]

#> f++(xp); // function upgrade by postfix ++ ans = [ 0 -16 ]

#> f''(-1); f''(1); f''(3); ans = 36.000003 ans = -12.000001 ans = 35.999995


#> double f(x) = -3*x^5 + 5*x^3;

#> f''(-1); f''(0); f''(1); ans = 30.000002 ans = 0 ans = -30.000002


#> plot.x(-8,8) ( 2*(x^2-9)/(x^2-4), 0 );


#> plot.x(0,20) ( 2*x^(5/3)-5*x^(4/3) , 0 );

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#> plot.x(0,5) ( x^4-12*x^3+48*x^2-64*x , 0 );


#>minfind .x{1} ( sqrt(x^4-3*x^2+4) ); ans = [ 1.22474 ] // minimum point [ 1.32288 ] // f_min

#> sqrt(3/2); ans = 1.2247449


#> minfind .x{1} ( sqrt(x^2+144)+sqrt(x^2-60*x+1684) ); ans = [ 8.99981 ] [ 50 ]

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Chapter 4 Integration

■ Page 261

#> matrix[20] .i ( 20*i ).sum1; // prob 15 ans = 4200

#> matrix[15] .i ( 2*i-3 ).sum1; // prob 16 ans = 195

#> matrix[20] .i ( (i-1)^2 ).sum1; // prob 17 ans = 2470

#> matrix[10] .i ( i^2-1 ).sum1 ; // prob 18 ans = 375

#> matrix[15] .i ( i*(i-1)^2 ).sum1; // prob 19 ans = 12040

#> matrix[10] .i ( i*(i^2+1) ).sum1; // prob 20 ans = 3080

// problem 34 ans = 20#> for(n=1000; n<=10000; n+=1000 ) matrix[n] .i ( (1+2*i/n)^3*(2/n) ).sum1; ans = 20.026008 ans = 20.013002 ans = 20.008668 ans = 20.006501 ans = 20.0052 ans = 20.004334 ans = 20.003714 ans = 20.00325 ans = 20.002889 ans = 20.0026

■ Page 263 #> p =[ 1,0,0 ].poly.sigma ; p = poly( 0 0.166667 0.5 0.333333 ) = 0.333333x^3 +0.5x^2 +0.166667x

#> p.ratio; ans = (1/6) * poly( 0 1 3 2 )

#> p.solve;

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ans = [ 0 ] [ -0.5 ] [ -1 ]

p ( x )=13x (x+ 1

2 )( x+1)

■ Page 273 problem 43.. #> double f(x) {if( -4 <= x && x <= -2 ) return (x+2)/2;else if( -2 <= x && x <= 2 ) return -sqrt(4-x^2);else if( 2 <= x && x <= 4 ) return x-2;else if( 4 <= x && x <= 6 ) return -(x-6);return 0;} #> double g(a,b) = int.x(a,b) ( f(x) );

#> g(0,2); ans = -3.1416212

#> g(2,4) + g(4,6); ans = 4

#> g(-4,-2)+g(-2,2); ans = -7.2833467

#> g(-4,2)+g(-2,2)+g(2,4)+g(4,6); ans = -9.5678161

#> -g(-4,2)-g(-2,2)+g(2,4)+g(4,6); ans = 17.567816

■ Page 276

example 2#>int.x(0,1/2) ( |2*x-1| ) + int.x(1/2,2) ( |2*x-1| ) ans = 2.5

example 3#> int.x(0,2) ( 2*x^2 -3*x +2 ); ans = 3.3333333#> ans.ratio ans = (10/3)

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■ Page 279

example 2#> matrix s(double x) = [ 0, 11.5, -4*x + 341; 11.5, 22, 295; 22, 32, 3/4*x+278.5; 32, 50, 3/2*x+254.5; 50, 80, -3/2*x + 404.5 ]; // first two columns for interval

#> int.x(0,80) ( s.spl(x) ); // evaluation of piecewise functions ans = 24640

// for more exact evaluation, it is necessary to divide intervals

#> int.x(0,11.5) ( s.spl(x) ) + int.x(11.5,22) ( s.spl(x) ) + int.x(22,32) ( s.spl(x) ) + int.x(32,50) ( s.spl(x) ) + int.x(50,80) ( s.spl(x) ); ans = 24640

It is found that the given distribution is smooth enough for integration.

■ Page 280 example 6#> double f(x) = int.t(0,x) ( cos(t) );#> f++( [ 0, pi/6, pi/4, pi/3, pi/2 ] ); ans = [ 0 0.5 0.707107 0.866025

■ Page 297 problem 59#> int.x(1,9) ( 1/(sqrt(x)*(1+sqrt(x))^2) ); ans = 0.5

problem 69#> int.x( pi/2, 2*pi/3 ) ( sec(x/2)^2 ); ans = 1.4641016

■ Page 300 example 1int.x[n+1](a,b) ( f(x) ).poly(kth);.poly(kth) for kth-order polynomial approximationn*kth is the number of subintervals

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#> int.x[5](0,pi) ( sin(x) ).poly(1); ans = 1.8961189#> int.x[9](0,pi) ( sin(x) ).poly(1); ans = 1.9742316

■ Page 302 example 2int.x[n+1](a,b) ( f(x) ).poly(kth);n*kth is the number of subintervals

#> int.x[3](0,pi) ( sin(x) ).poly(2); ans = 2.0045598#> int.x[5](0,pi) ( sin(x) ).poly(2); ans = 2.0002692

Chapter 5 Logarithmic, Exponential and Other Transcendental Functions

■ Page 326 #>int.x(0,pi/4) ( sqrt(1+tan(x)^2) ) ans = 0.88137359

■ Page 375 problem 79dydx

=x

#> plot.x[15](-5,5).y[15](-5,5) ( 1, x ).phase[0.3,1];

problem 79

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dydx

=−xy

#> plot.x[16](-5,5).y[16](-5,5) ( 1, -x/y ).phase[0,10,0];

problem 82dydx

=0.25 x(4− y )

#> plot.x[16](-5,5).y[16](-5,5) ( 1, -x/y ).phase[0,10,0];

Chapter 6 Applications of Integration

■ Page 437 example 2#> double f(x) = x^3/6 + 1/(2*x);#> int.x(1/2,2) ( sqrt( 1+ f'(x)^2 ) );ans = 2.0625

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#> ans.ratio; ans = (33/16)

■ Page 438 example 5#> double f(x) = log(cos(x));#> int.x(0,pi/4) ( sqrt( 1+ f'(x)^2 ) ); ans = 0.88137359

■ Page 441 example 6#> double f(x) = x^3;#> int.x(0,1) ( f(x) * sqrt( 1+ f'(x)^2 ) ) * (2*pi) ; ans = 3.5631219

■ Page 459 6-6 example 4.hold; plot .line(-2,0, 2,0); plot .x(2,-2) ( 4-x*x ) .link .int1da(area) .intyda(ym); plot .line(0,-1, 0,4); plot;area / (32/3); ym;ym/area;

ans = 0.99960011 ym = 17.055292 ans = 1.5995733#> 256/15; 8/5; ans = 17.066667 ans = 1.6

■ Page 460 6-6 example 5.hold;

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plot .x(-2,1) ( x+2 ); plot .x(1,-2) ( 4-x*x ) .link .int1da(area).intxda(xm).intyda(ym); plot .line(0,-1, 0,4); plot;xm/area;ym/area; ans = -0.49999979 ans = 2.3997597

Chapter 7 Integration Technique, L'Hopital's Rule, and Improper Integrals

■ Page 491 example 2#> int.x(pi/6,pi/3) ( cos(x)^3/sqrt(sin(x)) ) ans = 0.23852538

Chapter 8 Infinite Series

■ Page 583 example 4#> matrix[100].n ( (n+1).pm * 1/n.fact ).sum1 ans = 0.63212056

■ Page 621 problem 54#> 1 + matrix[100] .n ( n.pm/3^n/(2*n+1) ).sum1 ; pi/2/sqrt(3) ; ans = 0.90689968 ans = 0.90689968

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Chapter 9 Conics, Parametric Equations, and Polar Coordinates

■ Page 661 selected project

void epicycloid(A,B) { plot.@t[1001](0,24*pi) ( // @ for parametrized plot (A+B)*cos(t)-B*cos((A+B)/B*t), (A+B)*sin(t)-B*sin((A+B)/B*t) );;}

epicycloid(8,3);

epicycloid(24,3);

epicycloid(24,7);

■ Page 666 example 5double xt(t) = t/2000/pi*cos(t);double yt(t) = t/2000/pi*sin(t);

s = int.t(1000*pi,4000*pi) ( sqrt( (xt'(t))^2 + (yt'(t))^2 ) );s*_in/_ft;

s = 11764.509 ans = 980.37576

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■ Page 677 rose curve#> plot.t[1000](0,2*pi) ( 2*sin(5*t) ).cyl;

■ Page 680 butterfly curve#> plot.t[1001](0,10*pi) ( exp(cos(t))-2*cos(4*t)+(sin(t/12))^5 ).polar;

Chapter 10 Vectors and the Geometry of Space

■ Page 730 cross product#> u = < 1, -2, 1 >; v = < 3, 1, -2 >; u = < 1 -2 1 > v = < 3 1 -2 >

#> u^v ; ans = < 3 5 7 >

#> v^u ; ans = < -3 -5 -7 >

#> v^v ;

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ans = < 0 0 0 >

■ Page 734 triple scalar product#> u = < 3,-5,1 >; v = < 0,2,-2 >; w = < 3,1,1 >; u = < 3 -5 1 > v = < 0 2 -2 > w = < 3 1 1 >

#> u*(v^w); ans = 36

■ Page 740 intersection of two planes

#> n1 = < 1,-2,1 >; n2 = < 2,3,-2 >; n1 = < 1 -2 1 > n2 = < 2 3 -2 >

#> |n1*n2|/(|n1|*|n2|); ans = 0.59408853

■ Page 757 Klein bottle

plot .u(0,2*pi).v(0,2*pi) ( << a = 2-cos(u), b = a*cos(v), xu = 3*cos(u)*(1+sin(u)), yu = -8*sin(u) >>, u < pi ? xu + b*cos(u) : xu-b, u < pi ? yu - b*sin(u) : yu, -a*sin(v) );

■ Page 764 revolutioned body

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#> plot.x(0,3) ( x ).yrev(0,2*pi);

Chapter 11 Vector-Valued Functions

■ Page 771 example 4#> plot.@t(-2,2) ( t,t^2,sqrt((24-2*t^2-t^4)/6) );

■ Page 774 problem 21#> plot.@t[1000](0,6*pi) ( 4*cos(t), 4*sin(t), t/4 );

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■ Page 803 example 1#> plot.@t[1000](0,2) ( t, 4/3*t^(3/2), 1/2*t^2 ).intdl(1)(arclen);#> arclen; arclen = 4.8157022

Chapter 12 Functions of Several Variables

■ Page 824 revolution

#> plot.x(0,2) ( x ) .yrev(0,2*pi);#> plot+.x(0,1) ( x^2+1 ).yrev(0,2*pi);#> plot+.x(1,2.05) ( sqrt(x-1)*1.9 ).yrev(0,2*pi);

■ Page 825 3d plot#> plot.x(-3,3).y(-3,3) ( (x^2+y^2)*exp(1-x^2-y^2) );

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■ Page 826 3d plot#> plot.x(-2,2).y(-2,2) ( (1-x^2-y^2)/sqrt(|1-x^2-y^2|) );

■ Page 872.del ( y-sin(x) ) ( -pi,0,0 );.del ( y-sin(x) ) ( -2*pi/3,-sqrt(3)/2,0 );.del ( y-sin(x) ) ( -pi/2,-1, 0 ); ans = < 1 1 0 > ans = < 0.5 1 0 > ans = < -7.854e-006 1 0 >

Chapter 13 Multiple Integration

■ Page 916 example 1

#> int.x(1,2).y(1,x) ( 2*x^2*y^-2 + 2*y ) ; ans = 3

■ Page 920 example 6

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#> int.x(1,2).y(-3*x+6,4*x-x^2) ( 1 ) + int.x(2,4).y(0,4*x-x^2) ( 1 ); ans = 7.5

■ Page 928 example 3#> int.x(-2,2).y(-sqrt(|4-x^2|/2),sqrt(|4-x^2|/2)) (4-x^2-2*y^2 );

ans = 17.771505#> 4*sqrt(2)*pi;

ans = 17.771532

■ Page 936 example 2#> int.t(0,2*pi).r(1,sqrt(5)) ( (r^2*cos(t)^2+r*sin(t))*r ); ans = 18.849556#> 6*pi; ans = 18.849556

■ Page 958 example 1#> int.x(0,2).y(0,x).z(0,x+y) ( exp(x)*(y+2*z) ); ans = 65.797355

#> 19*(_e^2/3+1); ans = 65.797355

■ Page 971 example 5#> int.rho(0,3).t(0,2*pi).phi(0,pi/4) ( (rho*cos(phi))*rho^2*sin(phi) ); ans = 31.808626

#> 81*pi/8; ans = 31.808626

■ Page 973 section project%> wrinkled sphere in spherical coordinate (R,P,T)#> plot .P[21](0,pi) .T[51](0,2*pi) ( 1+0.2*sin(8*T)*sin(P),P,T ).sph;

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%> bumpy sphere #> plot .P[41](0,pi).T[51](0,2*pi) ( 1+0.2*sin(8*T)*sin(4*P),P,T ).sph;

Chapter 14 Vector Analysis

■ Page 984 Umbilic Torus NC#> plot .u(-pi,pi) .v(-pi,pi) ( << a = u/3-2*v, b = u/3+v >>, sin(u)*(7+cos(a)+2*cos(b)), cos(u)*(7+cos(a)+2*cos(b)), sin(a)+2*sin(b) );

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■ Page 1037 open-ended project#> plot .u(-pi,pi) .v(-pi,pi) ( << a = 3*u-2*v, b = 3*u+v >>, 3+sin(u)*(7-cos(a)-2*cos(b)), 3+cos(u)*(7-cos(a)-2*cos(b)), sin(a)+2*sin(b) );

■ Page 1049 Mobius strip#> plot .u(0,pi) .v(-1,1) ( << a = 4-v*sin(u) >>, a*cos(2*u), a*sin(2*u), v*cos(u) );

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Documents

Web viewHighSchool-Calculus (Larson et al.) We selected the textbook "Calculus with Analytic Geometry 6th edition" written by . Roland E. Larson . Robert P. Hostetler