Exercises

Find the equation of the tangent plane to the given surface at the given point.

1. x2+y2+z2 = 11 at ( 1,1,3) 2. x2+y2+z2 = 9 at ( 2,1,2) 3. xy+z2 = 4 at ( 1,2,2) 4. x2y+z2 = 4 at ( 1,2,2) 5. 3x+4y+2z = 13 at ( 1,2,1) 6. 3x-2y+4z = -4 at ( 2,1,-2) 7. x2+y2-z2 = 1 at ( 1,1,1) 8. x2-y2-z2 = 2 at ( 2,1,1) 9. xey+z = 2 at ( 1,0,1) 10. sin(xy) +z = 2 at ( p,1,2)

Find the equation of the tangent plane to r(u,v)  at the point r( p,q) for the given ( p,q). 

11. r = á vsin( u), vcos( u), v ñ   12. r = á vcos( u), vsin( u), v ñ   ( p,q) = ( p/4, 2)     ( p,q) = ( p/2, 1) 13. r = á cos( u), sin( u), v ñ   14. r = á cos( u), sin( u), v ñ   ( p,q) = ( p/4, 3)     ( p,q) = ( p/2, 1) 15. r = á vsin( u), vcos( u), uv ñ 16. r = á vsin( u), v2, vcos( u) ñ ( p,q) = ( p/3, 1) ( p,q) = ( p/4, 1) 17. r = á sin( v)sin( u), cos( v) sin( u), cos(u) ñ 18. r = á sin( v)sin( u), cos( v) sin( u), cos(u) ñ ( p,q) = ( p/4, p/4) ( p,q) = ( p/3, p/6) 19. r = á evsin( u), evcos(u), e-v ñ 20. r = á sin( u) cosh( v), sin( u) sinh(v), cos( u) ñ ( p,q) = ( p, 1) ( p,q) = ( p, ln2)

Find the tangent plane to the graph of f( x,y)  at ( p,q,f( p,q) )  by (a) using methods from section 2-5, (b) considering the surface z-f( x,y) = 0, and (c) finding the tangent plane to a parameterization of the form

r( u,v) = á u,v,f( u,v) ñ

The answer should be the same in all 3 cases.  

21. f( x,y) = x2-y,  ( p,q) = ( 2,1) 22. f( x,y) = x2+y2,  ( p,q) = (1,1) 23. f( x,y) = 3x+2y+1,  ( p,q) = ( 0,0) 24. f( x,y) = xy, ( p,q) = ( 1,1) 25. f( x,y) = xexy,  ( p,q) = ( 2,0) 26. f( x,y) = ln( x2+y2) ,  (p,q) = ( 1,1)

27. Is the surface r( u,v) = ávsin( u), vcos( u) ,v ñ , u in [ 0,2p] , v in [ 1,2] , orientable?  Explain.

28.  Consider the surface parameterized by

r( u,v) = vcos æ è    u 2 ö ø ,  sin( u),  vsin æ è    u 2 ö ø

for u in [ 0,2p] and v in [ -0.3,0.3] . Find r_u and r_v and then calculate

ru( 0,v) ,  ru( 2p,0) ,  rv( 0,v) , and  rv( 2p,0)

Use these to calculate

n( 0,v) =  ru( 0,v) ×rv( 0,v) || ru(0,v) ×rv( 0,v) ||   and  n( 2p,v) =  ru( 2p,v) ×rv( 2p,v) || ru( 2p,v) ×rv( 2p,v) ||

Is n( 0,v) = n( 2p,v)?  Is the surface orientable?

29.  Find the equation of the tangent plane to the cone x²+y² = z² at ( 1,0,1) using (a) a level surface and (b) the parameterization

r( u,v) = á vcos( u) ,vsin( u) ,v ñ

at r( 0,1) . The answer should be the same in both cases.

30.  Find the equation of the tangent plane to

r( u,v) = á uev,ue-v,u ñ

at r( 1,0) . Then find the equation of the tangent plane again using a level surface representation of r(u,v) . The answer should be the same in both cases.

31. Show that the equation of the tangent plane to an elliptic paraboloid

z c =  x2 a2 +  y2 b2

at a point ( m,n,p) on the elliptic paraboloid is of the form

z+p 2c =  mx a2 +  ny b2

 

32. Find the equation of the tangent plane to x²+y² = z² at the point ( m,n,p) , and then show that it must pass through the line mx+ny = 0 in the xy-plane. 

33. If ( m,n,p) is a point on a sphere of radius R centered at the origin, what is the equation of the tangent plane to the sphere at ( m,n,p) ?

34. Show that the normal vector to a sphere of radius R at a point ( m,n,p) is parallel to the radius vector ám,n,p ñ .  What does this say about the relationship between the radius vector á m,n,p ñ and any vector tangent to the sphere at ( m,n,p) ?

35. Show that any tangent plane to z = x²-y² intersects the surface in two perpendicular lines.

36. Show that any tangent plane to x²+y²-z² = 1 intersects the surface in two lines.

37. Determine the longitude q₀ and latitude j₀ of your present location. Assume that earth is a sphere with latitude-longitude parameterization of

r( j,q) = á 3960cos(j) cos( q) ,3960cos( j)sin( q) ,3960sin( j) ñ

What is the equation of the tangent plane to the earth at your location? (note: this problem assumes an xyz-coordinate system at the center of the earth with z-axis through the poles and x-axis at 0^° longitude).

38. A parabolic mirror is in the shape of a paraboloid with equation

4pz = x2+y2

where p > 0 is a number. Show that a vertical line through a point (a,b,c) on the paraboloid forms the same angle with the tangent plane at (a,b,c) as does the line through ( a,b,c) and (0,0,p) . (i.e., that a vertical ray of light is reflected by the parabolic mirror to the focus at ( 0,0,p) ).

Show that A = B (click and drag on center of plane to change reflection point)

39. Explain why if n denotes the surface normal of a surface r( u,v) , then n( u,v) is a parameterization of a section of the unit sphere. What section of the unit sphere is parameterized by the surface normal n to the right circular cylinder

r( u,v) = á cos( u) ,sin(u) ,v ñ

40. What section of the unit sphere is parameterized by the surface normal n to the surface

r( u,v) = á vcos( u) ,vsin( u) ,cosh( v) ñ

for v ³ 0 and u in [ 0,2p] . (see exercise 39).

41. Write to Learn: Write a short essay which explains why the tangent plane to a plane is the plane itself. In particular, demonstrate using a parametric representation of a plane

r( u,v) = áa+mu+nv,b+pu+qv,c+su+tv ñ

for constants a,b,c,m,n,p,q,s, and t, demonstrate using a level surface representation

a( x-a) +b( y-b) +g( z-c) = 0

for a, b, and g constant, and also demonstrate using a functional form by solving for z when g ¹ 0.

42. Write to Learn: Suppose that two smooth surfaces F(x,y,z) = k and G( x,y,z) = l both contain the point ( x₀,y₀,z₀) and that ÑF(x₀,y₀,z₀) = cÑG( x₀,y₀,z₀) for some number c. Write a short essay which explains why the two surfaces are tangent at ( x₀,y₀,z₀) .