Chabot Mathematics
§7.2 Partial
Derivatives
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Review §
7.1
 Any QUESTIONS About
• §7.1 → MultiVariable Functions
 Any QUESTIONS
About
HomeWork
• §7.1 → HW-03
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
§7.2 Learning Goals
 Compute and interpret Partial Derivatives
 Apply Partial Derivatives to study marginal
analysis problems in economics
 Compute Second-Order partial derivatives
 Use the Chain Rule for
partial derivatives to
find rates of change
and make incremental
approximations
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
OrdinaryDeriv→PartialDeriv
 Recall the Definition of an “Ordinary” Derivative
operating on a 1Var Fcn
df  z 
f z  h   f z 
f  z  
 lim
h 0
dz
h
 The “Partial” Derivative of a 2Var Fcn with respect to
indep Var x
f  x, y 
f  x  h, y   f  x, y 
f x  x, y  
 lim
h 0
x
h
 The “Partial” Derivative of a 2Var Fcn with respect to
indep Var y
f  x, y 
f  x, y  h   f  x, y 
f y  x, y  
 lim
h 0
y
h
Chabot College Mathematics
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Partial Derivative GeoMetry
 The “Partials” compute the SLOPE of
the Line on the SURFACE where either
x or y are held constant (at, say, 19)
• The partial
derivatives of f
at (a, b) are
the Tangent-Line
slopes of the Lines
of Constant-y (C1)
and Constant-x (C2)
Chabot College Mathematics
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Surface Tangent Line
 Consider z = f(x,y)
as shown at Right
 At the Black Point
Tangent Line
z
Graph of
with Slope :
x
z  f  x,0.2 
y  0.2
• x = 1.2 inches
• y = −0.2 inches
• z = 8 °C
• ∂z/∂x = −0.31 °C/in
 Find the Equation of
the Tangent Line
Chabot College Mathematics
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Pt 1.2,  0.2, 8
Surface Tangent Line
 SOLUTION
 Use the Point Slope
Equation
z  z1  m x  x1  
 In this case
0.31 C
z
x  7.628 C
in
Graph of
z  f  x,0.2 
Pt 1.2,  0.2 ,8
z
x  x1 
x
 0.31 C 
z  8 C   
 x  1.2 in 
in 

 Use Algebra to
Simplify:
0.31 C
z
x  7.628 C
in
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Partial Derivative Practically
 SIMPLE RULES FOR FINDING
PARTIAL DERIVATIVES OF z=f(x, y)
 To find ∂f/∂x, regard y as a constant and
differentiate f(x, y) with respect to x
• y does NOT change → z x
 2. To find ∂f/∂y, regard x as a constant
and differentiate f(x, y) with respect to y
• x does NOT change → z y
Chabot College Mathematics
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Example  2Var Exponential
 For z  f x, y   e
(
)
(
)
x2 2 y
¶ x 2 +2 y
¶ 2
x 2 +2 y
x 2 +2 y
e
=e
× ( x + 2y)= e
× ( 2x + 0)
¶x
¶x
z
x2 2 y
 2x  e
x
¶ x2 +2 y
x 2 +2 y ¶
2
x 2 +2 y
e
=e
× ( x + 2y)= e
× ( 0 + 2)
¶y
¶y
z
x2 2 y
 2e
y
Chabot College Mathematics
9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Example  Another Tangent Line
 Find Slope for
Constant x at (1,1,1)
z
y
 f y  x, y , z 
x  const
z

4  x2  2 y2  0  0  4 y
y


 Then the Slope at
(1,1,1) z
m
y&z Change;
x does NOT
Chabot College Mathematics
10
y
x 1
 41  4
y 1
 Then the Tan Line Eqn
z  1  4 y  1
 z  4 y Bruce
5 Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
• Bruce Mayer,
PE
Example MTH16
Another
Tangent
Line
4
3
z = 4 - x2 - 2y2
2
1
0
-1
-2
-3
-4
-2
-1
-0.5
0
0.5
Chabot College Mathematics
11
y
0
MTH16 Sec7 2 multi3D 1419.m
1
2
xBruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
12
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
MATLAB Code
% Bruce Mayer, PE
% MTH-16 • 19Jan14
% Sec7_2_multi3D_1419.m
%
clear; clc; clf; % clf clears figure window
%
% The Domain Limits
xmin = -2; xmax = 2; % Weight
ymin = -sqrt(2); ymax = sqrt(2); % Height
NumPts = 20
% The GRIDs) **************************************
xx = linspace(xmin,xmax,NumPts); yy = linspace(ymin,ymax,NumPts);
[x,y]= meshgrid(xx,yy);
xp = ones(NumPts); % for PLANE
xL = ones(1,NumPts); % for LINE
xt = 1; yt =1; zt = 1; % for Tangent POINT
% The FUNCTION SkinArea***********************************
z = 4 -(x.^2) - (2*y.^2); %
zp = 4-xp.^2-2*y.^2
zL = 5-4*y %
% the Plotting Range = 1.05*FcnRange
zmin = min(min(z)); zmax = max(max(z)); % the Range Limits
R = zmax - zmin; zmid = (zmax + zmin)/2;
zpmin = zmid - 1.025*R/2; zpmax = zmid + 1.025*R/2;
%
% the Domain Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
mesh(x,y,z,'LineWidth', 2),grid, axis([xmin xmax ymin ymax zpmin zpmax]), grid,
box, ...
xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y'), zlabel('\fontsize{14}z = 4
- x^2 - 2y^2'),...
title(['\fontsize{16}MTH16 • Bruce Mayer, PE',]),...
annotation('textbox',[.73 .05 .0 .1 ], 'FitBoxToText', 'on', 'EdgeColor',
'none', 'String', 'MTH16 Sec7 2 multi3D 1419.m','FontSize',7)
%
hold on
mesh(xp,y,zp,'LineWidth', 7)
plot3(xt,yt,zt,'pb', 'MarkerSize', 19, 'MarkerFaceColor', 'b')
plot3(xL,y,zL, '-k', 'LineWidth', 11), axis([xmin xmax ymin ymax zpmin zpmax])
%
Bruce Mayer, PE
Chabot
College
hold
off Mathematics
ReCall Marginal Analysis
 Marginal analysis is used to assist
people in allocating their scarce
resources to maximize the benefit of
the output produced
• That is, to Simply obtain the most value
for the resources used.
 What is “Marginal”
• Marginal means additional, or extra, or
incremental (usually ONE added “Unit”)
Chabot College Mathematics
13
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Example  Chg in Satisfaction
 A Math Model for a utility function,
measuring consumer satisfaction with a pair
of products: U x, y   xye2 x 3 y
• Where x and y are the unit prices of product A
and B, respectively, in hecto-Dollars, $h
(hundreds of dollars), per item
 Use marginal analysis to approximate the
change in U if the price of product A
decreases by $1, product B decreases by $2,
and given that A is currently priced at $30
and B at $50.
Chabot College Mathematics
14
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Example  Chg in Satisfaction
 SOLUTION:
 The Approximate Change, ΔU
ΔU =
[Change
due to Δx]
+
[Change
due to Δy]
 Using Differentials

U
U
U 
x 
y & U  xye2 x 3 y
x
y





 ye 2 x 3 y  xye 2 x 3 y   2 x  xe 2 x 3 y  xye 2 x 3 y   3 y
Chabot College Mathematics
15
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Example  Chg in Satisfaction
 Simplifying ΔU
= ye-2 x-3y (1- 2x ) Dx + xe-2 x-3y (1- 3y) Dy
U  e 2 x 3 y  y1  2 x x  x1  3 y y 
 Now SubStitute in
DU » e
• x = $0.30h &
Δx = −$0.01h
• y = $0.50h &
Δy = −$0.02h
-2(0.3)-3(0.5)
éë(0.5) (1- 2(0.3)) (-0.01) + 0.3(1- 3(0.5)) (-0.02)ùû
 0.000122
Chabot College Mathematics
16
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Example  Chg in Satisfaction
 Thus DROPPING PRICES
• Product-A: $30→$29
 A −1/30 = −3.33% change (a Decrease)
• Product-B: $50→$48
 A −2/50 = −1/25 = −4.00% change (a Decrease)
 IMPROVES Customer Satisfaction by
+0.00012 “Satisfaction Units”
 But…is +0.00012 a LOT, or a little???
Chabot College Mathematics
17
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Example  Chg in Satisfaction
 Calculate the PreChange, or Original
Value of U, Uo(xo,yo)
20.330.5 
U o xo , yo   U 0.3,0.5  0.30.5e
U o xo , yo   0.01837
New  Baseline 100%
 ReCall the
% 

Δ% Calculation
Baseline
1
 Thus the Δ% for U
U 100%  0.00012
U % 


 0.6532%
U0
1
0.01837
Chabot College Mathematics
18
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Example  Chg in Satisfaction
 The Avg Product-Cost = (30+50)/2 = 40
 The Avg Price Drop = (1+2)/2 = 1.5
 The Price %Decrease = 1.5/40 = 3.75%
 Thus 3.75% Price-Drop Improves
Customer Satisfaction by only 0.653%;
a ratio of 0.653/3.75 = 1/5.74
• Why Bother with a Price Cut? It would be
better to find ANOTHER way to Improve
Satisfaction.
Chabot College Mathematics
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
2nd Order Partial Derivatives
 If z=f (x, y), use the following notation:
 f x x
 f x y
f 
y
x
f 
y
y
Chabot College Mathematics
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 f xx
  f 
2 f
2z




2
x  x  x 
x 2
 f xy
  f 
2 f
2 z




y  x  yx
yx
 f yx
  f 
2 f
2z







x  y  xy
xy
 f yy
  f 
2 f
2z





2


y  y  y 
y 2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Clairaut’s Theorem
 Consider z = f(x,y) which is defined on
Domain, D, that contains the point
(a, b). If the functions ∂2f/∂x∂y and
∂2f/∂y∂x are both continuous on D, then
2z
f yx a, b  
xy
xa
y b
2z

yx
xa
y b
 f xy a, b 
 That is, the “Mixed 2nd Partials” are
EQUAL regardless of Sequencing
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Example  2nd Partials
¶
2
¶x
2
¶
2
¶y
2
¶
¶xy
¶2
¶yx
2
(e )
(e )
(e )
x 2 +2 y
x 2 +2 y
x 2 +2 y
(e )
x 2 +2 y
¶
=
¶x
¶
=
¶y
¶
=
¶x
¶
=
¶y
(2xe ) = 2e
(2e ) = 2e
(2e )  2e
x 2 +2 y
x 2 +2 y
x 2 +2 y
x 2 +2 y
x 2 +2 y
 2 x  4 xe
x2 2 y
22
x2 2 y
 2  4 xe
 The last two “mixed” partials are equal as
Predicted by Clairaut’s Theorem
Chabot College Mathematics
×(2x)
× ( 2)
x2 2 y
(2xe )  2 xe
x 2 +2 y
+ 2xe
x 2 +2 y
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
x2 2 y
Game Plan  09Feb16
 HandOut Exam-1 Study Guide
• Note that the Sp14 Exam (the Study
Guide) is TOO HARD – The Sp16 edition
will be slightly less “Pressure Packed”
 File: MTH16_Lec-05_Sp16_sec_72_Partial_Derivatives.pptx
• Slides24-31 → PD ChainRule, Incremental
Approx., Chain Rule WhtBd Example
 File: MTH16_Lec-06_Sp16_sec_73_2Var_Optimization.pptx
Chabot College Mathematics
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
The Chain Rule (Case-I)
 Let z=f(x, y) be a differentiable function
of x and y, where x=g(t) and y=h(t) and
are both differentiable functions of t.
Then z is a differentiable function of t
such that:
dz z dx z dy


dt x dt y dt
• Case-I is the More common of the 2 cases
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Example  Chain Rule (Case-I)
x y
 Let z 
x2
2
x  3t
 Then Find dz/dt →
yt
2
2
d
d  3t   t 2 
z 

dt
dt  3t  2 
dz z dx z dy
   
dt x dt y dt
2
(x + 2)× 2x - x + y ×1
1
=
× 3+
× 2t
2
(x + 2)
x+2
3 x2  4x  y
2t


2
( x  2)
x2
(

Chabot College Mathematics
25
)

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
The Chain Rule (Case-II)
 Let z=f(x, y) be a differentiable function
of x and y, where x=g(s, t) and y=h(s, t)
are differentiable functions of s and t.
Then
z z x z y


s x s y s
z z x z y


t x t y t
• Case-II is the Less common of the 2 cases
Chabot College Mathematics
26
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Chabot College Mathematics
27
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Chabot College Mathematics
28
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Another Chain Rule Example
Chabot College Mathematics
29
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Incremental Approximation
 Let z = f(x,y)
 Also Let
• Δx denote a small change in x
• Δy denote a small change in y,
 then the Corresponding change in z is
approximated by
z
z
z  x  y
x
x
Chabot College Mathematics
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
∆𝒛 Example based on P7.48
For Δ𝐾 = 3 & Δ𝐿 = 2
Chabot College Mathematics
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Linearization in 2 Variables
 The incremental Approximation Follows
from the Mathematical process of
Linearization
 In 3D, Linearization amounts to finding
the Tangent PLANE at some point of
interest
• Note that Two Intersecting
Tangent Lines Define
the Tangent Plane
Chabot College Mathematics
32
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Linearization in 2 Variables
 Suppose f has continuous partial
derivatives. An equation of the tangent
plane to the surface z=f (x,y) at the pt
P(xo,yo,zo) is given by z−z0=Σm(u-u0)
z  z0  f x ( x0 , y0 )( x  x0 )  f y ( x0 , y0 )( y  y0 )
z  z0    f x x0 , y0  x  x0    f y x0 , y0  y  y0 

f

z   
x

Chabot College Mathematics
33


  x   f
 y
x0 
y0 



  y 
x0 
y0 

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Linearization in 2 Variables
 Now the Linear Function whose graph
is Described by the Tangent Plane
f
Lx, y   f a, b  
x
f
 x  a  
xa
x
y b
xa
y b
  y  b
 The above Operation is called the
LINEARIZATION of f at (a,b) (constants)
 The Linearization produces the Linear
Approximation of f about (a,b)
f
f x, y   f a, b  
x
Chabot College Mathematics
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f
 x  a  
xa
x
y b
  y  b
xa
y bBruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Linearization in 2 Variables
 In other words, NEAR Pt (a,b)
f
f x, y   f a, b  
x
f
 xa 
x a
x
y b
xa
y b
 yb  f a, b   f x  f y
 The Above is called the Linear
Approximation or the Tangent Plane
Approximation of f at (a,b)
f x  0
 Note that lim
x 0
a
Chabot College Mathematics
35
lim f y  0
yb 0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
ReCall in 2D dx&dy vs Δx&Δy
dy
y  ya 
 xa
dx x a
Chabot College Mathematics
36

f x   f a   f a  x  a 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
in 3D dz vs Δz
Linear
Approximation
z  z a ,b 
z
z
 xa 
x x a
y
y b
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xa
y b
 yb  z  f a, b   f x a, b   x  a    f y a, b    y  b 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
WhiteBoard Work
 Problems From §7.2
• P62 → Hybrid AutoMobile Demand
Chabot College Mathematics
38
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
All Done for Today
Partial
Derivatives
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
2a
–
Chabot College Mathematics
40
BMayer@ChabotCollege.edu
2b
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Chabot College Mathematics
41
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Chabot College Mathematics
42
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Chabot College Mathematics
43
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Chabot College Mathematics
44
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx
Chabot College Mathematics
45
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx