Web viewHighSchool-Calculus (Larson et al.) We selected the textbook "Calculus with Analytic...
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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
[100] 060 revised on 2012.10.14 cemmath
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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
Page numbers and examples are all refered from the above textbook and therefore readers can compare the original contents with the present descriptions.
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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
#> 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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
#> 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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
■ 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
■ Page 183 Example 4
#> double f(x) = -3*x^5 + 5*x^3;
#> f''(-1); f''(0); f''(1); ans = 30.000002 ans = 0 ans = -30.000002
■ Page 197 Example 1
#> plot.x(-8,8) ( 2*(x^2-9)/(x^2-4), 0 );
■ Page 199 Example 4
#> plot.x(0,20) ( 2*x^(5/3)-5*x^(4/3) , 0 );
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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
■ Page 200 Example 5
#> plot.x(0,5) ( x^4-12*x^3+48*x^2-64*x , 0 );
■ Page 207 Example 2
#>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
■ Page 208 Example 4
#> minfind .x{1} ( sqrt(x^2+144)+sqrt(x^2-60*x+1684) ); ans = [ 8.99981 ] [ 50 ]
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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
■ 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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
#> 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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
#> 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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
■ 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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
#> 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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
■ 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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
■ 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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
#> 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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
%> 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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[100] 061 HighSchool-Calculus (Larson et al) www.msharpmath.com
■ 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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