PROBLEMS IN MATHEMATICAL ANALYSIS
CONTENTS 1 EXPLORING THE RIEMANN AND DE?NITE INTEGRAL EXAMINING THE INTEGRABILITY OF FUNCTIONS 12 FINDING RIEMANN INTEGRALS BY DE?NITION FINDING LIMITS OF CERTAIN SEQUENCES 14 CALCULATING VARIOUS INTERESTING DE?NITE INTEGRALS EXPLAINING SEVERAL APPARENT PARADOXES USING THE (SECOND) MEAN VALUE THEOREM EXERCISES FOR INDEPENDENT WORK EXAMINING IMPROPER INTEGRALS 21 INVESTIGATING THE CONVERGENCE OF INTEGRALS BY DE?NITION USING DIFFERENT CRITERIA USING INTEGRAL TEST FOR CONVERGENCE OF SERIES APPLYING ONE-DIMENSIONAL INTEGRALS TO GEOMETRY AND PHYSICS FINDING LENGTHS OF CURVES CALCULATING AREAS OF SURFACES FINDING VOLUMES AND SURFACE AREAS OF SOLIDS OF REVOLUTION FINDING VARIOUS PHYSICAL QUANTITIES DEALING WITH FUNCTIONS OF SEVERAL VARIABLES FINDING IMAGES AND PREIMAGES OF SETS EXAMINING LIMITS AND CONTINUITY OF FUNCTIONS INVESTIGATING DERIVATIVES OF MULTIVARIABLE FUNCTIONS CALCULATING PARTIAL AND DIRECTIONAL DERIVATIVES EXAMINING DIFFERENTIABILITY OF FUNCTIONS EXAMINING HIGHER DERIVATIVES, DIFFERENTIAL EXPRESSIONS, AND TAYLOR’S FORMULA VERIFYING THE EXISTENCE OF THE SECOND DERIVATIVES TRANSFORMING DIFFERENTIAL EXPRESSIONS AND OPERATORS EXPANDING FUNCTIONS EXAMINING EXTREMES AND OTHER IMPORTANT POINTS LOOKING FOR GLOBAL MAXIMA AND MINIMA OF FUNCTIONS ON COMPACT SETS EXAMINING LOCAL EXTREMES AND SADDLE POINTS OF FUNCTIONS EXAMINING IMPLICIT AND INVERSE FUNCTIONS INVESTIGATING THE EXISTENCE AND EXTREMES OF IMPLICIT FUNCTIONS FINDING DERIVATIVES OF INVERSE FUNCTIONS SOLVING DIFFERENTIAL EQUATIONS OF THE FIRST ORDER FINDING SOLUTIONS OF SEPARABLE EQUATIONS SOLVING HOMOGENEOUS EQUATIONS 93 SOLVING SEVERAL SPECI?C EQUATIONS SOLVING EXACT EQUATIONS 10 SOLVING DIFFERENTIAL EQUATIONS OF HIGHER ORDERS 101 SOLVING LINEAR EQUATIONS WITH CONSTANT COEF?CIENTS 102 USING VARIOUS SPECI?C METHODS 103 EXERCISES FOR INDEPENDENT WORK 11 SOLVING SYSTEMS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 111 USING THE METHOD OF ELIMINATION OF VARIABLES 112 SOLVING SYSTEMS OF LINEAR EQUATIONS WITH CONSTANT COEF?CIENTS 113 EXERCISES FOR INDEPENDENT WORK 12 INTEGRATING IN MANY DIMENSIONS 121 EXAMINING THE INTEGRABILITY OF FUNCTIONS 122 CALCULATING INTEGRALS ON GIVEN DOMAINS 123 CHANGING THE ORDER OF INTEGRATION 124 EXERCISES FOR INDEPENDENT WORK 13 APPLYING MULTIDIMENSIONAL INTEGRALS TO GEOMETRY AND PHYSICS 131 FINDING AREAS OF FLAT SURFACES 132 CALCULATING VOLUMES OF SOLIDS 133 FINDING CENTER-OF-MASS LOCATIONS 134 CALCULATING MOMENTS OF INERTIA 135 FINDING VARIOUS PHYSICAL QUANTITIES 136 EXERCISES FOR INDEPENDENT WORK