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graphs of, 190
key facts, 191 Maple, 256
inverses, some functions do not have, 43 Mathematica, 256
maxima and minima, applied, 88
maximum, derivative vanishing at, 77
Leibniz, Gottfried, 108
maximum/minimum problems, 86
lвЂ™HГґpitalвЂ™s Rule, 123вЂ“127
minimum, derivative vanishing at, 87
limit
money, depreciation of, 144
as anticipated value rather than actual
value, 59 motion, 1
-Оґ deп¬Ѓnition of, 57
informal deп¬Ѓnition of, 57 natural logarithm as log to the base e, 163
non-existence of, 62 natural numbers, 1
rigorous deп¬Ѓnition of, 57 Newton, Isaac, 108
uniqueness of, 62 non-repeating decimal expansion, 2
limits, 57 numerical approximation, 253
of integration, 106
one-sided, 60 open interval, 3
properties of, 61 opposite side of a triangle, 26
line
equation of, 13 parabola, 15, 18
key idea for п¬Ѓnding the equation of, 15 parallel lines have equal slopes, 12
point-slope form for, 13 partial fractions
two-point form for, 14 products of linear factors, 203
lines, graphs of, 7 quadratic factors, 206
loci in the plane, 15 repeated linear factors, 205
locus period of a trigonometric function, 25
of points, 39 perpendicular lines have negative reciprocal
plotting of, 7 slopes, 12
logarithm pinching theorem, 62
basic facts, 49 points in the plane, plotting, 5
formal deп¬Ѓnition of, 148 points in the reals, plotting, 3
graph of, 151 polynomial functions, 147
natural, 49, 149 powers, derivatives of, 167
of the absolute value, 152 principal angle, associated, 25
342 Index

quotient, writing a product as, 128 Tangent function, 185
tangent function, 26
range of a function, 31 calculation of, 69
rate of change and slope of tangent line, 70 slope of, 67
rates of change, 1 terminal point for an angle, 22
rational numbers, 2 transcendental functions, 147
real numbers, 2 trapezoid rule, 252, 254
reciprocals error in, 254
of linear functions, integrals of, 202 trigonometric expressions, integrals of, 210
of quadratic expressions, integrals of, 202, trigonometric functions
rectangles, method of, 253 fundamental identities, 29
related rates, 91 inverse, 180
repeating decimal expansion, 2 table of values, 28
Riemann sum, 104 trigonometric identities, useful,
rise over run, 10 210
trigonometry, 19
secant function, 26 classical formulation of, 25
set builder notation, 3
sets, 30 union of sets, 30
SimpsonвЂ™s rule, 256, 257 unit circle, 19
error in, 257 u-substitution, 207
sine and cosine, fundamental properties of,
23 vertical line test for a function, 35
odd powers of, 211 volume
Sine function, 182 by slicing, 219
sine function, principal, 182 calculation of, 217
sine of an angle, 22 of solids of revolution, 224
slope
washers, method of, 225
of a line, 8
water
undeп¬Ѓned for vertical line, 12
pumping, 236
springs, 234
weight of, 249
substitution, method of, 207
work, 233
surface area, 243
calculation of, 234
calculation of, 245

Steven G. Krantz is the Chairman of the Mathematics Department at Washington
University in St. Louis. An award-winning teacher and author, Dr. Krantz has
written more than 30 books on mathematics, including a best-seller.

343

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