CALCULUS REV IS ITED
PART 2
A Self-study Course STUDY GUIDE
Block 3
Partial Derivatives
Herbert I. Gross Senior Lecturer Center for Advanced Engineering Study Massachusetts Institute of Technology Copyright @ 1972 by
Massachusetts Institute of Technology Cambridge, Massachusetts All rights reserved. No part of this book may be reproduced in any form or by any means without permission in writing from the Center for Advanced Engineering Study, M.I.T. CONTENTS Studv Guide Block 3: Partial Derivatives Pretest
Unit 1:
Unit 2:
Unit 3:
Unit 4:
Unit
Unit
Unit
Unit
Quiz
5:
6:
7:
8:
Functions of More Than One Variable An Introduction to Partial Derivatives Differentiability and the Gradient The Directional Derivative in n-Dimensional Vector Spaces (Optional) The Chain Rule, Part 1 The Chain Rule, Part 2 More on Derivatives of Integrals The Total Differential Solutions Block 3: Partial Derivatives Pretest
Unit 1:
Unit 2:
Unit 3:
Unit 4:
Unit
Unit
Unit
Unit
Quiz
5:
6:
7:
8:
Functions of More Than One Variable An Introduction to Partial Derivatives Differentiability and the Gradient The Directional Derivative in n-Dimensional Vector Spaces (Optional) The Chain Rule, Part 1 The Chain Rule, Part 2 More on Derivatives of Integrals The Total Differential iii
Study Guide
I
BLOCK 3: PARTIAL DERIVATIVES Study Guide Block 3: Partial Derivatives Pretest 2Xy 2, (x,y) # (0.0). Show that
x + Y
lim
f(x,y) depends on the path by which (x,y) approaches (x,y)+(O,O) 1. Let w
=
f (x,y) =
2. Find the equation of the plane which is tangent to the surface
x4 + y6 z + xyz5 = 3 at ~ ~ , ~ , ~ ~ .
I. ,.
Suppose w depends on r but not on 8, say w = h(r), and that h is a
rn
twice-differentiable function of r.
L
Determine a
'
w + a ;, expressed
a~
ax
in terms of r.
4. Find the equation of the curve C if C passes through the origin and has its slope at each point (x,y) given by Given that g(y) =
i1
b dx where y > b > -1, determine gl(y). Study Guide Block 3: Partial Derivatives Unit '1: Functions of More Than One Variable 1.
Lecture 3.010 Study Guide
Block 3 : P a r t i a l D e r i v a t i v e s
Unit 1: F u n c t i o n s o f More Than One V a r i a b l e
2. Read Supplementary Notes, Chapter 4 .
3 . Read Thomas, S e c t i o n 15.1.
4. ( O p t i o n a l )
Read Thomas, S e c t i o n s 12.10 and 12.11.
(These s e c t i o n s
w i l l h e l p you f e e l more a t home w i t h e q u a t i o n s of s u r f a c e s . The
i d e a i s t h a t j u s t a s t h e graphs of f u n c t i o n s o f a s i n g l e v a r i a b l e
a r e c u r v e s i n t h e xy-plane,
t h e g r a p h s of f u n c t i o n s o f two r e a l
v a r i a b l e s a r e s u r f a c e s i n space.
Except f o r any peace-of-mind
t h a t you g e t i n f e e l i n g a t home w i t h t h e v a r i o u s e q u a t i o n s , it
s h o u l d be n o t e d t h a t w e can s u r v i v e t h e remainder of t h i s c o u r s e
w i t h o u t r e c o u r s e t o a c c u r a t e graphs j u s t a s was t h e c a s e i n f u n c t i o n s
of a s i n g l e r e a l v a r i a b l e . )
5. E x e r c i s e s :
Define
a s t h e Minkowski m e t r i c .
t h e n IlxIl
- = m a x ~ l x l l .....I
That i s , i f x = (xl,.
.., x n ) ,
xnIl.
a . Show t h a t
1.
2-
1k1) >/ 0
IIx + yll
3 . IlaxII =
for a l l x and llxll
x = 0.
- = 0 i f and o n l y i f -
lkll
l a l IkII
4
+
Ikll
x * ~ l,lXllr and llrll (where we a r e s t i l l u s i n g t h e Minkowski
b. Compute m e t r i c ) i f 5 = ( 2 , 4 , 1 ) and 4~ = ( 4 , 4 , 5 )
From t h i s conclude t h a t
i t need n o t be t r u e t h a t ~ -X - ~ 6I l l ~ l l Ilu_ll.
.
Mimic t h e proof of t h e corresponding 1-dimensional c a s e t o prove
lim
1i m
that i f x and a belong t o E" and x-rg f
= L1 w h i l e x,a
g (5) = L2,
-then
(x)
lim
[f
x-rg
-
(x) + g ( x ) I
= L1
+ L2
Study Guide
Block 3: P a r t i a l D e r i v a t i v e s
U n i t 1: F u n c t i o n s o f More Than One V a r i a b l e
a.
Using t h e Minkowski m e t r i c , suppose E > O i s g i v e n ; f i n d 6 such t h a t
f o r t h i s c h o i c e of 6
b.
I n t e r p r e t t h e answer i n ( a ) g e o m e t r i c a l l y and e x p l a i n why t h e same
v a l u e of 6 a s i n ( a ) would have s u f f i c e d had w e used t h e Euclidean
m e t r i c r a t h e r t h a n t h e Minkowski m e t r i c .
3.1.4(L)
Let x = (xl,x2,x3,x4)
f ( x ) = x12
+
2x2
+
x3
and l e t 1 = (1,1,1,1). Define f by
3
+
x4
2
.
Prove t h a t f i s continuous a t
=
3.1.5
x and 1 be a s i n E x e r c i s e 3.1.4.
Let f , -
For a given E > O ,
such t h a t
3.1.6(L)
L e t f be d e f i n e d by
a.
b.
c.
Is f continuous a t ( 0 , 0 ) ?
lim lim
lim limf ( x , y ) l .
Compute b o t h y+Q [x+o f ( x , y ) l and x+O
ry+(,
I n v e s t i g a t e t h e b e h a v i o u r of
i n more d e t a i l by i n t r o d u c i n g p o l a r c o o r d i n a t e s .
find 6
1.
Study Guide
Block 3 : P a r t i a l D e r i v a t i v e s
Unit 1: Functions of More Than One Variable
L e t f be defined by
a . Show t h a t l i m
f ( x , y ) depends on t h e p a t h by which ( x I y )
(x,y)+(O,O) approaches (0,O
.
b. Compute
1i m f (x,y) i f
(x,y)+(O,O)
71 r a y t3 = 4
( x , y ) approaches ( 0 , O ) along t h e
.
c. Show t h a t i f (x,y) approaches (0,O) e i t h e r along t h e x-axis o r
t h e y-axis then l i m f ( x , y ) = 0.
Define g by
r
l.
a . Show t h a t g i s n o t continuous a t (0,O).
lim
g ( x , y ) = g(O,O) i f
(x,y)+(O,O)
approach ( 0 , O ) along e i t h e r a x i s . (x,y) i s allowed t o
L e t t h e f u n c t i o n £ : E ~ - + E
be continuous.
Prove t h a t f cannot be 1-1.
b. Show t h a t
Comment
The following two e x e r c i s e s a r e o p t i o n a l .
They may be omitted
w i t h o u t l o s s of c o n t i n u i t y t o our p r e s e n t d i s c u s s i o n .
Their
main purpose i s t o supply t h e i n t e r e s t e d r e a d e r w i t h a few c l u e s
a s t o how a n a l y t i c p r o o f s a r e c a r r i e d o u t i n n-dimensional v e c t o r
spaces (with n g r e a t e r than t h r e e ) using t h e o r d i n a r y p r o p e r t i e s
of r e a l number a r i t h m e t i c .
Study Guide
lock 3: P a r t i a l D e r i v a t i v e s
U n i t 1: F u n c t i o n s of More Than One V a r i a b l e
4
.
Prove t h a t o u r d e f i n i t i o n of a = b_
Let a and b belong t o E
i s an e q u i v a l e n c e r e l a t i o n because a£ t h e f a c t t h a t " o r d i n a r y "
e q u a l i t y i s an e q u i v a l e n c e r e l a t i o n on t h e s e t of r e a l numbers.
.
4
With t h e d o t p r o d u c t a s
c be e l e m e n t s of E
L e t a, 2 and d e f i n e d i n o u r supplementary n o t e s , prove t h a t
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