S-I-R model
SARS, 456
threshold value, 456
Δ, Delta, notation, 9
∫, 297
, 250
, 255
λ, Lagrange multiplier, 390
Σ, Sigma notation, 246
d/dx notation, 103
dy/dt = ky, 424
f′ (x) notation, 98
dy/dt = k(y − A), 431
acceleration, 106
adult and child tickets, 402
age distribution in US, 332
air pressure, 169
amplitude, 72
analytical solution, 458
ancestors, 481
annuity, 470
present value, 470
antiderivative, 292
constant multiples, 299
finding formula for, 297
graphical, 292
numerical, 292
of 1/x, 300
of ekx, 300
of xn, 298
of constant, 298
of cos (kx), 300
of cos x, 300
of sin (kx), 300
of sin x, 300
properties of, 299
sums, 299
apple orchard yield, 329
approximation, see estimation
area
and definite integral, 255
between two curves, 257
asymptote, 68
of exponential decay function, 40
definition of, 202
marginal cost and, 204
minimizing, 204
not marginal cost, 202
visualizing, 202
average rate of change, 16
average value of function, 272
visualizing on a graph, 272
average velocity, 20
baby boom, 217
Barbra Streisand, 479
base
of exponential, 39
of natural logarithm, 46
Bay of Fundy, 164
bell-shaped curve, 346
best fit line, 486
bicycles, 23
bladder, 295
bonds, 473
trading at a discount, 473
trading at a premium, 473
break-even point, 30
calculators
degree vs. radian mode, 72
windows for, 176
Captain Cook, 230
carbon dioxide level, 302
cardiac output, 375
carrying capacity, 114, 216, 412, 503
cartel pricing, 309
causation, 487
cephalexin, antibiotic, 478
chain rule, 150
derivative formulas and, 172
integration and, 316
change, 16
relative, 21
total
from rate of change, 260
visualizing on a graph, 18
Chernobyl, 58
chlorofluorocarbons, 7, 53, 112, 130
closed form for geometric series, 465
closed interval, 2
CO2
in pond water, 283
coal production in US, 263
Cobb-Douglas production function, 201, 361, 398
and Lagrange multipliers, 393
contour diagram of, 362
Cobb-Douglas short-run cost curve, 207
competition, 442
compliance, 109
composite function, 60
composition of functions, 60
compound interest, 53, 56, 86, 492
continuous compounding, 54, 425, 494, 495
concavity, 19
and inflection point, 183
and second derivative, 113, 293
power functions and, 141
consols, 479
constant function, derivative of, 99
constant of proportionality, 65, 424
constrained optimization, 387
Lagrangian function, 392
constraint, 388
consumer surplus, 306
as definite integral, 307
consumption smoothing, 275
continuity, 133
definition of, 134
numerically, 133
continuous dosage, 476
continuous growth/decay rate, 48, 51
formula for, 363
contour diagram
of Cobb-Douglas function, 362
partial derivative and, 372
reading, 358
contour line, 359
convergence
of geometric series, 467
of improper integral, 304
corn production in US, 360, 374
cornea, 367
Coroner's Rule of Thumb, 169
correlation, 486
causation and, 487
cosine function, 72
antiderivative of, 300
derivative of, 162
graph of, 72
definition of, 202
minimizing, 204
visualizing, 202
marginal, 28, 105, 119, 196, 204, 237, 302
definition of, 120
total, 28
and definite integral, 269
from marginal cost, 269
graph of, 118
cough suppressant, 429
coupon, 473
credit card debt, 498
credit multiplier, 473
critical value, 177
not always a maximum or minimum, 179
saddle point, 384
second derivative test for, 385
cross-section, 355
cumulative distribution function, 336
density function and, 337
probability and, 338
properties of, 337
curve
indifference, 394
level, 359
dairy industry, 309
data
linear, 11
decay factor, 42
decay rate, 42
decreasing, 10
definite integral, 250
and average value, 272
and consumer surplus, 307
and producer surplus, 307
and total cost, 269
by substitution, 319
changing limits of, 319
definition of, 250
estimate, 252
from graph, 251
from table of values, 251
multiples, 288
notation for, 260
overestimate, 252
properties, 288
sums, 288
underestimate, 252
units of, 260
degree of polynomial, 68
Delta, Δ, notation, 9
elasticity of, 208
definition, 208
cumulative distribution function and, 337
probability and, 338
properties of, 334
depreciation, 81
depreciation function, 31
derivative
as a function
formulas for, 100
at a point, 92
chain rule, 150
critical point, 177
estimating graphically, 93, 98
estimating numerically, 93, 94, 99
finding using definition, 134
from table of values, 94
function, 97
definition of, 98
graphical interpretation, 99
Leibniz notation for, 103
local maxima/minima
test for, 178
of composite functions, 150
of constant function, 138
of constant multiple, 139
of cosine function, 162
of exponential function, 145, 172
of linear function, 138
of periodic function, 161
of polynomial, 140
of product of functions, 156, 173
of quotient of functions, 158, 173
of sine function, 162
of sums and differences of functions, 139
of velocity, 106
partial, see partial derivative
product rule, 156
quotient rule, 158
second, see second derivative
second derivative
test, 179
slope of curve, 92
slope of tangent line, 92
units of, 104
visualizing, 92
dialysis, 444
difference quotient, 10
instantaneous rate of change and, 91
partial derivative and, 369
differentiable, 131
differential equations, 414
S-I-R model, 445
SARS, 456
analytical solution, 458
and implicit functions, 420
equilibrium solution, 430, 434
existence of solutions, 421
family of solution curves, 419
formula for solution, 415
graphical solution, 419
initial conditions, 415
initial-value problem, 416
logistic, 412
Newton's Law of Cooling (Heating), 435
numerical solution, 414
particular solution, 416
predator-prey model, 439
separation of variables, 458
slope fields and, 418
solution, 414
uniqueness of solutions, 421
discrete doses, 476
disease, 445
and S-I-R model, 445
distance
estimating from velocity, 242
visualizing on velocity graph, 242
distribution function
cumulative, 336
distribution of resources, 328
divergence
of geometric series, 467
of improper integral, 304
dollar bills, 473
dose-response curve, 218
double angle formulas, 321
doubling time, 53
Rule of Seventy, 54
drug
repeated doses, 464
drug concentration, 355
drug concentration curve, 223, 262
drug desensitization, 482
duality, production and cost, 402
Dubois formula, 83
Ebbinghaus model, 438
economy of scale, 118
economy, expanding, 481
effective annual rate, 493
cross-price, of demand, 213
income, of demand, 213
of cost, 213
of demand, 208
definition, 208
elementary functions, 316
endocrinologist, 266
energy
electrical, 276
and S-I-R model, 445
equilibrium point, 32
equilibrium price/quantity, 32, 306
equilibrium solution
of differential equation, 430, 434
stable, 434
unstable, 434
erythromycin, 225
estimation
of definite integral, 251
of derivative at a point, 93, 94
of derivative function, 98, 99
of function values using linear approximation, 106
of function values using partial derivative, 371
of partial derivative, 370, 372
Ethanol fuel, 59
half-life, 53
as solution to differential equation, 418, 458
domain, 41
two forms of, 48
exponential growth, 39, 40, 424, 458
doubling time, 53
linear and, 497
exponential regression, 487
exponential rule, 146
danger of, 485
extrema
local, 177
family of functions, 12
as solution of differential equation, 418
exponential, 42
Gompertz growth, 235
linear, 12
finite geometric series, 465
sum of, 466
firebreaks, 237
first derivative
test, 178
fiscal policy multiplier, 109
flipped classroom, ix
fluoxetine, antidepressant, 478
fog, 357
fox density in England, 397
function, 2
average rate of change, 16
change in, 16
Cobb-Douglas, 201, see Cobb-Douglas production function
composite, 60
composition of, 60
constant
derivative of, 99
cosine
graph of, 72
cumulative distribution, see cumulative distribution function
density, see density function
depreciation, 31
derivative at a point, 92
differentiable, 131
economic applications, 28
elementary, 316
exponential decay, 40
graph of, 2
implicit, 420
input, 2
inside, 60
joint cost, 393
Lagrangian, 392
local extrema of, 177
local maxima/minima of, 177
logistic, 213
notation, 3
of two variables, 354
output, 2
outside, 60
periodic, 71
power, 66
graph of, 67
profit, 30
proportional, 65
quadratic, 68
representation, 2
shifting, 62
sine
graph of, 72
stretching, 61
Fundamental Theorem of Calculus, 268, 302
Second, 288
continuous compounding, 56
formula for, 56
of income stream, 312
gains from trade, 306
GDP, 169
genetics
population, 455
geometric series, 500
finite, 465
sum of, 466
infinite, 466
converge, 467
diverge, 467
sum of, 467
partial sum, 467
Gini's index of inequality, 329
global extrema, 381
global maxima/minima, 381, 388
gold, 109
Gompertz equation, 423
Gompertz growth, 235
Grand Canyon flooding, 283
graphing
inappropriate window, 176
Greenland Ice Sheet, 112
growth curve
logistic, 412
growth model
inhibited, 213
growth of children, 266
growth rate, 42
percent, 42
Harley-Davidson, 315
Harrod-Hicks model, 481
harvesting and logistic growth, 455
heater in room, 401
height velocity graph, 266
hematocrit, 193
histogram, 332
HIV, 58
horizontal line, 10
hybrid cars, 249
hyperinflation, 496
implicit function, 420
as solution to differential equation, 420
improper integral, 304
convergence/divergence of, 304
income stream, 312
increasing, 10
increasing function, 4, 9, 354
indefinite integral, 297
finding formula for, 297
properties of, 299
infinite geometric series, 466
converge, 467
diverge, 467
partial sum, 467
sum of, 467
inflation, 27
inflection point
definition of, 183
information
spread of, 504
inhibited growth model, 213
input into a function, 2
inside function, 60
instantaneous
integral
definite, see definite integral
definite vs. indefinite, 297
improper, see improper integral
indefinite, see indefinite integral
integrand, 250
integration
limits of, 250
reduction formula, 326
techniques
by parts, 321
interest
formula for, 494
continuously compounded, 54, 425, 494, 495
effective annual rate, 493
nominal rate, 493
the number e, 495
internal rate of return, 502
interpolation, 485
interval notation, 2
inversely proportional, 66
investment comparisons, 501
ion channel, 221
island species, 407
isotherms, 358
joint cost function, 393
Keeling Curve, 170
kidney function and antibiotics, 367
Lagrange multiplier, 389
meaning of, 391
Lagrangian function, 392
lambda, λ, see Lagrange multiplier
lapse rate, 109
leading term, 68
leaf decomposition, 452
left-hand sum, 245
Leibniz, Gottfried Wilhelm (1646–1716), 103, 461
level
curve, 359
set, 359
limit
definite integral as, 250
intuitive notion of, 91
meaning of, 132
limits of integration, 250
substitution and, 319
line
best fit, 486
contour, 359
equation of, 10
in fitting formula to data, 484
least-squares regression, 403, 486
linear approximation, 107, 372
linear data, 11
derivative of, 138
intercept of, 10
slope of, 10
linear regression, 484
correlation, 486
least-squares, 486
loading curve, 234
loan, 499
local extrema, 381
local linearity, 104, 107, 372
local maxima/minima, 177, 381, 388
partial derivative and, 383
test for, 177
logarithm
base e, 46
natural, 46
properties of, 47
logistic differential equation, 233, 412
carrying capacity, 216
definition of, 215
exponential and, 503
point of diminishing returns, 216
population growth and, 114
logistic growth curve, 412
harvesting, 455
logistic regression, 215
Lotka-Volterra equations, 439
lottery winnings, 501
lung, 109
Madrid, Spain, 275
malaria, 464
Malthus, Thomas (1766–1834), 497
marginal
cost, 28, 31, 105, 119, 196, 204, see cost, marginal
costs/revenues, 302
product of labor, 201
market stabilization, 472
Mauna Loa Observatory, 77, 170
maxima/minima
concavity and, 179
on (a, b), 189
on [a, b], 189
first derivative test, 178
second derivative test, 179, 385
mean, 344
graphical interpretation of, 345
of normal distribution, 346
median, 343
cumulative distribution function, 344
density function, 344
meteorology, 109
method of Lagrange multipliers, 388
model, Harrod-Hicks, 481
model, mathematical, 484
money supply, 381
Monod growth curve, 161
monotonic, 181
Montgolfier brothers, 267
mortgage, 500
multiplier effect, 471
multiplier, fiscal policy, 109
multiplier, Lagrange, 389
natural gas reserves, 478
natural logarithm, 46
graph of, 47
properties of, 47
solving equations and, 47
new or used?, 502
Newton, Isaac (1642-1727), 131
Law of Cooling (Heating), 167, 435
nicotine, 57, 221, 225, 438, 469, 478
nominal rate, 493
normal distribution, 346
standard, 347
notation
Δ, Delta, 9
d/dx, 103
f′(x), 98
definite integral, 260
function, 3
partial derivative, 370
Sigma, Σ, 246
numerical derivatives, 99
objective function, 388
oil consumption
Okun's law, 15
open interval, 2
optimization, 189
constrained, 387
functions of many variables, 381
maximizing/minimizing averages, 190
output of a function, 2
outside function, 60
ozone depleting gas index, 112
μ and σ of normal distribution, 346
of logistic function, 215
of surge function, 222
partial derivative, 369
contour diagram and, 372
definition of, 370
difference quotient and, 369
estimating function values with, 371
formula for, 376
graphical, 372
Leibniz notation for, 370
local maxima/minima, 383
mixed, 380
rate of change and, 370
second order, 378
units of, 373
partial sum of a geometric series, 467
parts
integration by, 321
PCB pollution, 17
pendulum, 166
period of, 66
penicillin, 226
pennies, 472
percent growth rate, 42
period, 72
period of pendulum, 66
periodic function, 71
amplitude, 72
parameter, 73
period, 72
pH, 489
trajectories, 441
point of diminishing returns, 216
point-slope form, 10
polynomial, 68
degree, 68
derivative of, 140
leading coefficient, 68
leading term, 68
quadratic, 68
population
of China, 51
of Hungary, 149
of Kenya, 52
of McAllen, Texas, 273
of Mexico, 150
of Nevada, 147
of Tokyo, 270
of US, 58, 96, 150, 168, 407, 487, 489
baby boom, 217
exponential model 1790-1860, 214
logistic model 1790-1940, 215
logistic model 1790-2010, 216
of world, 25, 43, 44, 58, 59, 149, 165, 248, 263, 275, 325, 490
outstrips food supply, 497
population center of US, 87
population genetics, 455
population growth, 213
logistic model, 114
Portland tides, 74
power function, 66
concavity of, 141
graph of, 67
zero or negative powers, 68
power rule, 140
predator-prey model, 439
equilibrium values, 441
present value, 55, 313, 479, 501
annuity, 470
continuously compounded, 56
formula for, 56
of income stream, 312
pressure
air, 169
price control, 309
principal, 473
probability
cumulative distribution function and, 338
density function and, 338
histogram, 332
triangular distribution, 351
producer surplus, 306
as definite integral, 307
product rule, 156
derivative formulas and, 173
production
and price of raw materials, 238
production function, 201
Cobb-Douglas, 201
profit, 30
profit function, 30
proportional, 65
inversely, 66
pulmonologist, 109
Quabbin Reservoir, 328
quadratic function, 68
quarantine, 456
quotient rule, 158
derivative formulas and, 173
range, 2
average, 16
average and instantaneous, 91
rate, effective annual, 493
rats and formaldehyde, 370, 375
real wage, 202
reduction formula, 326
regression
exponential, 487
least-squares, 486
correlation, 486
slope of, 486
linear, 484
logarithmic, 487
logistic, 215
relative change, 21
relative growth rate, 42, 503, 504
relative rate, 41
and logarithms, 154
relative rate of change, 107
rent controls, 311
repeating decimal, 469
revenue
elasticity and, 210
definition of, 120
total, 29
graph of, 119
Ricker curve, 193
Riemann sum, 250
and consumer surplus, 307
and producer surplus, 307
right-hand sum, 245
rise, 10
robins and worms, 439
roller blades, 374
Rule of Seventy, 54
run, 10
saddle point, 384
sales predictions, 217
sales tax, 33
secant line, 18
second derivative, 112
interpretation of, 114
maxima/minima, 179
second derivative test, 179, 385
Second Fundamental Theorem of Calculus, 288
second-order partial derivative, 378
sensitivity of a drug, 230
separation of variables, 458
exponential differential equation, 458
justification, 460
Sigma, Σ notation, 246
sine function, 72
antiderivative, 300
derivative of, 162
graph of, 72
Sioux Lakes and EPA, 279
slope
of graph at a point, 92
of line, 10
units of, 10
slope field, 418
solar panels, 27
Solow model, 414
Soviet-German pipeline, 326
species density of birds, 365
species interaction, 442
species on islands, 407
speed of sound, 168
speed vs. velocity, 20
stable equilibrium solution, 434
standard deviation of normal distribution, 346
Streisand, Barbra, 479
strontium-90, 58
substitution
integration by, 316
into definite integrals, 319
into indefinite integrals, 316
sum
left-hand, 245
Riemann, 250
right-hand, 245
summation notation, 246
surplus, consumer/producer, 306
sustainable yield, 455
symbiosis, 442
Syracuse snowfall, 2
table, 2
tangent line, 92
Tangent Line Approximation, 104, 107
tax
sales, 33
specific, 33
tax cut, 479
temperature and illumination, 129
theophylline, 349
theophylline and asthma, 438
topographical maps, 359
total change
total cost, 28
and definite integral, 269
from marginal cost, 269
total revenue, 29
total utility, 117
trajectories, 441
trigonometric identity
double angle, 321
tuition payments, 502
units
of average value, 272
of definite integral, 260
of derivative, 104
of partial derivative, 373
of slope, 10
unstable equilibrium solution, 434
urologist, 295
utility, 368
vaccination, 448
valproic acid and epilepsy, 428, 474
variable
continuous, 2
discrete, 2
discrete vs. continuous, 9
variable cost per unit, 28
velocity
average, 20
vs. speed, 20
Verhulst, Pierre-François (1804–1849), 503
Vermeer (1632–1675), 58
volcanic explosion, 357
wage control, 309
warfarin, anticoagulant, 430, 451
weather map, 358
wind power, 46
winning probability, 481
Zeno (5th century, B.C.), 131
zero, 3
3.138.35.255