MATH-1411-notes-part-1.docx - Math 1411 Calculus I Andrew...

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Unformatted text preview: Math 1411, Calculus I Andrew Pownuk 1 Table of contents Table of contents.........................................................................................................................................2 1 Introduction.......................................................................................................................................22 2 (*) Sequences - Finding Limits Graphically and Numerically..............................................................23 2.1 (*) Sequence, examples.............................................................................................................23 2.1.1 Example 2.1.2 Example 1 lim 0 n n ...........................................................................................................24 1 0 n n 1 .....................................................................................................26 lim 2.1.3 Example 1 0 n n 3 ........................................................................................................27 lim 2.1.4 Example 2.1.5 Example 2.1.6 Example ..............................................................................................28 1 lim 1 3 1 n n 1 lim 2 3 2 n n sinn 0 n n2 .............................................................................................29 .....................................................................................................30 lim 2.1.7 Example sin n lim 2 2 2 0 2 n n ...........................................................................33 2 2.1.8 Example n n n lim 1 exp sin 2 2 n 50 8 ......................................................34 2.1.9 Example ..............................................................................................35 n lim 1 DNE n 2.1.10 Example limsin n DNE n 2.2 (*) Sequence – formal definition of the limit.............................................................................40 2.2.1 Definition...........................................................................................................................40 2.2.2 Example 2.3 3 ...........................................................................................36 1 lim 0 n n ...........................................................................................................45 (*) Sequence – Evaluating Limits Analytically – polynomials and rational functions..................48 2.3.1 Example 2.3.2 Example n2 lim n n ............................................................................................................49 n 2 2n 5 lim 2 n 2n 3n 7 ................................................................................................49 Limit of the function..........................................................................................................................52 3.1 Definition...................................................................................................................................52 3.2 Evaluating Limits Analytically – polynomials and rational functions..........................................62 3.2.1 Introduction.......................................................................................................................62 3.2.2 Example ...............................................Error! Bookmark not defined. lim x 2x 3 x0 3.2.3 2 Example ................................................................................................................66 2 x x 0 x lim 3 3.2.4 Example x x 0 x ..................................................................................................................67 lim 3.2.5 Example x 1 x 1 x 1 ...........................................................................................................63 2 lim 3.2.6 Example x 1 x1 x 1 ...........................................................................................................65 2 lim 3.2.7 Example x 9 x3 x 3 ..........................................................................................................68 2 lim 3.2.8 Example lim x 0 3.2.9 Example lim x0 3.2.10 Example ...................................................................................................69 x x 11 3x ...............................................................................................72 x2 2 f x h f x f ' x lim h f x 2x 3 .........................................................................74 h 0 3.2.11 Example 3.2.12 Example x3 1 lim x 1 x 1 ...........................................................................................................75 x1 lim 3 x1 x 1 ...........................................................................................................76 4 3.2.13 Example x 27 x1 x 3 3 .........................................................................................................76 lim 3.2.14 Example x7 lim x 7 x1 3.2.15 Example x 7 x 7 Example 3.2.17 Example x 4 lim x 1 5 x4 x4 x 0 Example lim Example 2 x 2 x x x x2 x 0 3.2.19 .....................................................................................................80 .............................................................................................82 lim 3.2.18 .....................................................................................................78 x7 lim 3.2.16 .....................................................................................................78 2 .............................................................................................83 2 x 2 x 2x 8 x 2 x2 x 2 2 ................................................................................................85 lim 3.2.20 3.3 Example x 5 lim x 1 6 x 5 x5 ....................................................................................................89 Evaluating Limits Analytically - trigonometric functions............................................................90 3.3.1 Introduction.......................................................................................................................90 3.3.2 Limit from the graph sinx 1 x 0 x ..................................................................................91 lim 5 3.3.3 (*) Geometrical proof sin x 1 x 0 x ................................................................................92 lim 3.3.4 (*) Proof 1 cosx 0 x 0 x ............................................................................................95 lim 3.3.5 Example sin2x ? x0 x ...................................................................................................96 lim 3.3.6 Example sin8x ? x 0 x ...................................................................................................97 lim 3.3.7 Example lim x 0 3.3.8 Example lim x 0 3.3.9 Example lim x 0 3.3.10 Example x ? sin7x 4x ? sin8x ...................................................................................................97 ...................................................................................................98 7x tan5x ? sin8x 3x x x 0 cosx ....................................................................................103 ........................................................................................................104 lim 3.3.11 Example lim x 3.3.12 Example x cosx tan x lim x0 x ........................................................................................................104 ...............................................................Error! Bookmark not defined. 6 3.3.13 Example 2x x 0 tan4x ......................................................................................................106 lim 3.3.14 Example x x 0 sin x .........................................................................................................107 lim 3.3.15 Example .................................................................................................107 2 2x x0 1 cos2 5x lim 3.3.16 Example 3.3.17 Example 3.3.18 Example 3.3.19 Example 1 cosx lim x 0 x2 sin x lim x x ...........................................................................................................109 sin x sin 2 1 2 lim x x 2 2 2 Example 3.3.21 Example .............................................................................109 .....................................................................109 sin x x x 1 lim 2x 1 x 0 x x 1 x 3.3.20 ...................................................................................................108 3x 4x tan4x lim x 0 sin2x cos5x 7x ...............................................................................110 2x x x x1 x 1 2 x0 sin3x x 3 3x x1 x1 ..................................................112 2 lim 7 3.3.22 Example .................................................................................................113 cosx ? x 0 x lim 3.4 (*) Epsilon-delta definition of limit..........................................................................................135 3.4.1 Example limx 1 ..........................................................................................................143 x1 3.4.2 Example limx 2 .........................................................................................................153 x2 3.4.3 Example limx 1 2 ..................................................................................................154 x1 3.4.4 Example lim2x 4 .......................................................................................................143 x2 3.4.5 Example 3.4.6 Example .......................................................................................................148 1 lim x 1 x2 2 lim2x 1 3 .................................................................................................156 x1 3.4.7 Example lim3x 2 4 .................................................................................................158 x2 3.4.8 Example lim3x 1 7 .................................................................................................159 x2 3.4.9 Example limx2 4 .......................................................................................................160 x 2 3.4.10 Example limx2 4 .......................................................................................................169 x 3 8 3.4.11 Find f x x2 3.4.12 4 x2 we have where f x 4 1 .....................................................................................................................................173 Find f x x such that for all x such that such that for all x such that x2 we have where f x 4 .....................................................................................................................................175 2 One side limits – examples (section 1.4)..........................................................................................114 4.1 (*) Formal definition................................................................................................................114 4.2 Example .....................................................................................................................118 x lim x x0 4.3 Example 1 x0 x .......................................................................................................................120 lim 4.4 Example 4.5 Example 1 lim x1 x 1 Example lim x1 4.7 lim x 2x 1 x1 4.6 .................................................................................................................122 Example 2 ............................................................................................................128 1 x 1 2 ...........................................................................................................129 2 lim x 3 .....................................................................................................126 x x 3 2 9 4.8 Example 4.9 Example x lim 1 x 0 x ............................................................................................................130 x2 1, x 0 f (x) x , lim f x ? x0 2x 1, x 0 ......................................................................131 4.10 Example sin2x ,x 0 f (x) x ,lim f x ? x 0 x 1, x 0 x ....................................................................132 4.11 Example x 1, x 0 f (x) x ,lim f x ? x 0 sin2x , x 0 x ....................................................................133 5 Continuity........................................................................................................................................135 5.1 Example ....................................................................................................181 f x x , x0 2 5.2 Example ...............................................................................188 5.3 Example 2 sin2x ,x 0 x x f (x) 1, x 0 , x0 0 x 2, x 0 sin2x ,x 0 x x f (x) 1, x 0 , x0 0 x 1, x 0 ...............................................................................189 10 5.4 Example 5.5 Example 5.6 Example sin2x ,x 0 x x f (x) 2, x 0 , x0 0 x 2, x 0 sin2x ,x 0 f (x) x , x0 0 x 1, x 0 x ...............................................................................191 ...............................................................................193 ..............................................................................195 x 9 ,x 3 f x x3 , x0 3 6, x 3 2 5.7 Example ..............................................................................195 x2 9 ,x 3 f x x3 , x0 3 1, x 3 5.8 Example ...............................................................................199 x2 1 ,x 1 f x x 1 , x0 1 2, x 1 5.9 Example 2 x x ,x 0 x f x 2 ,x 0 4 ..............................................................................200 5.10 (*) Example x2 1 f x x 1 , x 1, x0 1 2, x 1 .......................................................................201 11 5.11 Example .....................................................................................202 x ,x 0 f x x , x0 0 ?, x 0 5.12 Example x f x x ,for x 0, x0 0 0,for x 0 ...............................................................................203 5.13 Example x , for x 0 f x x , x0 0 0,for x 0 ...............................................................................204 5.14 Example ....................................................................................................205 f x 5.15 Example x x , x0 0 x f x tan 2 . Find the x such that the function is not continuous..................206 5.16 Find the constant � such that the function is continuous over the entire real line sin 4x , x 0 f x x 2x a, x 0 5.17 Find the constants �,b such that the function is continuous over the entire real line sin 4x , x 0 x f x 2b 8, x 0 2x 2a, x 0 .................................................................................................................207 ..................................................................................................................211 12 5.18 Find the constants �,b such that the function is continuous over the entire real line sin 8x , x 0 2x f x x a b, x 0 2x a b, x 0 5.19 ..............................................................................................................213 Find the constants �,b such that the function is continuous over the entire real line x2 , x 0 2 1 cos 2x f x x a b, x 0 2x a b, x 0 6 .........................................................................................................215 Infinite Limits (section 1.5 textbook)...............................................................................................217 6.1 Introduction.............................................................................................................................217 6.2 Table........................................................................................................................................222 6.3 Example 6.4 Example 6.5 Example 6.6 Example 6.7 Example x1 lim x x 1 .................................................................................................................230 6x 1 lim x 2x 1 ..............................................................................................................232 6x2 1 lim x 2x 1 6x 1 lim 2 x 2x 1 ...........................................................................................................232 ............................................................................................................233 6x 1 lim 4 x 2x 1 ...........................................................................................................233 13 6.8 Example 2x 2x 1 x x2 3x 2 2 ......................................................................................................235 lim 6.9 Example 2x 2x 1 x x3 3x 2 2 ......................................................................................................237 lim 6.10 Example 2x 2x 1 x x2 3x 2 3 ......................................................................................................237 lim 6.11 Example 2x 2x 1 x 8x2 3x 2 3 ...................................................................................................238 lim 6.12 Example 16x 3x 2x 1 x 8x4 5x3 3x 2 4 3 ......................................................................................239 lim 6.13 Example 16x 16x 3x 2x 1 x 4x5 8x4 5x3 3x 2 5 4 3 .........................................................................239 lim 6.14 Example lim x 6.15 x 6.16 Example lim x 6.17 4x2 3x 2 Example lim Example lim x ................................................................................................239 4x 2 ........................................................................................241 8x2 x 2 4x4 x3 4x 2 x1 x x1 2 3x 1 lim x x1 (!!!)...............................................................242 x x1 2 ..........................................................................................................244 9x2 x 14 6.18 Example 4x x 2 lim x 6.19 4x3 3x 2 Example 2 3 6.20 ..............................................................................................256 1 3 4x x 2 lim x ................................................................................................254 3 2 3 1 8x2 3x 2 1 Example ..................................................................................258 4x 3x 2 x 1 4 lim x 6.21 Example 2 4x4 5x 2 2x 1 2x x 1 x 5x 3 2 .......................................................................................................258 lim 6.22 Examples 6.23 Example 6.24 Example 6.25 Example 1 lim 2 x0 x 1 lim x1 x 1 2 lim x1 1 x lim Example lim x 6.27 .................................................................................................................261 2 .......................................................................................................262 1 x .................................................................................................263 x 1 x x 1 x 1 Example lim x .................................................................................................................259 2 x 1 x1 6.26 ....................................................................................................................259 2 2 ........................................................................................265 15 7 8 Asymptotes......................................................................................................................................266 7.1 Introduction...................................................................................Error! Bookmark not defined. 7.2 Horizontal asymptotes...................................................................Error! Bookmark not defined. 7.3 Vertical asymptote.........................................................................Error! Bookmark not defined. 7.4 Whole in the graph........................................................................Error! Bookmark not defined. 7.5 Slant asymptote.......................
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  • Spring '14
  • STAFF
  • Math, Calculus, Limits, lim, ........., Natural logarithm, Logarithm, Inverse trigonometric functions

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