Index

A

  • absolute extrema, 62, 69–72
  • acceleration, 84, 86, 89–90
  • antidifferentiation, 119–121, 128–135
  • arc length, 168–170
  • area between two curves, 160–162
  • area of rectangle, 10
  • area under a curve
    • approximating, 105–116
    • integration, 12–13, 103–105, 121–124
  • asymptotes
    • horizontal, 18–19, 30–31
    • vertical, 18, 171–173
  • average value of a function, 159

C

  • calculus, 5–8. See also differentiation; integration; limits
  • chain rule, 54–58
  • change, calculus as mathematics of, 6–8, 14, 33
  • closed interval, absolute extrema for, 69–71
  • coefficients, equating, 155–156
  • composite functions, differentiating, 54–58
  • concavity of function, 63, 66–69
  • concavity points, 73–75, 77
  • conjugate multiplication, 27–28
  • constant multiple rule, 50–51
  • constant rule, 49
  • continuous functions, 16, 17, 21–22, 26
  • convergence, improper integral, 172
  • cosecants, integration of, 147
  • cosines, integration of, 141–144
  • cotangents, integration of, 147
  • critical points, 62–65
  • curve. See also parabola
    • area between two curves, 160–162
    • area under, approximating, 105–116
    • area under, integration for, 12–13, 103–105, 121–124
    • derivative of, 39–40
    • length of, 13–14, 168–170
    • as locally straight, 11
    • slope of, 12–14, 34–35
    • zooming in on. See limits

D

  • decreasing derivative, 62–63
  • definite integral, 116–117
  • derivative. See differentiation
  • difference quotient, 40–46
  • difference rule, 51
  • differentiation
    • absolute extrema, 62, 69–72
    • acceleration, 84, 86, 89–90
    • antidifferentiation, 119–121, 128–135
    • chain rule, 54–58
    • of composite functions, 54–58
    • concavity points, 73–75, 77
    • constant multiple rule, 50–51
    • constant rule, 49
    • critical points, 62–65
    • of a curve, 39–40
    • decreasing derivative, 62–63
    • described, 8–9, 23, 34–39, 46
    • difference quotient, 40–46
    • difference rule, 51
    • of exponential functions, 52–53
    • First Derivative Test, 65–66
    • graph of, 75–78
    • implicit, 59–60
    • increasing derivative, 62–63
    • inflection points, 73–75, 77–78
    • of a line, 36
    • linear approximation, 97–100
    • of logarithmic functions, 52–53
    • not existing, 47–48
    • optimization problems, 81–83
    • power rule, 49–50
    • product rule, 53–54, 178
    • quotient rule, 54, 178
    • of radical functions, 50
    • as rate, 36–38
    • related rates, 91–96
    • Second Derivative Test, 66–69
    • as slope, 34–35
    • sum rule, 51
    • of trigonometric functions, 52
    • velocity, 84–88
  • disk method, 163–164
  • displacement, 87–88
  • distance traveled, 88–89
  • divergence of improper integral, 172
  • division by zero, 26, 175. See also undefined fraction

E

  • equating coefficients, 155–156
  • exponential functions, differentiating, 52–53
  • extrema, absolute, 62, 69–72
  • extrema, local, 62–69, 77, 86–87

F

  • factor pattern, 175
  • factoring, 27
  • First Derivative Test, 65–66
  • Fundamental Theorem, 124–128

G

  • graphs of derivatives, 75–78
  • guess-and-check method, 130–132

H

  • height, maximum and minimum, 86–87
  • hole in a function, limit for, 22–23
  • horizontal asymptotes, limits with, 18–19, 30–31

I

  • icons used in this book, 3
  • implicit differentiation, 59–60
  • improper integrals, 171–174
  • increasing derivative, 62–63
  • indefinite integral, 120–121
  • infinity
    • limits at, 18, 29–31, 171–172
    • limits not existing as, 18–19
    • over infinity, 31
  • inflection points, 73–75, 77
  • instantaneous rate, 46–47
  • instantaneous speed, 19–21
  • integration
    • antidifferentiation, 119–121, 128–135
    • approximations of, 105–116
    • arc length, 168–170
    • area between two curves, 160–162
    • area under a curve, 12–13, 103–105, 121–124
    • average value of a function, 159–160
    • definite integral, 116–117
    • described, 9–10, 101–103
    • equating coefficients, 155–156
    • Fundamental Theorem, 124–128
    • improper integrals, 171–174
    • indefinite integral, 120–121
    • Mean Value Theorem, 158–159
    • partial fractions method, 152–155
    • by parts, 137–140
    • rules for, 128
    • of trigonometric functions, 141–151
    • volumes of solids, 162–168

K

  • Kasube, Herbert E. (mathematician), 139

L

  • left sums, 105–108
  • length of curves, 13–14, 168–170
  • length of hypotenuse, 13–14
  • LIATE acronym, 139–140
  • limits
    • conjugate multiplication for, 27–28
    • of continuous functions, 16, 17, 21–22, 26
    • described, 11–14
    • factoring for, 27
    • of functions with holes, 22–23
    • with horizontal asymptotes, 18–19, 30–31
    • at infinity, 18, 29–31, 173–174
    • instantaneous speed using, 19–21
    • list of, common, 25–26
    • one-sided, 17
    • simplification for, 28–29
    • substitution for, 16, 27
    • two-sided, 15–17
    • with vertical asymptotes, 18
  • line
    • differentiation of, 36
    • secant, slope of, 41–42
    • slope of, 8, 35–36
    • tangent, absence of, 47
    • tangent, slope of, 41–44
  • linear approximation, 97–100
  • local extrema, 62–69, 75, 86–87
  • local maximums and minimums, 62
  • logarithmic functions, differentiating, 52–53

M

N

  • negative area, 105

O

  • one-sided limits, 17
  • optimization problems, 81–83

P

  • parabola, 39–40. See also curve
  • partial fractions method, 152–155
  • parts, integration by, 137–140
  • power rule, 49–50
  • product rule, 53–54, 176
  • Pythagorean identity, 142
  • Pythagorean theorem, 13–14

Q

R

  • radical functions, differentiating, 50
  • rate
    • area swept out under curve, 122–124
    • average, 46–47
    • derivative as, 36–38
    • instantaneous, 46–47
    • related rates, 91–96
    • relationship to slope, 38–39, 46–47
  • rationalizing, 27–28
  • related rates, 91–96
  • reverse rules, 128–130
  • Riemann sums, 112–116
  • right sums, 108–110
  • rise or run, 35–36
  • Russian matryoshka doll method, 166–168

S

  • secant line, slope of, 40–42
  • secants, integration of, 144–146, 151
  • Second Derivative Test, 66–69
  • sigma (summation) notation, 112–116
  • simplification, 28–29
  • sines, integration of, 141–144, 150–151
  • slope
    • of a curve, 12–13, 34–35
    • derivative as, 34–36
    • of a line, 8, 35–36
    • relationship to rate, 38–39, 46
    • of secant line, 41–42
    • of tangent line, 42–43
  • SohCahToa mnemonic, 175
  • solids, volumes of, 162–168
  • speed, 19–21, 86, 88–89
  • stationary points, 62
  • steepness. See slope
  • substitution
    • in antidifferentiation, 132–135
    • for limits, 16, 26
    • trigonometric, 147–151
  • sum rule, 51
  • summation (sigma) notation, 112–116

T

  • tangent line. See also linear approximation
    • absence of, 47
    • slope of, 42–44
  • tangents, integration of, 144–146, 148–150
  • trigonometric functions
    • differentiating, 52
    • integration of, 141–151
    • SohCahToa mnemonic for, 175
  • trigonometric substitution, 147–151
  • trigonometric values, common, 176
  • two-sided limits, 15–17

U

  • undefined fraction. See also division by zero
    • causing hole in function, 23
    • described, 175
    • resulting from substitution, 27

V

  • velocity, 84–86, 88
  • vertical asymptotes
    • improper integrals with, 171–173
    • limits with, 18
  • vertical inflection point, 47
  • volumes of solids, 162–168

W

  • washer method, 165–166

X

  • x-axis, 30
  • x-coordinate, point, 76
  • x-intercept, 77

Y

  • y-coordinate, point, 77
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