TEACH YOURSELF VISUALLY - CALCULUS

Index denominators, 43–45, 47
derivatives. See also differentiation
A
alternate notations for, 70–71
acceleration function (a(t)), 57, 188, 247, 249 analyzing motion on objects with, 57
acceleration problems. See word problems analyzing rates of change with, 57, 72–73
algebraic functions, 99 defined, 1
algebraic substitution, 237–239 definition forms, 60–63
antiderivatives, 195–196 differentiability, 74–76
antidifferentiation, 195 of exponential functions, 125–128
approximations, 9 finding equation of lines tangent to curves with, 67
arcsins, 110, 232, 235 finding horizontal tangents with, 68–69
area finding points where relative maximums/minimums occur with, 56
finding slope of tangent lines with, 56, 58–59
bound regions, 205–208 formulas, 66, 97, 114, 119
curves, 9–13, 250–259 of logarithmic functions, 113–122
asymptotes. See horizontal asymptotes; vertical asymptotes Mean Value Theorem, 91–92
average radius, 277 optimizing word problems with, 57
Rolle’s Theorem, 89–90
B rules, 78–88, 93–95, 123–124
second, 71, 165, 200
balloon rate of change problem, 72–73, 139–140 of specific functions at specific numbers, 63–66
bound regions, 205–208, 250–259. See also solids of revolution of trigonometric functions, 96–111
bus company fare problem, 181–182 differential calculus, 1
differential equations, 240–245
C differentiation. See also derivatives
concavity, 165–168
Chain Rule, 96, 104–109 and continuity, 76
change of base property, 114 critical numbers, 146–147
change of variable technique, 216–217 defined, 55, 60, 65, 80
circumscribed rectangles, 9, 11 extrema, 155–164, 172–175
closed intervals, 31, 155–159 finding tangent lines to graphs of functions at points, 143
“combo” technique, 235 horizontal tangents, 144–145
common definition forms, 60–63 implicit, 129, 134–141
common denominators, 43–45 increasing/decreasing functions, 148–154
common logs, 119 inflection points, 168–171
complicated natural log expressions, 117 versus integration, 195–196
composite continuity property, 30 logarithmic, 129–133
composite functions, 96, 104–109 overview, 142–175
composite limit property, 24 rules of, 283–284
concavity, 165–168 when functions fail to have, 74–75
conditional functions, 18–19 word problems, 176–193
conical water tank problem, 183–184 direct substitution, 36–37
Constant Multiple Rule, 79 discontinuity, 28–29
Constant Rule, 78 disk method, 260–267
continuity, 26–31, 76 dividing by largest power of variables, 40–42
cosecant, 100–101 Double-Angle Identities, 286
cosine, 97–99, 202
cotangent, 100–101 E
critical numbers, 146–147
curves Ε, 14–16
e, 113
area, 9–13, 250–259 equations
tangents, 6–8, 67–69
cylindrical can construction problem, 179–181 differential, 240–245
graphs of, 250–251
D of lines tangent to curves, 67
written in implicit form, 134
∆–Ε definition of limits of functions, 14–16
decreasing functions. See increasing/decreasing functions 287
definite integrals, 13, 203, 205–208

explicit form, 134 overview, 48
exponential functions of rational functions, 49–52
horizontal rectangles, 256–259
continuous, 30 horizontal tangents, 68–69, 144–145
derivatives of, 125–128, 283
First Fundamental Theorem, 203–204 I
integral formulas, 201, 284
integrals of, 202, 220–222, 284 implicit differentiation, 129, 134–141
expressions implicit form, 134
logarithmic, 113–114, 117 increasing/decreasing functions, 148–154
rational, 37, 43–45 indefinite integrals, 197–200
extrema, 155–159, 160–164, 172–175. See also maximums/minimums indeterminate forms, 38–47, 93–95
Extreme Value Theorem, 34 infinite discontinuity, 29
infinite series, 2–3
F infinity, limits at, 48–54
inflection points, 146–147, 168–171
f, 14 initial conditions, 244
factor and reduce technique, 39–40 inner radius (r), 139, 269
first derivative test, 160 inscribed rectangles, 9–10
First Fundamental Theorem, 203–204 integrable, defined, 198
formulas. See derivatives; integrals integral calculus, 1
functions. See also specific functions by name integrals. See also integration

algebraic, 99 antiderivatives, 195–196
composite, 96, 104–109 definite, 205–208
conditional, 18–19 of exponential functions, 220–222
power, 201 First Fundamental Theorem, 203–204
formulas, 201–202, 226, 232, 284–285
G indefinite, 197–200
“look-alike”, 236
General Power Rule overview, 1, 194
and arcsins, 235 Second Fundamental Theorem, 209–210
with natural logarithmic functions, 116 that result in inverse trigonometric functions, 232–234
overview, 84–85, 214–219 that result in natural logarithmic functions, 223–225
with radical trigonometric functions, 101 of trigonometric functions, 226–231
integrands, 197, 235
geometric formulas, 66, 139 integration. See also antidifferentiation; integrals
graphs of functions algebraic substitution, 237–239
“combo” technique, 235
concavity for, 166–168 definition, 211
determining limits from, 20–22 differential equations, 240–245
finding maximums/minimums on, 56 versus differentiation, 195–196
finding tangent lines to at points, 143 finding area between curves, 250–259
with “holes”, 75 finding volume of solids of revolution, 260–282
with “jumps”, 75 General Power Rule, 214–219
polynomial, 144 limits of, 203, 217–218
with sharp turns, 74 overview, 211–245, 246
that have two horizontal asymptotes, 53–54 problems, 247–249
trigonometric, 145 Simple Power Rule, 212–214
with vertical tangent lines, 74 Intermediate Value Theorem, 32–33
when intersect once or more, 251–259 intervals, 31–33, 148–150, 155–159
inverse trigonometric functions. See trigonometric functions
H
J
h (height), 139
Half-Angle Identities, 286 “jumps”, 27, 29, 75
height (h), 139
highs, 155. See also maximums/minimums
“holes”, 27–28, 75, 268–269
horizontal asymptotes

functions whose graphs have two, 53–54
functions with x-axis as, 49

288

L negative sign, 141
negative velocity, 191
L, 14 nth derivatives, 71
ladder sliding down side of building problem, 140–141 numerators, rationalizing, 46
large radius (R), 272–273
L’Hôpital’s Rule, 93–95, 102–103, 123–124, 127–128 O
light pole and shadow problem, 185–187
limits one-sided limits, 17–19
open intervals, 31
calculating with algebraic methods, 35–54 optimization problems. See word problems
calculating with properties of, 23–25 outer radius (R), 266
continuity of functions, 26–31
determining from graphs of functions, 20–22 P
Extreme Value Theorem, 34
of functions, 4–5, 14–16 particle moving along straight line problem, 190–193
indeterminate forms, 93–95 polynomial functions
of integration, 203, 217–218
Intermediate Value Theorem, 32–33 continuous, 30
L’Hôpital’s Rule, 93–95, 123–124 critical numbers of, 146–147
one-sided, 17–19 direct substitution to find limits of, 36
overview, 1 extrema of, 156–157, 161–162, 172–174
Riemann Sums, 9–13 finding derivatives of, 63–64, 80
slopes of lines tangent to curves, 6–8 First Fundamental Theorem, 204
of sums of infinite series, 2–3 increasing/decreasing functions for, 149–151
terms of infinite series, 2 inflection points, 169–170
line tangent. See tangents integrals, 201
linear functions, 205–206 logarithmic functions of, 121
log of a power property, 114, 117, 122 one-sided limits for, 19
log of a product property, 113, 117 position function, 188
log of a quotient property, 114, 117 position problems. See word problems
logarithmic differentiation, 129–133 positive velocity, 191
logarithmic expressions, 113–114, 117 power functions, 201
logarithmic functions. See also natural logarithmic functions power limit property, 24
continuous, 30 Power Rule, 78
derivatives of, 119–122 powers, 24, 114–116, 201
differentiation, 283 problems. See word problems
integration, 284 product limit property, 23
L’Hôpital’s Rule and, 123–124 Product Rule, 81–83, 101
“look-alike” integrals, 236 properties of continuity, 30
lower approximations, 9 properties of limits, 23–25
lows, 155. See also maximums/minimums Pythagorean Identities, 99, 286

M Q

maximums/minimums quotient continuity property, 30
on closed intervals, 155 quotient limit property, 24
finding with critical numbers, 146 Quotient Rule, 86–87, 116
on graphs of functions, 56 quotients
relative, 56, 160–164, 172
word problems, 177 continuous, 30
derivatives of, 99, 116, 121–122
Mean Value Theorem, 91–92 limited, 24
motion on objects, analyzing, 57 of radical functions, 87–88
multiples, scalar, 30 of rational expressions, 37

N R

natural logarithmic functions r (inner/small radius), 139, 269
derivatives, 113–118, 119 R (outer/large radius), 272–273
integrals, 202, 223–225, 284
products involving, 151–152

289

radical functions sine, 97–99, 202
continuous, 30 slope, 6–8, 56, 58–59
critical numbers of, 147 small radius (r), 139, 272–273
derivatives of at specific numbers, 64–65 solids of revolution, finding volume of
derivatives of trigonometric, 101
direct substitution to find limits involving, 36 disk method, 260–267
finding derivatives of quotient of, 87–88 shell method, 275–282
inflection points, 170–171 washer method, 268–274
limits involving, 42 special trigonometric limit property, 24–25
logarithmic functions of, 122 Sum and Difference Identities, 286
natural logarithmic functions of, 115 sum of (• ), 3
sum or difference continuity property, 30
radius sum or difference limit property, 23
average, 277 Sum/Difference Rule, 80
inner/small, 139, 272–273 sums, 23, 30, 98
measuring, 266
outer/large, 272–273 T
representative disk, 261
t. See time
rates of change, 57, 72–73, 139–140 tangents
rational expressions, 37, 43–45
rational functions derivatives of, 97–99
to graphs of trigonometric functions, 143
continuous, 30 horizontal, 68–69, 144–145
horizontal asymptotes of, 49–52 slope of, 56, 58–59
indeterminate forms involving, 38 vertical, 74
limits of, 39–41 terms of infinite series, 2
one-sided limits for, 18 thickness, representative disk, 261
relative extrema of functions, 162–164 third derivatives, 71
rationalizing, 46–47 three-dimensional solids. See solids of revolution
reciprocals, 38 time (t), 137–139, 183–187
rectangles, 9–11, 250, 256–259 trigonometric functions
rectilinear motion problem, 190–191 continuous, 30
regions. See bound regions definite integrals, 206
related rates problems. See word problems derivatives of, 96–111, 283
relative extrema. See extrema evaluated at natural logarithmic functions, 118
relative maximums/minimums. See maximums/minimums extrema of, 158–159, 174–175
removable discontinuity, 28 First Fundamental Theorem, 203
representative disks, 261 graphs of, 143
representative rectangles, 250, 256–259 indeterminate forms involving, 38
reversed differentiation formulas. See integrals integrals, 202, 226–231, 285
revolved bound regions. See bound regions inverse, 110–111, 232–234, 284–285
Riemann Sums, 9–13 limits of, 22, 37, 40
rocket problem, 188–190 products of, 153–154
Rolle’s Theorem, 89–91 special, 25
rotations. See bound regions trigonometric identities, 99, 286

S U

• (sum of), 3 unit circles, 97, 285
scalar multiple continuity property, 30 unknown variables, 136–137
scalar product limit property, 23 upper approximations, 9
secant, 58, 100–101 u-substitution technique, 216–217, 237
second derivative test, 172–175
second derivatives. See derivatives V
Second Fundamental Theorem, 209–210
sharp turns, 74 V (volume), 139
shell method, 275–282 variables, 40–42, 136–137, 240–245
Simple Power Rule, 212–214

290

velocity, 191. See also word problems W
velocity functions, 57, 188
vertical asymptotes, 27, 29, 49, 51 washer method, 268–274
vertical tangent lines, 74 word problems
volume (V), 139
volume of box problem, 177–179 implicit differentiation, 139–141
volume of solids of revolution. See solids of revolution optimization, 57, 177–182
position, velocity, and acceleration, 188–193, 247–249
rate of change, 72–73
related rates, 183–187

291

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