Tangent Plane

1
Tangent Plane

• Assume that the function is
  differentiable at and let
  denote the corresponding point on the
  graph of Let T denote the graph of the local
  linear approximation
• to at Then a line is tangent at
  to a curve C on the surface if
  and only if the line is contained in

2
Definition of Tangent Plane

• If is differentiable at the point
  then the tangent plane to the surface
  at the point or
  is the plane
• The normal line to the surface is the line
  through the point parallel to the vector
  is perpendicular to the tangent plane. It may be
  represented by the parametric equations

3
Tangent Plane to Level Surface

• Assume that has continuous
  first-order partial derivatives and let
  If then
  is a normal vector to the surface
  at the point and
  the tangent plane to this surface at is the
  plane with equation

4
Tangent Lines to Intersecting Surfaces

• In general, the intersection of two
  surfaces and
  will be a curve in 3-space.
• If is a point on this surface,
  then will be normal
  to the surface at
  and will be normal to
  the surface at
  Thus, if the curve of intersection can be
  smoothly parametrized, then its unit tangent
  vector at will be
  orthogonal to both and

5
Maxima and Minima of Functions of Two Variables

• A function of two variables is said to be a
  relative maximum (relative minimum) at a point
  if there is disk centered at
  such that
  for all points that lie
  inside the disk, and is said to have an
  absolute maximum (absolute minimum) at
  if
  
  for all points in the domain of

6
Open and Closed Sets

• If D is a set of points in 2-space, then a point
  is called an interior point of D if
  there is some circular disk with positive radius,
  centered at and containing only
  points in D.
• A point is called a boundary point of D
  if every circular disk with positive radius and
  centered at contains both points of D
  and points not in D.
• A set D is called open if it contains none of its
  boundary points, and closed if it contains all of
  its boundary points.

7
Extreme Value Theorem

• If is continuous on a closed and
  bounded set R, then has both an absolute
  maximum and an absolute minimum on R.
• Note If any of the conditions fail to hold, then
  there is no guarantee that an absolute maximum or
  absolute minimum exists on the region R.

8
Finding Relative Extrema

• If has a relative extremum at a point
  and if the first-order partial derivative of
  exist at this point, then
• A point in the domain of a function
  is called a critical point of the function
  if and or
  if one or both partial derivatives do not exist
  at

and
9
Saddle Point

• A surface has a saddle point at
  if there are two distinct vertical planes
  through this point such that the trace of the
  surface in one of the planes has a relative
  maximum at and the trace in the other
  has a relative minimum at

10
The Second Partials Test

• Let be a function of two variables with
  continuous second-order partial derivatives in
  some disk centered at a critical point and
  let
• a) If and then
  has a relative minimum at
• b) If and then
  has a relative maximum at
• c) If then has a saddle point.
• d) If then no conclusion can be drawn.

11
Finding Extrema

• If a function of two variables has an
  absolute extrema (either an absolute maximum or
  an absolute minimum) at an interior point of its
  domain, then this extremum occurs at a critical
  point.

12
Finding Absolute Extrema on Closed and Bounded
Sets

• 1) Find the critical points of that lie in
  the interior of the closed and bounded set R.
• 2) Find all boundary points at which the absolute
  extrema can occur.
• 3) Evaluate at the points obtained in
  the preceding steps. The largest of these values
  is the absolute maximum and the smallest the
  absolute minimum.

13
Lagrange Multipliers Used for Min(Max) With
Constraints

• Three-Variable Extremum Problem with One
  Constraint
• Maximize or minimize the function
  subject to the constraint
• Two-Variable Extremum Problem with One Constraint
• Maximize or minimize the function
  subject to the constraint

14
Constrained-Extremum Two Variables and One
Constraint

• Let and be function of two variables with
  continuous first partial derivatives on some open
  set containing the constraint curve
  and assume that at any point on
  this curve. If has a constrained relative
  extremum, then this extremum occurs at a point
  on the constraint curve at which the
  gradient vector and
  are parallel that is, there is some number
  such that

15
Constrained-Extremum Three Variables and One
Constraint

• Let and be function of two variables with
  continuous first partial derivatives on some open
  set containing the constraint surface
  and assume that at any point on
  this surface. If has a constrained relative
  extremum, then this extremum occurs at a point
  on the constraint curve at which the
  gradient vectors and
  are parallel that is, there is some
  number such that