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# Note for Applied Mathematics-1 - M-1 by Bhavik Yoganandi

• Applied Mathematics-1 - M-1
• Note
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Calculus (2110014) – B.E. Semester-I Darshan Institute of Engineering and Technology Name : _____________________________________ Division : _____________________________________ Roll No : _____________________________________ 4e - 2017

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I N D E X UNIT 1 – INFINITE SEQUENCE & SERIES ........................................................1 1). METHOD – 1: CHARACTERISTICS OF SEQUENCE AND LIMIT ................................................. 1 2). METHOD – 2: CONVERGENCE OF SEQUENCE................................................................................. 3 3). METHOD – 3: GEOMETRIC SERIES ...................................................................................................... 5 4). METHOD – 4: ZERO TEST ........................................................................................................................ 7 5). METHOD – 5: INTEGRAL TEST ............................................................................................................. 8 6). METHOD – 6: DIRECT COMPARISION TEST .................................................................................... 9 7). METHOD – 7: LIMIT COMPARISION TEST ..................................................................................... 10 8). METHOD – 8: RATIO TEST ................................................................................................................... 14 9). METHOD – 9: RABBE’S TEST .............................................................................................................. 16 10). METHOD – 10: CAUCHY’S NTH ROOT TEST ................................................................................... 17 11). METHOD – 11: LEIBNITZ’S TEST ...................................................................................................... 18 12). METHOD – 12: ABSOLUTE CONVERGENT SERIES..................................................................... 20 13). METHOD – 13: CONDITIONALLY CONVERGENT SERIES ........................................................ 21 14). METHOD – 14: POWER SERIES .......................................................................................................... 23 15). METHOD – 15: 1ST FORM OF TAYLOR’S SERIES.......................................................................... 26 16). METHOD – 16: 2ND FORM OF TAYLOR’S SERIES ......................................................................... 27 17). METHOD – 17: MACLAURIN’S SERIES ............................................................................................ 28 18). METHOD – 18: FIND EXPANSION USING DERIVATIVE AND INTEGRATION .................. 33 19). METHOD – 19: TRIGONOMETRY SUBSTITUTION ...................................................................... 33 UNIT 2 – CURVE SKETCHING ........................................................................35 20). METHOD – 1: MONOTONIC FUNCTION .......................................................................................... 35 21). METHOD – 2: CONVEXITY & CONCAVITY AND POINT OF INFLECTION ........................... 36 22). METHOD – 3: CRITICAL POINT AND EXTREME VALUE .......................................................... 38 23). METHOD – 4: SYMMETRY, DOUBLE POINTS AND ASYMPTOTES OF CURVE ................. 41 DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » CALCULUS - 2110014

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I N D E X 24). METHOD – 5: CARTESIAN CURVE SKETCHING ...........................................................................42 25). METHOD – 6: POLAR COORDINATES ..............................................................................................44 26). METHOD – 7: POLAR CURVE SKETCHING .....................................................................................46 UNIT 3A – INDETERMINATE FORMS ..............................................................47 27). METHOD – 1: 0/0 FORM .......................................................................................................................47 28). METHOD – 2: ∞/∞ FORM ....................................................................................................................49 29). METHOD – 3: 0 × ∞ FORM ...................................................................................................................50 30). METHOD – 4: ∞ – ∞ FORM ..................................................................................................................51 31). METHOD – 5: 00, ∞0 & 1∞ FORM ........................................................................................................52 UNIT 3B – IMPROPER INTEGRALS .................................................................55 32). METHOD – 6: IMPROPER INTEGRAL OF FIRST KIND ...............................................................55 33). METHOD – 7: IMPROPER INTEGRAL OF SECOND KIND ..........................................................57 34). METHOD – 8: CONVERGENCE OF IMPROPER INTEGRAL OF FIRST KIND .......................59 35). METHOD – 9: CONVERGENCE OF IMPROPER INTEGRAL OF SECOND KIND ..................60 36). METHOD – 10: CONVERGENCE OF IMPROPER INTEGRAL OF THIRD KIND ...................62 UNIT 4 – APPLICATION OF INTEGRATION .....................................................63 37). METHOD – 1: VOLUME BY SLICING..................................................................................................63 38). METHOD – 2: VOLUME OF SOLID BY ROTATION USING DISK METHOD ..........................65 39). METHOD – 3: VOLUME OF SOLID OF REVOLUTION IN POLAR FORM ...............................68 40). METHOD – 4: VOLUME OF SOLID BY ROTATION USING WASHER METHOD .................69 41). METHOD – 5: VOLUME OF SOLID BY ROTATION USING CYLINDRICAL SHELL .............72 UNIT 5 – PARTIAL DERIVATIVES...................................................................74 42). METHOD – 1: LIMIT OF FUNCTION OF TWO VARIABLES .......................................................74 43). METHOD – 2: CONTINUITY OF FUNCTION OF TWO VARIABLES.........................................76 44). METHOD – 3: PARTIAL DERIVATIVES ............................................................................................78 45). METHOD – 4: CHAIN RULE ..................................................................................................................83 DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » CALCULUS - 2110014

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I N D E X 46). METHOD – 5: IMPLICIT FUNCTION ................................................................................................. 86 47). METHOD – 6: NEW NOTATION .......................................................................................................... 88 48). METHOD – 7: EULER’S THEOREM .................................................................................................... 90 49). METHOD – 8: MODIFIED EULER’S THEOREM ............................................................................. 93 50). METHOD – 9: JACOBIAN ....................................................................................................................... 96 51). METHOD – 10: TANGENT PLANE AND NORMAL LINE ............................................................ 98 52). METHOD – 11: LINEARIZATION..................................................................................................... 100 53). METHOD – 12: MAXIMA MINIMA .................................................................................................. 101 54). METHOD – 13: LAGRANGE’S MULTIPLIERS .............................................................................. 103 55). METHOD – 14: TAYLOR’S SERIES OF FUNCTION OF TWO VARIABLES ......................... 107 UNIT 6 – MULTIPLE INTEGRALS.................................................................. 109 56). METHOD – 1: DOUBLE INTEGRALS BY DIRECT INTEGRATION ........................................ 111 57). METHOD – 2: TRIPLE INTEGRALS BY DIRECT INTEGRATION .......................................... 114 58). METHOD – 3: D.I. OVER GENERAL REGION IN CARTESIAN COORDINATES ................ 117 59). METHOD – 4: D.I. OVER GENERAL REGION IN POLAR COORDINATES .......................... 121 60). METHOD – 5: D.I. BY CHANGE OF ORDER OF INTEGRATION ............................................ 123 61). METHOD – 6: D.I. BY CHANGE OF VARIABLE IN CARTESIAN COORDINATES ............. 126 62). METHOD – 7: D.I. BY CHANGE OF VARIABLE IN POLAR COORDINATES....................... 129 63). METHOD – 8: T.I. OVER GENERAL REGION IN CARTESIAN COORDINATES ................ 131 64). METHOD – 9: T.I. OVER GENERAL REGION IN CYLINDRICAL COORDINATES ............ 132 65). METHOD – 10: T.I. OVER GENERAL REGION IN SPHERICAL COORDINATES .............. 133 66). METHOD – 11: T.I. BY CHANGE OF ORDER OF INTEGRATION .......................................... 134 67). METHOD – 12: T.I. BY CHANGE OF VARIABLE OF INTEGRATION.................................... 136 MCQ OF CALCULUS ALL UNITS ................................................................... 137 68). UNIT – 1: MCQ OF INFINITE SEQUENCES AND SERIES ........................................................ 137 69). UNIT – 2: MCQ OF CURVE SKETCHING ........................................................................................ 141 DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » CALCULUS - 2110014