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Math 1320-9 Notes of 3/18/20
Chapter 11: Partial Derivatives
• basic topic: functions of several variables, their (partial)derivatives, and the use of those derivatives.
• Before diving into details: Here is an example of the cruxof the matter:
Supposef(x, y) = x2 sin(x+ y)
Then
• fx =
• fy =
• fxx =
• fxy =
• fyx =
• fyy =
• So we need to talk about functions of several variables,their limits, continuity, and then derivatives ...
Math 1320-9 Notes of 3/18/20 page 1
2
2x sink 14 12cos City
x'cos City
2SimCity 12 05414 1 2x cosarty Isin x 14
Ei csina.ms
2X cos lxty Isin City
x'sink 14
11.1 Functions of Several Variables
• Defined much like function of one variable, usually in termsof a mathematical expression, although it might also bedescribed verbally, numerically by a table of values, orvisually by a surface or a set of contour lines.
• We’ll focus on the algebraic form.
• Example 3:
f(x, y) =!
9! x2! y2
Math 1320-9 Notes of 3/18/20 page 2
Graphy of 2
is the set of Domain XM such that 9 H y o
all points Cx 1,4in 423 that satisfy 93
2 42ry rthe equation
2 frT2z2q X2 42 32 42 122 9 270upper half sphere
Range 0,3
Graph
12 with(plots);3 plotsetup(ps,plotoutput=‘45.ps‘,plotoptions=‘portrait,noborder,height=500,width=500‘);4 contourplot(sqrt(9-x**2-y**2),x=-3.1..3.1,y=-3.1..3.1,5 contours=[0.0,0.25,0.5,0.75, 1.0,1.25,1.5,1.75,2.0,2.25,2.5,2.75,3.0],thick-ness=3);
678 plotsetup(gif,plotoutput=‘45a.gif‘,plotoptions=‘portrait,noborder,height=500,width=500‘);9 plot3d(sqrt(9-x**2-y**2),x=-3..3,y=-3..3);
–3
–2
–1
1
2
3
y
–3 –2 –1 1 2 3
x
Figure 1. Contour lines for f(x, y) =!
9! x2! y2.
Math 1320-9 Notes of 3/18/20 page 3
Maple
Figure 2. Surface Drawing for f(x, y) =!
9! x2! y2.
• Some interactive drawings:
• f(x, y) = xy
• f(x, y) = sin(x+ y)
• f(x, y) = e!(x2+y2)
Math 1320-9 Notes of 3/18/20 page 4
c
L
Terminology
•
z = f(x, y)
• x, y are the independent variables.
• z is the dependent variable.
• The domain is the set of all points (x, y) where f can beevaluated.
• Natural Domain as before.
• The range is the set of all outputs
• Domains and Ranges of
f(x, y) =!
9! x2! y2
f(x, y) = ln(y ! x2)
Math 1320-9 Notes of 3/18/20 page 5
Domain Cay Atyeo
Range 0,3
Domain y xµ
Respiaogagoafaooe
42
Range C 00,00
Functions of Three Variables
• Same sort of ideas, except we have three independent vari-ables.
• What happens to the contour lines?
• Example:f(x, y, z) = x2 + y2 + z2.
Math 1320-9 Notes of 3/18/20 page 6
W
11.2 Limits and Continuity
• Recall that in one variable we considered limits like
limx!"c
f(x) = L (1)
and also one-sided limits like
limx!"c+
f(x) = L and limx!"c!
f(x) = L (2)
• The limit (1) exists if and only if the two one-sided limits(2) exist and are equal.
• In several variables we have infinitely many, rather thanjust two, directions in which we can approach a point.
• Example:
lim(x,y)!"(0,0)
sin(x2 + y2)
x2 + y2
Math 1320-9 Notes of 3/18/20 page 7
O
2 4,2
bing.oi o ezign.si
lim(x,y)!"(0,0)
1
x2 + y2
Math 1320-9 Notes of 3/18/20 page 8
DIVE
Legio I oo
• Example 1:
lim(x,y)!"(0,0)
x2! y2
x2 + y2
Math 1320-9 Notes of 3/18/20 page 9
L
Y x Fmx
Tp1 0 Egig i
y22 0 L I 8 0 L lying yr
I
Y x Le einF H O2 2
f ME LH Cmx 2Ft Czyz
l m
I 1 M2
m O I
m OO L I
Mei L O
• Example 2:
f(x, y) =xy
x2 + y2
Math 1320-9 Notes of 3/18/20 page 10
XYLimGil 710,0 X't 12
y Mx
KlumX Mt
70 X Cmx
Z mbin Mx
X 70 Itm X2 jam
• Example 3:
f(x, y) =xy2
x2 + y4
Math 1320-9 Notes of 3/18/20 page 11
2line XY
44 710,0 82 44
y Mx x nextbunx 70 541m44
m2 isline70 x m 4 4
m2bum70 I m4 2 0