Multivariable Calculus

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Multivariable Calculus
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Chapter 17
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FUNCTION OF TWO OR MORE VARIABLES The expression
z f ( x, y ) is a function of two variables if
a unique value of z is obtained from each ordered
pair of real numbers ( x, y ). The variables x
and y are independent variables, and z is the
dependent variable. The set of all ordered pairs
of real numbers ( x, y ) such that f ( x, y )
exists is the domain of f the set of all values
of f ( x, y ) is the range. Similar definitions
could be given for functions of three, four , or
more independent variables.
PARTIAL DERIVATIVES (INFORMAL DEFINITION) The
partial derivative of f with respect to x is
the derivative of f obtained by treating x as a
variable and y as a constant. The partial
derivative of f with respect to y is the
derivative of f obtained by treating y as a
variable and x as a constant.
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PARTIAL DERIVATIVES (FORMAL DEFINITION) Let z
f ( x, y ) be a function of two independent
variables. Let all indicated limits exist. Then
the partial of f with respect to x is
And the partial derivative of f with respect to
y is
If the indicated limits do not exist, then the
partials derivatives do not exist. Notice these
definitions are similar to those on the first
test.
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SECOND ORDER PARTIALS DERIVATIVES For a
function z f ( x, y ), if the indicated
partial derivatives exist then
This is read as the second partial of z with
respect to x the second time. The twos are not
exponents.
This is read as the second partial of z with
respect to x with respect to y.
This is read as the second partial of z with
respect to y with respect to x.
This is read as the second partial of z with
respect to y the second time.
Notice when using the ? symbol it is read and
found from right to left. However when using the
subscript notation it is found and read from left
to right.
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Finding partials is very similar to doing
implicit differentiation z y contains all the
terms that had dy/dx and z x contains all the
terms that did not have dy/dx. Each variable is
considered a constant when working with another
variable.
Find all the second partials for z f ( x, y )
5x 3 y 2 4x 2 y 4 7y 3 8x y 10.
First we must find the first partials of which
there are two.
z x 15 x 2 y 2 8 x y 4 8 z y 10 x 3 y
16 x 2 y 3 21 y 2 1
Study problems 21 24 and 33 36 page 989.
Just practice finding all of the second partials.
Next find all of the second partialsof which
there are four. z x x 30 x y 2 8 y 4 z x y
30 x 2 y 32 x y 3 z y x 30 x 2 y 32 x y 3 z
y y 10 x 3 48 x 2 y 2 42 y
Notice the z x y and z y x answers are the same.
They should always be the same for any problems
we work. Do not just copy the second answer.
Find the answer and compare in case you made a
mistake in finding the first one.
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MAXIMA AND MINIMA
RELATIVE MAXIMA AND MINIMA Let ( a, b ) be the
center of a circular region contained in the x
y-plane. Then, for a function z f ( x, y )
defined for every ( x, y ) in the region, f ( a,
b ) is a relative maximum if
f ( a, b ) f ( x,
y ) for all points ( x, y ) in the circular
region, and f ( a, b ) is a relative minimum if
f (
a, b ) f ( x, y ) for all points ( x, y ) in
the circular region.
LOCATION OF EXTREMA Let a function z f ( x, y
) have a relative maximum or relative minimum at
the point ( a, b ). Let f x ( a, b ) and f y (
a, b ) both exist. Then
f x ( a, b ) 0 and f y ( a, b ) 0.
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TEST FOR RELATIVE EXTREMA For a function z f
( x, y ), let f x x , and f y y, and f x y all
exist in a circular region contained in the x
y-plane with center ( a, b ). Further, let
f x ( a, b ) 0 and f
y ( a, b ) 0. Define the number D by
D f x x ( a, b ) f y y ( a, b )
f x y ( a, b ) 2. Then a. f ( a, b ) is a
relative maximum if D gt 0 and f x x ( a, b ) lt
0 b. f ( a, b ) is a relative minimum if D gt
0 and f x x ( a, b ) gt 0 c. f ( a, b ) is a
saddle point (neither a maximum nor a minimum,)
if D lt 0 d. If D 0, the test gives no
information. This test is similar to the second
derivative test used in test three.
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STEPS FOR FINDING RELATIVE EXTREMA Step 1. Find
the first partials f x ( x, y ) and f y ( x, y
). Step 2. Find the critical points ( a, b ) by
setting the first partials equal to zero and
solving the system of equations by either
elimination or substitution. Step 3. Find the
second partials f x x ( x, y ), f x y ( x, y ),
and f y y ( x, y ). Remember f x y
( x, y ) is the same answer as f y x ( x, y
). Step 4. Evaluate the second partials with the
critical point ( a, b ) f x x ( a, b ),
f x y ( a, b ), and f y y ( a, b ). Step
5. Determine the value of D D f x x ( a, b )
f y y ( a, b ) f x y ( a, b ) 2. Step 6.
Identify the extrema using criteria on the
previous slide. Remember to write
the answer correctly. Study problems 1 16 page
1001
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Find all points where the functions have any
relative extrema. Identify any saddle points.
Example 1 f ( x, y ) 5xy 7x 2 y 2 3x
6y 4
We are solving this system of equations by
elimination. This is a procedure you were
supposed to have learned in an algebra class.
The equations were rearranged using rules of
algebra.
Step 1. f x ( x, y ) 5y 14x 3
f y ( x, y ) 5x 2y 6
Step 2. f x ( x, y ) 5y 14x 3 0
f y ( x, y ) 5x 2y 6 0
You will not necessarily be using 2 and 5.
14x 5y 3 multiply by 2 5x 2y 6 multiply
by 5
Then substituting into either equation 5( 8 )
2y 6 40 6 2y 46 2y
23 y
28x 10y 6 25x 10y 30
3x 24 x 8
Critical Point ( 8, 23 )
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Step 3. f x x ( x, y ) 14 f
x y ( x, y ) 5 f y y ( x, y )
2
Step 4. f x x ( 8, 23 ) 14
f x y ( 8, 23 ) 5 f y y
( 8, 23 ) 2
Replace all of the xs with 8 and all of
theys with 23 in step 3 and evaluate.
Luckily there is nothing to evaluate.
Step 5. D f x x ( 8, 23 ) f y y ( 8,
23 ) f x y ( 8, 23 ) 2
D ( 14 )( 2 ) 5 2 D 3
Step 6. Because D gt 0 in step 5 and f x x (
8, 23 ) lt 0 in step 4, the critical point
found in step 2 is a relative
maximum.
Answer ( 8, 23 ) is a relative maximum
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Example 2 to be done later
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DOUBLE INTEGRAL The double integral of f ( x, y
) over a rectangular region R is written
And equals either
VOLUME Let z f ( x, y ) be a function that is
never negative on the rectangular region R
defined by c x d, a y b. The volume of
the solid under the graph of f and over the
region R is
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Step 1. Evaluate the inside integral first.
The dy means to integrate the y variable only.
Notice sign change.
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Step 2. Substitute this result into the original
problem.
Step 3. Evaluate the remaining integral.