INDEX

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

e, 42, 495

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

ampicillin, 225, 474, 477

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 e^kx, 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

average cost, 202, 204

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

bioavailability, 262, 323

biodiesel fuel, 27, 45

bladder, 295

bonds, 473

trading at a discount, 473

trading at a premium, 473

break-even point, 30

budget constraint, 34, 387

caffeine, 57, 413

calculators

degree vs. radian mode, 72

windows for, 176

cancer rates, 7, 80

Captain Cook, 230

carbon dioxide level, 302

carbon-14, 82, 166, 451

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

concave up/down, 19, 113

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

on an interval, 133, 134

continuous dosage, 476

continuous growth/decay rate, 48, 51

continuous variable, 2, 9

contour, 358, 359

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

coefficient of, 406, 486

cosine function, 72

antiderivative of, 300

derivative of, 162

graph of, 72

cost

average, 202, 204, 237

definition of, 202

minimizing, 204

visualizing, 202

fixed, 28, 118, 302

marginal, 28, 105, 119, 196, 204, 237, 302

definition of, 120

total, 28

and definite integral, 269

from marginal cost, 269

variable, 28, 269

cost function, 28, 118

graph of, 118

cough suppressant, 429

coupon, 473

credit card debt, 498

credit multiplier, 473

critical point, 177, 383

critical value, 177

how to find, 177, 383

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

demand, 31, 306

indifference, 394

level, 359

supply, 31, 306

dairy industry, 309

data

linear, 11

decay factor, 42

decay rate, 42

continuous, 48, 51

decreasing, 10

decreasing function, 4, 354

derivative of, 99, 113, 176

definite integral, 250

and average value, 272

and consumer surplus, 307

and producer surplus, 307

and total cost, 269

as area, 255, 256

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

demand, 31, 306

elasticity of, 208

definition, 208

demand curve, 31, 306

density function, 332, 333

cumulative distribution function and, 337

probability and, 338

properties of, 334

dependent variable, 2, 354

depreciation, 81

depreciation function, 31

derivative

as a function

formulas for, 100

at a point, 92

chain rule, 150

critical point, 177

definition of, 92, 131, 171

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 a^x, 146, 172

of e^x, 146, 171

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 ln x, 147, 172

of periodic function, 161

of polynomial, 140

of power function, 139, 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

exponential, 424, 458

family of solution curves, 419

formula for solution, 415

general solution, 416, 459

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

discount rate, 56, 501

discrete doses, 476

discrete variable, 2, 9

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

domain, 2, 354

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

elasticity, 22, 208

cross-price, of demand, 213

income, of demand, 213

of cost, 213

of demand, 208

definition, 208

revenue and, 209, 210

elementary functions, 316

endocrinologist, 266

energy

electrical, 276

epidemic, 445, 457

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

exponential decay, 40, 424

half-life, 53

exponential function, 39–41

as solution to differential equation, 418, 458

derivative of, 145, 172

domain, 41

formula for, 40, 42

two forms of, 48

exponential growth, 39, 40, 424, 458

doubling time, 53

linear and, 497

exponential regression, 487

exponential rule, 146

extrapolation, 9, 485

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

fixed cost, 28, 118, 302

flipped classroom, ix

flu, 1918 epidemic, 449, 504

fluoxetine, antidepressant, 478

fog, 357

Four, Rule of, ix, 2

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

continuous, 133, 134

cosine

graph of, 72

cost, 28, 118

critical point of, 177, 383

cumulative distribution, see cumulative distribution function

decreasing, 4, 354

derivative of, 99, 113

density, see density function

depreciation, 31

derivative at a point, 92

differentiable, 131

economic applications, 28

elementary, 316

exponential, 39–41, 424

exponential decay, 40

graph of, 2

implicit, 420

increasing, 4, 9, 354

derivative of, 99, 113

input, 2

inside, 60

joint cost, 393

Lagrangian, 392

linear, 8, 41

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

revenue, 29, 118

shifting, 62

sine

graph of, 72

stretching, 61

surge, 221, 223

table of values of, 2, 354

Fundamental Theorem of Calculus, 268, 302

Second, 288

future value, 55, 313

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

Great Lakes, 426, 429

Greenland Ice Sheet, 112

growth curve

logistic, 412

growth factor, 39, 42

growth model

inhibited, 213

growth of children, 266

growth rate, 42

continuous, 48, 51

percent, 42

relative, 42, 503, 504

half-life, 53, 428

Harley-Davidson, 315

Harrod-Hicks model, 481

harvesting and logistic growth, 455

heat index, 357, 362, 374

heater in room, 401

height velocity graph, 266

hematocrit, 193

histogram, 332

HIV, 58

horizontal line, 10

hybrid cars, 249

hydroelectric power, 27, 46

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

derivative of, 99, 113, 176

indefinite integral, 297

finding formula for, 297

properties of, 299

independent variable, 2, 354

indifference curve, 368, 394

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

rate of change, 91, 369

velocity, 90, 91

integral

as total change, 260, 268

definite, see definite integral

definite vs. indefinite, 297

improper, see improper integral

indefinite, see indefinite integral

integrand, 250

integration

by substitution, 316, 319

limits of, 250

reduction formula, 326

techniques

by parts, 321

intercept, 3, 9

interest

compound, 53, 56, 492

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

Kleiber's Law, 69, 144

koalas, 58, 213

Lagrange multiplier, 389

meaning of, 391

Lagrangian function, 392

lambda, λ, see Lagrange multiplier

lapse rate, 109

leading term, 68

leaf decomposition, 452

least-squares line, 403, 486

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

regression, 403, 486

linear approximation, 107, 372

linear data, 11

linear function, 8, 10, 41

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

derivative of, 147, 172

natural, 46

properties of, 47

logistic differential equation, 233, 412

logistic function, 114, 213

carrying capacity, 216

definition of, 215

exponential and, 503

point of diminishing returns, 216

population growth and, 114

logistic growth curve, 412

logistic model, 503, 504

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

analysis, 119, 195

cost, 28, 31, 105, 119, 196, 204, see cost, marginal

costs/revenues, 302

product of labor, 201

profit, 31, 194

revenue, 31, 119, 196

market stabilization, 472

Mauna Loa Observatory, 77, 170

maxima/minima

concavity and, 179

global, 189, 381

on (a, b), 189

on [a, b], 189

local, 177, 381

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

morphine, 413, 452, 477

mortgage, 500

multiplier effect, 471

multiplier, fiscal policy, 109

multiplier, Lagrange, 389

murder, 436, 439

natural gas reserves, 478

natural logarithm, 46

derivative of, 147, 172

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

number e, 42, 495

numerical derivatives, 99

objective function, 388

oil consumption

world, 247, 279, 477

oil reserves, 476, 477

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, 130, 429

ozone depleting gas index, 112

parameter, 12, 73

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

phase plane, 440, 446

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 India, 51, 150

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

exponential, 39, 425

logistic model, 114

Portland tides, 74

power function, 66

concavity of, 141

derivative of, 139, 140

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

maximizing, 194, 196

profit function, 30

proportional, 65

inversely, 66

pulmonologist, 109

Quabbin Reservoir, 328

quadratic function, 68

quarantine, 456

quinine, 43, 464, 469

quotient rule, 158

derivative formulas and, 173

range, 2

rate of change, 10, 370

average, 16

average and instantaneous, 91

instantaneous, 91, 369

relative, 107, 504

rate, effective annual, 493

rats and formaldehyde, 370, 375

real wage, 202

reduction formula, 326

regression

exponential, 487

least-squares, 486

line, 403, 484

correlation, 486

slope of, 486

linear, 484

logarithmic, 487

logistic, 215

quadratic, 487, 488

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

marginal, 119, 196

definition of, 120

maximizing, 198, 209

total, 29

revenue function, 29, 118

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 Four, ix, 2

Rule of Seventy, 54

run, 10

S-I-R model, 445, 504

saddle point, 384

sales predictions, 217

sales tax, 33

SARS, 220, 456

secant line, 18

second derivative, 112

and concavity, 113, 293

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

supply, 31, 306

supply curve, 31, 306

surge function, 221, 223

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

threshold value, 448, 456

topographical maps, 359

total change

from rate of change, 244, 260

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

dependent, 2, 354

discrete, 2

discrete vs. continuous, 9

independent, 2, 354

variable cost, 28, 269

variable cost per unit, 28

velocity

average, 20

instantaneous, 90, 91

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

Wikipedia, 57, 413

wind chill, 366, 395, 397

wind power, 46

winning probability, 481

zebra mussels, 44, 143

Zeno (5^th century, B.C.), 131

zero, 3