Unformatted text preview: or a maximum For a minimum Where is either the local maximum or minimum value Definition (Global Extrema): f has an absolute or global extrema
For a maximum For a minimum This is called the extreme maximum or minimum
Theorem: If f has a local extreme at and the first order partial derivatives exists, then Definition (Critical/Stationary Points): The critical points are the points Or the derivatives do not exist
Theorem: If is a local extreme point, then it is also a critical point 27 such that The Second Derivative Test
Given that is a critical point of the function
The second order partial derivatives of the function are continuous on a disk centered at Let We can arrive at some conclusions based on the value of D
If , two cases arise When
When , then
, then If , then If is a local maximum
is a local minimum is a saddle point , the test is inconclusive To find the local extremes and saddle points:
1) Find the critical points of the function
2) Use the second derivative to decide which of these points local extrema and/or saddle points
Maximum and Minimum Values
Recall in single variable calculus, for some closed interval
and the function ,
f ,is continuous over this interval then f has a global maximum and a global minimum on
Now for multivariable calculus, for some closed set (a set which contains its bounding points)
, we arrive at the Extreme Value Theorem
Theorem (Extreme Value Theorem):
is closed and bound, and
continuous then f attains an absolute (global) maximum and minimum (
respectively) at some points
and
in D is
and To Find Absolute Maximum and Minimum
1) Find all the critical points (that are not on the boundary of D) and find the values for the
function at these points 28 2) Find the extreme points on the boundary of D
3)
The largest values found for 1) and 2) are the global maximum
The smallest values found for 1) and 2) are the global minimum 29 14.8 Lagrange Multipliers
Goal: Find the extreme points of a function subject to a constant. More generally:
Goal: find the extreme values of subject to the constant So, how do we find them?
 Suppose
Take any curve has an extreme value at
that is on the surface and passes through the point Evaluate along:
 Since the function has an extreme value at then, Using the chain rule, we can evaluate at
Using the gradient vector: This shows that these two vectors are orthogonal and that the gradient is also orthogonal
to the tangent plane at the point to the surface
Thus
λ is defined as the lagrange multiplier
Method of Legrange Multiplier (Find all the maximum and minimum points subject to the
constraint
)
Step 1: Find all such that Step 2: Evaluate all the points from the first step and the largest value yielded is the
maximum and the smallest is the minimum
Two constraints:
Goal: Find the extreme values of subject to the constraints We recall that the gradient vector is orthogonal to the curve
30 and This means that the gradient vectors of the three curves must lie in the same plane Method:
Step 1: Solve Step 2: Evaluate at each point 31 Chapter 15: Multiple Integrals 32
15.1 Double Integrals Over Rectangles
Recall:
Area and simple integrals: Now:
Volumes and double integrals: Approximate the volume under the surface that lies above some subrectangle.
Take the contributions of all the subrectangles: Definition (Double Integral): The double integral of f over is defin...
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 Spring '08
 Dr.AnthonyDixon
 Calculus, Derivative, Gradient

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