Bear H.S. Understanding Calculus
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CHAPTER
41
41.1 1
41.2
-1
41.3 2
41.4 15
41.5
5f
41.6 !
41.7 2
41.8
-¥
CHAPTER
42
297
42.1
42.2
42.3
42.4
42.5
42.6
42.7
42.8
42.9
42.10
42.11
42.12
o
4
-3
o
1
271'
(¥)3
o+
f~
Idy + f?x dx
=!
571'
o
32
T
~71'
371'
CHAPTER 43
43.1
43.2
43.3
43.4
43.5
43.6
43.7
43.8
39
9 +
e
5
.JI-
12
x
2y
+ siny = c
cosy +
XeY
= c
x
3
y2_X
2
y3
= C
x
siny+
y=
c
X
2+xy+y2=C
A
absolute
convergence,
135
absolute value,1
acceleration of
gravity,
63
additionof vectors, 185
alternating series,135
amplitude of simpleharmonic
motion,
176
anglebetweenlines,vectors, 196,197
anti
derivative
, 64
approximation with differentials, 57, 222,223
approximation by tangentplane,222,223
arc length,84
area,69
as a lineintegral, 264
asymptotes, 8
auxiliary equation, 174
B
bacterial
growth,
44, 163
binomial
expansion,
158
C
catenary,
113
Cauchy's
Mean
Value
Theorem,
84
centroidof a triangle, 193
chainrules,26, 227-231
changeof
variable,
87
circles,8
closedcurve,259
comparison tests,
for integrals, 128
Index
for series,137
components of a
vector,
187
composite
function,
25
compound
interest, 58
concave
up,
concave
down,
53
conditional
convergence,
135
conditions for exactdifferential, 268,269
cone,214
volume
of, 76
continuous
function,
20
convergence
of powerseries,
141
coordinates, 1
coordinate planes,201
cosh
x, 44
criticalpoints,53, 239
crossproduct, 206
curly-d, 217,218
cycloid,
85
area
under,
266
cylindrical shells
method,
99
cylindrical surface, 212
D
D(J,D
u
'
233
definiteintegral, 69
del(V),235
derivative,
13
higherorder,52
determinant, 206,207
difference quotient, 14, 19
differential, 57, 223-225
299
300
differential equation,
161
differentiation, implicit, 54
differentiation rules, 19-23
directionangles,cosines, numbers, 203
directional
derivative,
233
dot product, 195,201
doubleintegral, 245
doubleyour
money,
58
E
e, 41,
121
ex,
graph and
derivative
of, 41
ellipse,7
ellipsoid, volumeof, 79
Eulerformula, 175
exactdifferential, 267
integralof, 267
exponential functions, 39, 40
F
factorial,
121
falling
body,
13
with air resistance, 164
first orderdifferential equation,
161
first order linearequations, 167
forcefield, 255
frequency of simpleharmonicmotion, 176
G
g, the acceleration of
gravity,
63
geometric series, 135
gradient, 235
Green'sTheorem, 259
H
harmonicseries, 134
heat equation, 220
homogeneous differential equation, 167
hyperbola, 7
hyperbolic identities, III
hyperbolic sine and cosine,44,
111
hyperboloid,
214-216
I
l, j, 186
implicitdifferentiation, 54
improperintegrals, 127
inclination of a line,3
indefiniteintegral,64
independent of path, 268
independent solutions, 174,179
indeterminate forms, 120
inflexion, point of, 53
initialcondition, 161, 174
integral,indefinite,64
definite, 69
integraltest for series, 138
integration by parts, 97
interest,
compound,
44
intervalof convergence, 142
inverse
functions, 41, 42, 45, 46
iteratedintegral,247
L
Laplace'sequation, 220
lawof cosines,199
LCR circuit, 173
lengthof a curve,84
levelcurve, 235,236
I'Hospital'sRule, 58, 120
limit, 14
at
00,
119
limitcomparison test, 138
lineardifferential equations, 167
linearoperator, 179
line integral,255
for area,264
lines in space,201, 203
localmaximumand minimum, 51
logx, 41, 42
M
maximumand minimum, 51, 239
maximumdirectional derivative, 235
max/minsufficientconditions, 239, 241
Mean
Value
Theorem, 51
measurement errors,223, 224
mixedpartials,219
midpointrule, 115
N
Newton'slawof motion,63
Newton'slawof cooling,163
Newton
method,
59
non-homogeneous equations, 179
norm
ofa
vector,
185
normalvector
to a plane,211
to a tangentplane,222
numerical integration, 115
o
octants,201
one-to-one function, 45
orientationof a curve,259
p
parabolas, 7
Index
Index
paraboloidof revolution, 213
parameterizations of a curve,256
parallellines,3
parametric equations, 81
tangentlinefor,83
partialderivative, 217
partialfractions, 92
partial sums, 133
pendulum, 36
periodicfunction, 31
perpendicularlines,
3
phase shift, 176
plane,equationof, 211
positiveseries, 136
powerseries,
141
projectionof A on B, 195
p-series,
138
Pythagorean Theorem, 200
Q
quadrants, 1
quadricsurface,211
R
radianmeasure,31
radiusof convergence, 142
rate of change,13,29
ratio test, 142
rationalfunction, 91
reducedform of a differential equation,167
relativeerror, 57, 224
Riemannsum, 69
for doubleintegral,245
rotationof coordinates, 198
S
saddlesurface, 239, 240
scalarproduct,195,201
second
derivative,
52
seconddirectional
derivative,
240
secondorderdifferential equation, 173
secondpartials,218
separable differential equation, 162
sequence,
121
sequence of partial sums, 133
series, 133
simpleclosedcurve,259
simpleharmonicmotion,176
simplyconnectedregion,270
Simpson's Rule, 116
sin
x and cosinex, 31
derivatives of, 34
sinecurve,37
sin:' x and cos:' x, 46
sinhx, 44
slopeof a line,
1
slopeinterceptform, 2
solidof revolution, 78, 245
sonicboom, 13
speed,
13
surfaces, 201, 211
of revolution, 213
T
tangentline, 14
tangentplane, 221-223
tangentvectors
t
x
'
t
y
'
221
tangents
on surfaces,221
to approximate a function, 57
tan
x, 31
Taylor
polynomials, 150
at
x
o'
153
Taylor
series, 155
terminal
velocity,
164
tractrix,113
trajectoryof a projectile, 81, 85
trigonometric functions, 31
trigonometric identities, 33,
101
trigonometric integrals,
101
trigonometric substitution, 107
U
unit
vector,
186
u-substitution, 65
V
variables separate, 162
vectors, 185
in space,201
norm of, 185
sumof, 185,201
volume,
76
volumeof revolution, 76, 78
volumeunder a surface,245
W
work,75, 255
alonga curve,255
301
About the Author
H. S. Bear
received
both his Bachelor of Arts
degree
and
doctorate
from the
University
of
California at
Berkeley.
Priorto
receiving
his
Ph.D.,
he taughtfor oneyearas an instructor at
the
University
of
Oregon.
He then returned to the
University
of California at
Berkeley
and
obtained
his
doctorate
while
teaching
there.
Dr.
Bearhas taughtat a geographically
diverse
groupof
schools,
including
permanent positions at the
University
of
Washington,
University
of California at Santa
Barbara,
andNew
Mexico
State
University.
He has also held
teaching
positions at
Princeton
University,
the
University
of California at San
Diego,
and the
Univer-
sityof
Erlangen-Nuremberg,
Germany.
Dr.
Bearspentmostofhis
academic
careeras profes-
sor of
mathematics
at the
University
of
Hawaii,
where
he alsoservedtermsas chairman and
graduate
chairman
of the
Mathematics
Department.
He now holds the title of professor
emeritus.
Throughout
his
career,
Dr.
Bear has been
active
in publishing books and articles in
mathematics.
His most recentpublications are two
advanced
textson mathematical
analysis
with
Academic
Pressanda
Dover
Publications
reprintof an earliertext on differential
equa-
tions.
With
the publication of
Understanding
Calculus:
A
User's
Guide,
Dr.
Bear will
have
contributed morethan40
pieces
of
mathematical
literature.