3811.sagews

html("<b>Problem #1:</b> Find equation of the line normal to the surface that contains points (0,0,0), (-1,2,3), (1,1,2).")

P1fig1=point([(0,0,0),(-1,2,3),(1,1,2)],size=15,color='red'); P1fig1_labels= text3d('(0,0,0)',(0,0,0.25),fontsize=15,alignment='left') + text3d('(-1,2,3)',(-1,2.2,2.75),fontsize=15,alignment='left') + text3d('(1,1,2)',(1,1.25,1.75),fontsize=15,alignment='left');(P1fig1 + P1fig1_labels).show(aspect_ratio=[1,1,1])

Problem #1: Find equation of the line normal to the surface that contains points (0,0,0), (-1,2,3), (1,1,2).

P1fig1_vecs = arrow3d((0,0,0),(-1,2,3),2,color='red') + arrow3d((0,0,0),(1,1,2),2,color='green');(P1fig1+ P1fig1_labels +P1fig1_vecs).show(aspect_ratio=[1,1,1]);

V=VectorSpace(RR,3);v1=V([-1,2,3]);v2=V([1,1,2]);

v3=v1.cross_product(v2);v3

P1fig1_normvec=arrow3d((0,0,0),v3,2); P1fig1_labels += text3d('n = <1,5,-3>',.5*v3+V([.5,.5,.5]),fontsize=15,alignment='left'); (P1fig1 + P1fig1_labels + P1fig1_vecs + P1fig1_normvec).show(aspect_ratio=[1,1,1])

x,y,z = var('x y z');
P1PlaneEq=1/3*(x+5*y);
P1fig1_Plane= plot3d(P1PlaneEq,(x,-5,5),(y,-5,5),opacity=0.5);
(P1fig1 + P1fig1_vecs + P1fig1_normvec + P1fig1_Plane).show(aspect_ratio=[1,1,1])

html("<b>Problem #2:</b> Find angle betwing the tangent plane to the sphere\n <center>x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = 6</center> \nat the point (1,2,1) and the x-z plane.")


x,y,z=var('x y z')

P2fig1_sphere = sphere(size=sqrt(6), opacity=0.7); #Sphere of radius 2sqrt(3) centered at the origin. Colored in BLUE
P2fig1_point = point((1,2,1),color='red',size=15); #Point (1,2,1) lying on the sphere. Colored in RED
P2fig1_labels= text3d('(1,2,1)',(1.1,2.2,1.1),size=15,alignment='left'); #Labels for the point
P2fig1_xzplane = implicit_plot3d(y==0,(x,-5,5),(y,-5,5),(z,-5,5),color='green',opacity=0.7); # xz-plane (y=0). Colored in GREEN
P2fig1_tangentplane = implicit_plot3d((x-1)+2*(y-2)+(z-1)==0,(x,-5,5),(y,-5,5),(z,-5,5),color='orange',opacity=0.5);
P2fig1_nvec = arrow3d((1,2,1),(2,4,2),3,color='red')

(P2fig1_sphere+P2fig1_point+P2fig1_labels + P2fig1_xzplane + P2fig1_tangentplane + P2fig1_nvec).show(aspect_ratio=[1,1,1])

Problem #2: Find angle betwing the tangent plane to the sphere x² + y² + z² = 6 at the point (1,2,1) and the x-z plane.

html("<b>Problem #3:</b> Approximate\n <center>e<sup>1.01<sup>2</sup> + 1.97</sup></center> \nusing total differentials.")

Problem #3: Approximate e^{1.012 + 1.97} using total differentials.

P3fig1=contour_plot(exp(x^2+y),(x,1,2),(y,1,2),cmap='rainbow',fill=False,labels=True,colorbar=True,contours=20); P3fig1_point=point((1.01,1.97),color='red',size=20);
(P3fig1_point+P3fig1).show(axes_labels=['$x$','$y$'])

html("The value of e<sup>1.01<sup>2</sup> + 1.97</sup> = ")

exp(1.01^2+1.97)

The value of e^{1.012 + 1.97} =

19.8876711592456

html("<b>Problem #4:</b>")

contour_plot(x^2*y/(x^4+y^2),(x,-2,2),(y,-2,2),cmap='rainbow',colorbar=True,labels=True,plot_points=1000,fill=False,contours=20)

Problem #4:

html("<b>Problem #6:</b> \nThe region is inside the cylinder and bounded above and below by the sphere.")

P6_cylinder = implicit_plot3d(x^2+y^2==1,(x,-2,2),(y,-2,2),(z,-5,5),opacity=0.55)
P6_sphere = sphere(size=4,color='red',opacity=0.5)

(P6_cylinder+P6_sphere).show(aspect_ratio=[1,1,1])

Problem #6: The region is inside the cylinder and bounded above and below by the sphere.