Chapter 15
MATH 2433 - 13897
Annalisa Quaini
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Definition
Absolute maximum and local minimum
Let f be a function of several variables with domain D:
f is said to take on an absolute maximum at x0 if
f (x0 ) ≥ f (x)
for all x ∈ D;
f is said to take on an absolute minimum at x0 if
f (x0 ) ≤ f (x)
for all x ∈ D.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Theorem
If f is continuous on a bounded closed set D, then f takes on an
absolute maximum value and an absolute minimum value.
How to find absolute max and min
1 Find the critical points in the interior of D.
2
Find the extreme points on the boundary of D.
3
Evaluate f at the points found in Step 1 and 2.
4
The largest number found in Step 3 is the absolute maximum
value of f and the smallest is the absolute minimum.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
How to find maxima and minima with side conditions
Suppose g is a continuously differentiable function of two or three
variables defined on a subset of the domain of f . If x0 maximizes
(or minimizes) f (x) subject to the side condition g (x) = 0, then
∇f (x0 ) and ∇g (x0 ) are parallel. Thus, if ∇g (x0 ) 6= 0, then there
exists a scalar λ such that
∇f (x0 ) = λ∇g (x0 ).
λ is called Lagrange multiplier.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Example
Maximize f (x, y ) = xy on the ellipse 4x 2 + 9y 2 = 36.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
(continue)
Maximize f (x, y ) = xy on the ellipse 4x 2 + 9y 2 = 36.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Example
Minimize f (x, y , z) = x + 2y + 4z on the sphere x 2 + y 2 + z 2 = 7.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
(continue)
Minimize f (x, y , z) = x + 2y + 4z on the sphere x 2 + y 2 + z 2 = 7.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Example
A rectangular box has three of its faces on the coordinate planes
and one vertex in the first octant on the paraboloid
z = 4 − x 2 − y 2 . Determine the maximum volume of the box.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
(continue)
A rectangular box has three of its faces on the coordinate planes
and one vertex in the first octant on the paraboloid
z = 4 − x 2 − y 2 . Determine the maximum volume of the box.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Differentials
Section 15.8
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
What is the definition of increment and differential in 1D?
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
What is the definition of increment and differential in 1D?
Increment
∆f (x) = f (x + h) − f (x)
Differential
df (x) = ∇f (x) · h
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Geometric representation in 2D
As in 1D: ∆f ∼ df for small h.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Example
Find the differential of
f (x, y , z) = xy + yz + xz.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Example
Find the differential of
f (x, y ) = sin(x + y ) + sin(x − y ).
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Example
Calculate ∆u and du for u(x, y ) = x 2 − 3xy + 2y 2 at x = 2,
y = −3, ∆x = −0.3, ∆y = 0.2.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Example
Use differentials to approximate
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√
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Reconstructing a Function from
its Gradient
Section 15.9
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Given a function f (x, y ), we know how to find the gradient
∇f (x, y ) = fx (x, y )i + fy (x, y )j.
So, gradient are vectors in the form
P(x, y )i + Q(x, y )j
with a condition on P(x, y ) and Q(x, y ): which?
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Example
Show that f (x, y ) = y i − xj is not a gradient.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Example
Show that f (x, y ) = (y 3 + x)i + (x 2 + y )j is not a gradient.
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Now, given a vector function P(x, y )i + Q(x, y )j that can represent
the gradient of a function f (x), we want to reconstruct f (x).
Here’s the procedure:
1
set P(x, y ) = fx (x, y ) and Q(x, y ) = fy (x, y );
2
integrate fx (x, y ) with respect to x to get f (x, y )
(REMEMBER that y is a constant!);
3
derive f (x, y ) found at Step 2 with respect to y and compare
what you get with Q(x, y ).
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Example
Given
∇f (x, y ) = (e x + 2xy )i + (x 2 + sin y )j,
find f (x, y ).
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Chapter 15
Sect. 15.6 Sect. 15.7 Sect. 15.8 Sect. 15.9
Example
Given
∇f (x, y ) = (y 2 e x − y )i + (2ye x − x)j,
find f (x, y ).
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