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)

The following are surfaces in cylindrical coordinates. Find the equation of the tangent plane at the given point. The point is also in cylindrical coordinates and thus must be transformed using x = rcos(q) , y = rsin( q) .  

21. Cylinder: r = 2 at ( 2,p,3) 22. Cylinder: r = 5 at ( 5,0,0) 23. Cone: r = z at æ è 1, p 6 ,1 ö ø 24. Cone: r = z at ( 2,0,2) 25. z = r2 at ( 2,0,4) 26. z = r2 at (2,0,4) 27. r2+z2 = 25 at ( 3,0,4) 28. r2+z2 = 25at ( 3,0,4)

29. Use the chain rule to show that if U( r,q,z) = k is a level surface in cylindrical coordinates, then

Uy = sin( q) Ur+ 1 r Uqcos(q)

30. A level surface in spherical coordinates is of the form U( r, f, q) = k, where

x = rsin( f) cos( q) ,  y = rsin( f) sin( q) ,  and z = rcos(f)

Use the chain rule to express U_r in terms of U_x, U_y, and U_z.

31. Find the equation of the tangent plane to z = f(x,y) at a point ( p,q,f( p,q) ) in two ways:

1. By letting U( x,y,z) = z-f( x,y) and using the gradient.
2. By computing r_u( p,q) ×r_v( p,q) for r( u,v) = áu,v,f( u,v) ñ and using it as the normal vector to the tangent plane.

32. Find the equation of the tangent plane to the cone x²+y² = z² at the point (m,n,p) in two ways:

1. By letting U( x,y,z) = x²+y²-z² and using the gradient.
2. By explaining geometrically why every tangent plane to the cone must pass through the line mx+ny=0, and then using the line and the point to find the equation of the plane..

33. Explain why r = 1 is the equation of a right circular cylinder in cylindrical coordinates, and then find the unit surface normal using

ÑU = Ur er+ 1 r Uq eq+Uz ez

Sketch n, e_r, and e_q at the point ( 1, 0, 1) on the cylinder.

34. Show that equation of the right circular cone in cylindrical coordinates is r-z = 0, and then find the unit surface normal using

ÑU = Ur er+ 1 r Uq eq+Uz ez

Sketch n, e_r, and e_q at the point ( 1, p/2, 1) on the right circular cone.

35. If ( x₀,y₀,z₀) 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 ( x₀,y₀,z₀) ?

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

z c =  x2 a2 +  y2 b2

at a point ( x₀,y₀,z₀) on the elliptic paraboloid is of the form

z+z0 c =  2xx0 a2 +  2yy0 b2

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

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

39. 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).

40. 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)

41. 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

42. 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 41).

43. 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.

44. 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) .