Question (A)

The surface z=f(x,y)=5/(1+x^2+y^2) is a hill. A bug walks along the path with x(t)=2+3cos(t) and y(t)=-1+2sin(t)

and z(t) is on the surface (0≤t≤2pi).

Plot the surface z for -4≤x≤7, -4≤y≤4 and includethe path (in thick black) of the bug on the surface.

(this will require 2 graphing commands plus a display 3d. Think spacecurve for the bug's path.)

Animate a solid...

SURF is the surface z=2x^2+y^2 for x=0 .. 2, y=0 ..4 and view= -1 ..16.

L is a straight line laser beam x=7-2t, y=t, z=4t-1 which hits SURF at the point P=(1, 3, 11).

Graph SURF, the laser path in red (t=0 to 3), the normal vector at P in black, and the reflected path in green.

Include the equations of the normal vector and the reflected path.

The surface z=f(x,y)=5/(1+x^2+y^2) is a hill. A bug walks along the path with x(t)=2+3cos(t) and y(t)=-1+2sin(t)

and z(t) is on the surface (0≤t≤2pi).

Plot the surface z for -4≤x≤7, -4≤y≤4 and includethe path (in thick black) of the bug on the surface.

(this will require 2 graphing commands plus a display 3d. Think spacecurve for the bug's path.)

plot part of the surface f(x,y)=10-x^2-3y^2:

plot3d(10-x^2-3y^2, x=-1 .. 4, y=1 .. 2, view=0 .. 11, axes= normal); (press enter)

Determine the normal vector and the equation of the tangent plane at the point (2,1,3) on the surface.

Display the surface , a black vector and the tangent plane, and give the equation for each of them

label the normal vector and the tangent plane, and give the equation for each of them

In 2D, graph a blue ellipse x(t)=3cos(t), y(t)=2sin(t) for 0≤t≤2∏. For t=0.5 graph a green tangent line to the ellipse and a red osculating circle. Also, give the curvature, the equation of the tangent line and the center of the osculating circle