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```1.  Functions  and  G raphs  ......................................  1
1.1 Functions and Graphs ................. .................. 1
1.2 Domain and Range ................. ................... 2
1.3 Plotting Graphs using MAPLE ........................... . 5
1.4 Odd and Even Functions . ................................ 7
1.5 Composite Functions ................. ................... 9
1.6  Some Elementary Functions  ....  ...........................  11
1.6.1 Polynomials ....................................... 11
1.6.2  Rational Functions  ............................ . .   .  17
1.6.3 The Modulus Function ............................. 18
1.6.4  Trigonometric Functions  .............. ..  ..........  20
1.6.5 Exponential and Logarithmic Functions ............... 23
1.6.6  Hyperbolic Functions  .............................  .  26
1.6.7 Trigonometric and Hyperbolic Identities .............. 29
1.7  Inverse  Functions . ......... ..... ............. ...........  29
1.7.1  Increasing and Decreasing Functions ..................  33
1.7.2 Inverse Trigonometric Functions............... ....  35
1.7.3  Inverse Hyperbolic Functions  ........................  39
1.8  Piecewise Functions. ..................  ...................  40
2. Limits of Functions ...................................... 47
2.1 What are Limits? ....................................... 47
2.2 One-sided Limits ...................................... 52
2.3  Infinite Limits and Limits at Infinity ........................  53
2.4 Algebraic Rules for Limits . .............................. 57
2.5 Techniques for Finding Limits ............................ . 58
2.5.1  Squeezing  .......... .. ............ .... ...... .  59
2.5.2  Algebraic Manipulation  ...........................  .  61
2.5.3  Change of Variable  ................... ............  67
2.5.4  L'H8pital's  Rule....................................  68
2.6 An Interesting Example . ................................ 71
2.7  Limits using  M APLE  ........................ ...........  72
2.8 Limits with Two Variables .............................. . 72
3.  Differentiation  .......... .................... .............  79
3.1 The Limit Definition ........ ........................... 79
3.2  Using  the  Lim it Definition  .................................  81
3.3  Basic Rules of Differentiation  ............................  .  84
3.4  The  Chain  Rule  .........................................  85
3.5  Higher Derivatives........................................  88
3.6  Differentiation using MAPLE  .............................  90
4. Techniques of Differentiation .................. . ....... 93
4.1 Implicit Differentiation . ................................. 93
4.2  Logarithmic Differentiation  ........................... . .  .  96
4.3  Parametric Differentiation  ............................ . .  .  98
4.4  Differentiating Inverse Functions  ................... ...... . 100
4.5 Leibniz Theorem .................  ........................103
5. Applications of Differentiation ............................. 111
5.1 Gradients and Tangents ............................... 111
5.2 Maxima and Minima .................................... 113
5.3 Optimisation Problems ................................ 118
5.4  The Newton-Raphson Method  ...........................  . 121
5.5 Motion in a Straight Line. ............................... 123
5.6 Growth and Decay.................................... 127
6. Maclaurin and Taylor Expansions ........................... 133
6.1 Linear Approximation ................  .................. 133
6.2 The Mean Value Theorem .............................. 135
6.4 Taylor Polynomials ....... ............................ . 141
6.5 Taylor's Theorem.. ..................................... 146
6.6 Using MAPLE for Taylor Series .............. ........ .. .148
7. Integration ..............................................153
7.1  Integration  as Summation  ............................ . .   . 153
7.2  Som e  Basic  Integrals ...................................... 155'
7.3 The Logarithmic Integral . .............................. 160
7.4  Integrals with Variable Limits ......................... . .  . 161
7.5 Infinite Integrals ......................................163
7.6 Improper Integrals......................................168
8.  Integration  by  Parts .......................................173
8.1  The Basic Technique ......................... ...........  173
8.2 Reduction Formulae ................ ................... 175
8.3  Integration  using  MAPLE  ................................. 178
8.4 The Gamma Function ...............   .................. 179
8.5  A  Strange Example ...................................... 181
9. Integration by Substitution . ............................. 185
9.1 Some Simple Substitutions. .............................. 186
9.2 Inverse Substitutions ...............  ................... 188
9.3 Square Roots of Quadratics ........................ .. .189
9.4  Rational Functions of cos and sin ........................... 195
9.5  Substitution  using MAPLE  ............................. . 197
10. Integration  of Rational Functions ........................... 201
10.1 Introduction ..........................................201
10.2  Partial Fractions  ........................................ 202
10.3 The Integration Process ...............  ................ 207
10.4  Examples ..............................................210
11. Geometrical Applications of Integration .................... 217
11.1 Arc Length ............................................217
11.2  Surface Area of Revolution  ............................ .   . 220
11.3 Volumes by Slicing ................................... 222
11.4 Volumes of Revolution ................................... 224
11.4.1  The Disc Method...................................224
11.4.2  The Cylindrical Shell Method  ................... ....  226
11.5 Density and Mass ...............  ................... . 229
11.6 Centre of Mass and Centroid . ........................... 233

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Library of Congress subject headings for this publication: Functions of real variables