Life of Fred
Calculus
Stanley F. Schmidt, Ph.D.
Polka Dot Publishing
What Calculus Is About
I
remember standing in the college bookstore at the beginning of my
freshman year. I pulled a beginning calculus textbook off the shelf and
opened it. What a frightening sight it was.
The pages were filled with strange symbolism like
and
I
2
(4 – x 2) dx
x=0
MN
= M. It might as well have been in Turkish. No one else in my family had ever
Mx
studied calculus, so there was no one to give me an overview of what lay ahead. All I was
told was that anyone who wanted to study any of the topics that I was even remotely
interested in would want to have a grounding in calculus. Even business majors going on
for a master's degree were required to study it.
But that didn't tell me what it was. I looked at some of the problems in that old
textbook:
11. Find dy/dx for y = sin x.
18. Determ ine the eccentricity of (y + 5)² + 4(x – 5)² = 1.
22. Solve y' = tanh x
From my trig class I recognized “sin x” and knew that it didn't have theological overtones
in this context. But I was still at a loss as to what calculus was about or why I needed to
learn it. W as this stuff useful? W ould I find a need for it in my everyday life?
Yes. The book you now hold in your hands shows that every aspect of calculus
can arise in the course of daily living. If you've ever fallen into a vat of cheese soup
(Chapter 19) or tried to run a thousand pounds of ammo through a custom's station
(Chapter 9) you know what I mean.
So what's calculus? In a sentence:
If it moves at a varying speed,
if it has a curvy shape,
if it has a maximum that you'd like to find,
if it involves adding up an infinite number of terms,
then you're probably looking at calculus.
7
Contents
(Topics in small italics are covered in the Further Ado section of the book at
the page numbers indicated.)
Chapter 1
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
range, onto, 1-1 correspondence, inverse functions
Chapter 2
Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
,-* definition 399
Chapter 3
Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Average Speed vs. Instantaneous Speed
Chapter 4
Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Tangent Lines
Chapter 5
Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Maximums/Minimums
Product/Quotient/Chain Rules
proofs of the product/quotient/chain rules 404
Chapter 6
Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Second Derivatives
Asymptotes
Chapter 7
Trig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Tests for Extrema
Chapter 8
Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Implicit Differentiation
Explicit/Implicit/Parametric
Chapter 9
Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Mean Value Theorem
proof of the MVT 412
L'Hospital Rule 414
Acceleration
Antiderivatives
Chapter 10
Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Fundamental Theorem of Calculus
proof of FTC 415
Chapter 11
Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Parametric Forms for Area and Length
Improper Integrals
Chapter 12
Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Solids of Revolution
Torque
Chapter 13
Centroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Differentials
Average Value of a Function
Integration by Parts
Moments of Inertia
Chapter 14
Logs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Probability Density Functions
definition of e 420
bounded increasing sequences 422
Chapter 15
Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Hydrostatic Force
oblique asymptotes 426
Chapter 16
Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Tests for Convergence
Chapter 17
Solids of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Trig Substitutions
Surface Area
Arc Length
Chapter 18
Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Alternating Series
Power Series
Evaluating Integrals Using Substitutions
partial fractions 439
Maclaurin and Taylor Series
remainder formula for Taylor 441
Chapter 19
12
Hyperbolic Trig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Separating the Variables in Differential Equations
Numerical Integration
Chapter 20
Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Chapter 21
Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Chain Rule with Intermediate Variables
Lagrange Multipliers 445
Chapter 22
Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Cylindrical Coordinate System
spherical coordinates 447
Chapter 23
Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
Gradient
Directional Derivative
Line Integrals
Green's Theorem
flux of a vector through a surface 453
Divergence Theorem 461
Stokes's Theorem 466
Chapter 24
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 338
Variables Separable
exact and integrating factors 468
Orthogonal Trajectories
First Order Linear
Bernoulli's equation 396
Second Order
Your Turn to Play . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Further Ado . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
13
Chapter One
Functions
nce upon a time, a long time ago, on the western slopes of the
Siberian mountains there lived Fred's parents. However, they
weren't called Fred's parents since Fred hadn't been born yet.
But one day, more recently than a long
time ago, the stork delivered Fred. The lucky
couple, Mr. & Mrs. Gauss, discovered that they
were his parents.
At least Mrs. Gauss (rhymes with
“house”) thought she was a parent. Staring at
Fred, she chattered, “Oh, isn't our baby beautiful!”
Mr. Gauss frowned and said, “He doesn't look a bit like me.” Mrs.
Gauss didn't get the drift of what her husband was saying. She responded,
“Of course he doesn't. He's just a little baby, all red and wrinkly, and,
besides, he was very young when he was born.” Fred's father rolled back
his eyes, turned and left the room.
Mrs. Gauss carried him around a while and then, not knowing what
to do with him, put him back in his crib. She had high hopes that her little
tyke would grow up to be a country western singer. After she tucked him
in, she handed him a new toy. It was a box with three buttons on it. Each
button had an animal printed on it.
O
When Fred hit the button with the dog on it, the box sounded,
“Bow-wow!” When he tried the lion, “Roar!” The duck, “Quack!” This
was Fred's first encounter with the idea of function. He found out that
EVERY time he touched the
he heard, “Quack!”
15
Chapter One
Functions
Here is how he summarized in his head what he knew about his
animal-toy function:
1. There are two sets involved: the set of animal buttons
{dog, lion, duck} and the set of animal sounds {Bow-wow, Roar,
Quack}.
2. Every time I hit the lion I get a sound and it's always
the same sound.
Fred was fascinated by this idea of function. You start with two
sets and for each element of the first set there is exactly one element of the
second set which corresponded to it. Fred looked around his study (crib)
and invented a new function. His first set contained the things in his crib
and the second set was colors. He saw his sheet and that matched up with
“white” in the second set. His
matched up with “yellow.” The bars
on his crib also matched up with “yellow.”
Can two different things be “yellow”? Yes. The only critical thing
for the idea of a function is that each element in the first set have exactly
one image in the second set. It’s okay if two different elements in the first
set have the same image.
Fred thought to himself, “This is baby stuff! I’m three days old
and I should be able to think of a more sophisticated example of a
function.” He thought of his diaper which was in the shape of a triangle.
He labeled one of the acute angles with the letter A and created the
following function: “For any acute angle A, draw a diaper—I mean a
triangle—with
one of the acute angles being A.
A
Then,” Fred continued, “measure the length of the side opposite and divide
that length by the length of the hypotenuse.” When Fred set angle A equal
to 35º, the result of using his function (namely, drawing a triangle with a
35º angle and dividing the opposite side by the hypotenuse) gave him a
result of 0.5735764. Fred was very good at measuring lengths. He called
this function that he invented the sine function and he wrote
sin(35º) = 0.5735764.
16
Chapter One
Functions
But what if he had used a bigger triangle? Would the answer come
out differently? No. He knew he’d get the same answer every time since
any two right triangles with 35º angles would be similar (something he had
read in his geometry book on the previous day) and similar triangles are
triangles in which the sides are proportional.
Now since every element in the first set, which is the set of all
acute angles, has a unique image in the second set, Fred knew that he was
dealing with a genuine function.
When Mrs. Gauss came in to see how Fred was doing, she found
that he had drawn triangles all over his bed sheets.
As she looked down into Fred’s study and made little “goo-googoo” sounds at him, he said, “Mom, let’s play a little game. We’ll call it,
Guess the Function.”
Fred continued, “I’m thinking of a function which I’m going to
call 'f ' and I’m going to give you some examples and you try and guess
what the function is. Are you ready?”
Mrs. Gauss nodded but wasn’t sure what Fred was talking about.
Then Fred wrote on a sheet:
f(7) = 15
f(3) = 7
f(6) = 13
f(100) = 201
Mrs. Gauss looked at what he had written. She looked at it for a
long time. Finally she said, “Are you hungry?”
17
Chapter One
Functions
(From page 447)
Spherical Coordinates
In rectangular coordinates III dV was III dx dy dz. Great
for everyday use. A perennial favorite.
In cylindrical coordinates, it was III r dr d2 dz. That's a real
handy way to approach a piece of pie even when the outer edge is wavy
(when r is a function of 2).
But there are times when spherical coordinates is the way to go.
The tip-off is when you have some surface and all the points between that
surface and the origin. It's like an ice cream cone
where the sides of the cone may be fluted.
To locate a point in space in spherical
coordinates, first measure the distance from the
origin to the point. Call that distance D (rho).
That's the Greek letter
for r . I know it looks
like the Latin letter p
but you and I know that
p in Greek is B. To write a D, start at the
bottom. Here's
the movie:
(Some books call that distance r but
that's asking for trouble, since we use r in
cylindrical coordinates for a different distance.)
Next, the angle that D makes with the z axis is called N (phi).
Finally, 2 keeps the same meaning it had with cylindrical coordinates
(viz., the angle between the x axis and the line from the origin to the
projection of the point on the x-y plane).
So we have the point (D, N, 2).
447
Index
M 282
L @ F 462
Lu 318
6 (curvature) 103
) (“change in”) 34
( (Euler's constant) 425
T (angular speed) 164
1-1 correspondence A-359
between the natural numbers and the
rational numbers B-398
not between the natural numbers
and the real numbers B-398
two sets have the same number of
elements in them B-397, C-499
absolute maximum 60
absolutely convergent series 233
commutative law does hold C-519
acceleration 110, A-371
alternating series 232, A-385
absolutely convergent series A-385
conditionally convergent series A385
convergence 232
truncating A-385
angular speed 164
and related rates 165, 182
antiderivative 111
guess-and-by-gosh method A-383
of 1/x 174
substitutions A-383
Trig Substitution A-384
applications of integration 222, 223
arc length 221, 222
cardioid 231
in parametric 222
parametric form 137
rainbow 139
rectangular form 137
Archimedes B-462
area A-373
ellipse 135
parametric form 136
Tinker Creek 131
under a curve 171
area under a curve 222
in parametric 222, A-374
Aristotle C-492
asymptote 189, A-364
def of horizontal asymptote B-409
horizontal 66
oblique 189, B-426
vertical A-364
average value of a function 160, 223, A376
BASIC program
for approximating e B-421
for integration 138
for partial sums 200
Bernoulli's equation A-396
binomial formula B-424
boundary conditions 341
cap 64
cardioid 231, A-385
arc length 231
catenary 252
center of gravity 158, A-375
center of mass 158
centroid 158, 223, A-375
chain rule 58, A-362
proof B-404
with several intermediate variables
291, 292
change of variable 159, 214
chart
delta process 37, 44, 52, 81
derivative of 1/x 175
derivatives 58
favorite integration substitutions
235
implicit vs. explicit 96
limit of sine 26
Maclaurin series 243
Taylor series 245
three tests for minimums 78
vertical asymptote 72
circle 186
closed curves 332
closed interval 108
codomain 18
Comparison Test 202
concave 64
concavity 64, A-364
conditionally convergent series 233
commutative law doesn't hold C-518
Index-539
Index
cone
definition 189
filling 50, 51
conic sections 187, A-381
in polar B-437
pointed the "wrong way" 192
rotated 191, A-381
translated 191
conservative field 329
constant of integration 167
continuous function 30
contrapositive B-430
convergent 198, A-382
convex 64
cosh x 251, A-387
derivative 254
critical points 60
cross product B-465
curl F B-467
curvature 102, 103, A-370
formula 104
formula (in parametric form) 105
cycloid 96
parametric form B-411
cylindrical coordinate system 309
deductive logic B-430
definite integrals 167, A-372
evaluating 123
steps to setting them up 124
del 319
delta 34
delta process 36, 81
for a cubic 43
for log x 175
for x to the nth power 52
for y = 6 B-404
for y = sin x 81
Your Turn to Play examples A-361
density
in a solid 149
probability density function
176-178
variable along a length 121
derivative 64
arc trig functions 80
definition 51
dy/dx 51
Index-540
hyperbolic trig functions 254, A388
in parametric form 97
partial 282
sine by delta process 81
trig functions 79, 80, A-365, A-366
trig functions (applications) A-367,
A-368
variable in the exponent 178
differential equations 90, A-388
boundary condition 341
exact B-468
first-order linear A-395
homogeneous A-393
integrating factors B-468, B-470, B471, B-473, B-474
mixing problem 346
orthogonal trajectory 344
second order 350
separating the variables 255, 339
variables separable A-393
what lies ahead B-474
differential form 159, B-418
direction of maximum change 319
directional derivative 322
distributive
finite case 202
formulas for vectors 275
infinite case 202, B-433
div F B-462
divergence B-462
Divergence Theorem B-461
divergent 198
diving board (off the) 110, 111
domain of a function 25
dot product of vectors 271
formulas 275
double integral 300, A-390
finding area 304
finding weight of area with variable
density 305
moment of inertia 306
torque 306
Dr. Johnson 224
ds 137
in polar 229
dS (surface) B-455
Index
in polar B-459
dummy variable 258
d²y/dx² 64
e 175
definition B-420, B-421
eccentricity
in polar B-438
in rectangular B-428
ellipse 187
eccentricity B-427
foci B-427
in polar B-437
parametric form 135
energy of rotation 164
epsilon-delta B-400, B-401
arguments using B-402, B-403, B406, B-407, B-423, B-426, C476, C-485, C-491
equation of a plane through a point and
normal to a vector 320
exact differential equation B-468
potential function B-468
explicit relation 94, A-369
exponential functions A-377
extrema 55
first derivative test 78
first moment 153
first-order linear 347, A-395
flux of a vector through a surface B-453
focus 187
folium of Descartes 94
function of two variables 281
functions 16, B-397
1-1 A-359
codomain 18
continuous A-360
domain 18, A-358
image under a function 19
increasing bounded functions B-422
inverses A-359
onto 19
ordered pairs definition A-359
range 19
Fundamental Theorem of Calculus 124,
219
needs continuous function 140
proof B-415-417
Gauss, Carl Friedrich B-463
Gauss's Theorem B-461
general operating rule for doing
integration in polar form 231
geometric series 197
sum 198, B-429
when convergent 199
Godel 197
golden mean C-492
Goldilocks C-492
grad u 318
gradient 318, A-391, A-392, B-448
length of the vector 321
normal to a surface B-453
gravity (motion under) 110, 111
Greek alphabet 258
Green, George 331
Green's Theorem 331
Green's Theorem in Space B-461
guess a function 18, A-357
guess-and-by-gosh method 215, A-383
homogeneous equations A-393
horizontal asymptote 66, B-409
hydrostatic force 184, 223, A-380
hyperbola 188
asymptotes 189
eccentricity B-427
foci B-427
in polar B-437
hyperbolic trig functions 251
derivatives 254
formulas 251
Ice Cream Cone Problem B-457, B-458,
B-460, B-461, B-463
image 16
implications (if–then) B-430
contrapositive B-430
converse B-430
inverse 200, B-430
implicit differentiation 94
implicit relation 94, A-369
finding tangent 94
improper integral A-374
incomplete elliptic integral of the
second kind 139
indefinite integral 167
inductive reasoning B-431, B-432
Index-541
Index
infinite series A-382
"last term" 199
absolutely convergent series 233
conditionally convergent series 233
convergent 198
distributive law 202
divergent 198
harmonic series 202
integral test 203
Limit Comparison Test 202, B-432
Maclaurin series A-387
partial sums 200
Ratio Test 208
rules for convergence 199, 202,
203, 208
Taylor series 246, A-387
initial condition 340
initial conditions 341
instantaneous speed 36
integers B-425
Integral Test 203
integrals
change of variables 214
definite 167, A-372
favorite integration substitutions
235
improper A-374
in polar form (general operating
rule) 231
indefinite 167
limits A-372
line 325, A-392, A-393
power series A-386
substitutions A-386
table of all the applications 222, 223
Trig Substitution 215-217, A-384
integrand 139
integrating factors B-470, B-471, B473, B-474
little black book B-473
integration
changing the order B-460
over a surface B-453
integration by parts 162, A-377
proof 163
used twice 164
intermediate variables 291
Index-542
interval of convergence 239
differentiating or integrating within
240
isothermally 320
iterated integral 300
kinetic energy 164
L'Hospital's rule B-414
Lagrange multipliers 288, B-445
8 B-446
Law of Cooling 340
least upper bound B-423
Leibnitz 198
lemniscate 229
in polar 229
length of a vector 270
length of the gradient vector 321
lim sin 2 / 2 81
limaçon A-385
limit 29
definition (epsilon–delta) B-399
of a function 25
of a product B-407
of a sum B-406
one-sided 140
limit comparison test 202
proof B-432
line integral 325, A-392, B-448
ln x 175
local maximum 60
logarithmic differentiation A-378
logistics curve 345
logs 175, A-377
definition B-419
long-time stories 204-206, 208
M (torque) 153
Maclaurin series 243, A-387
mappings 18
maximums 54
absolute 60
along curves in a three-dimensional
surface 288
angle of truck sign 79
dog's play area 120
height 110
hemp plant yield 68
local 60, 281
on an interval A-363
Index
tests for surfaces in three
dimensions 283, 287
Mean Value Theorem 108, A-371
proof B-412
Rolle's Theorem (lemma) B-412
used for approximation 112
minimums 54
bubbles lost 77
fetching the beer 56
paper used 69
shortest route across the tundra 85
sound reaching Fred's ears 109
mixed partial derivative 284
moment of inertia 163, 223, A-377
of a banana 164
using double integrals 306
MVT 108, A-371
proof B-412
used for approximation 112
My = Nx 329
exact differential equation B-468
Napier, John C-510, C-517
natural logs 175
Newton 198, 340, B-462
normal to a surface B-453
numerical integration 257, A-388
memory aids 262
Simpson's rule 261
trapezoidal rule 260
one-sided limit 140
one-to-one A-359
open interval 108
p-series 204
Pappus 217
Pappus' Theorem 217
parabola 188
in polar B-437
parametric representation 96, A-369
partial derivative 282
partial fractions 236
a brush-up B-439, B-440
path-independent 334
point of inflection 65, A-364
point-slope equation of the line 45
polar coordinates 229, A-385
a brush-up B-434, B-436
double integration 307, 308
finding volume 307, 308
formulas B-436-438
position vector B-450
potential function 330, B-448
power series 237, A-386
interval of convergence 239
Maclaurin 243
Taylor 246
probability density function 176, 223,
A-377
product rule 58, A-362
proof B-404
Product Rule song 65
projection of a vector 271
length 272
pure second partial derivative 284
quotient rule 58, A-362
proof B-404
rabbit and the wall B-422
radian measure 80
definition B-409
dimensionless 165
radius of convergence 239
range 19
rational numbers B-425
rationalizing the numerator 255
rectangular rule 259
related rates 90, A-368, A-369, A-373
and angular velocity 165, 182
distance to the Christmas tree 92
of the gas cloud 91
of the squab 89
silo 133
surface area 134
Tody's body 133
relative maximum 59
for surfaces in three dimensions 287
Rolle's Theorem B-412
Root Test 208
saddle point 284, 285
scalar 267
scalar field 324
scalar function 318
scalar multiplication 267
Schröder-Bernstein Theorem B-399
secant line 42
second derivative test 64, 78
Index-543
Index
second moment 163, 223, A-377
second order differential equations
lacking the dependent variable 350
lacking the independent variable
352
separating the variables 255, 339
simple paths 332
Simpson's rule 261
sinh x 251, A-387
derivative 254
slope of a line B-403
slope of tangent line 42, 43
slopes of tangents to curves A-361
smooth paths 332
snake (weight) 121, 123
solid of revolution 213
speed 35
instantaneous 36
spherical coordinates B-447
volume B-448
Stokes's Theorem B-466, B-467
surface area A-384
in polar 229
of a paint can 219, 220
of revolution 218, 223
surface integral B-453, B-454, B-456,
B-458-460
tangent to a curve 41
curve in implicit form 94
Taylor series 246, A-387
remainder formula B-441
Tinker Creek 131
torque 152, A-374
of a bar 222
of a pb&j sandwich 161, 163
tractrix curve 255
transcendental numbers B-425
trapezoidal rule 260
Trig Substitution 217
triple integrals A-391
unit tangent vector B-450
unit vector 272
i, j and k 274
variable density 121
variables separable A-393
vector 267
an algebraic view B-442
Index-544
components B-444
dot product 271
formulas 274, 275
length 270
position vector B-450
projection 271, 273
scalar product 271
subtraction A-388
unit 272
vector product B-465
zero vector 274, A-389
vector addition 268
an algebraic view B-443
associative 269
formulas 274, 275
vector field 324
velocity
hitting the ground 110
limiting 252
velocity vector B-450
vertical asymptote 72
vertically simple C-516
volume
cylindrical shells method 214
of revolution 148, 151, 222
polar coordinates 307, 308
solid of revolution 213, 223, A-374
solid with constant height 298
solid with variable height 299, 303
spherical coordinates B-448
water pressure 184, 223, A-380
weight of area with variable density 305
weight of length with variable density
222
wire skating 220
work 222, A-374
along a curved path 324
compressing a spring 146
defined 144
lifting Toddy to Oz 144, 146
pumping water out 148, 222
x-y-z axes 280
y'' 64