CALCULUS DEMYSTIFIED
CHAPTER BASICS INTRODUCTORY REMARKS NUMBER SYSTEMS COORDINATES IN ONE DIMENSION COORDINATES IN TWO DIMENSIONS THE SLOPE OF A LINE IN THE PLANE THE EQUATION OF A LINE LOCI IN THE PLANE TRIGONOMETRY SETS AND FUNCTIONS EXAMPLES OF FUNCTIONS OF A REAL VARIABLE GRAPHS OF FUNCTIONS PLOTTING THE GRAPH OF A FUNCTION COMPOSITION OF FUNCTIONS THE INVERSE OF A FUNCTION A FEW WORDS ABOUT LOGARITHMS AND EXPONENTIALS FOUNDATIONS OF CALCULUS LIMITS ONE-SIDED LIMITS PROPERTIES OF LIMITS CONTINUITY THE DERIVATIVE RULES FOR CALCULATING DERIVATIVES THE DERIVATIVE OF AN INVERSE THE DERIVATIVE AS A RATE OF CHANGE CONTENTS CHAPTER APPLICATIONS OF THE DERIVATIVE GRAPHING OF FUNCTIONS MAXIMUM/MINIMUM PROBLEMS RELATED RATES FALLING BODIES THE INTEGRAL INTRODUCTION ANTIDERIVATIVES AND INDE?NITE INTEGRALS THE CONCEPT OF ANTIDERIVATIVE THE INDE?NITE INTEGRAL AREA SIGNED AREA THE AREA BETWEEN TWO CURVES RULES OF INTEGRATION LINEAR PROPERTIES ADDITIVITY INDETERMINATE FORMS L’HÔPITAL’S RULE OTHER INDETERMINATE FORMS WRITING A PRODUCT AS A QUOTIENT THE USE OF THE LOGARITHM PUTTING TERMS OVER A COMMON DENOMINATOR OTHER ALGEBRAIC MANIPULATIONS IMPROPER INTEGRALS: A FIRST LOOK INTEGRALS WITH IN?NITE INTEGRANDS AN APPLICATION TO AREA MORE ON IMPROPER INTEGRALS THE INTEGRAL ON AN IN?NITE INTERVAL SOME APPLICATIONS TRANSCENDENTAL FUNCTIONS LOGARITHM BASICS A NEW APPROACH TO LOGARITHMS THE LOGARITHM FUNCTION AND THE DERIVATIVE EXPONENTIAL BASICS FACTS ABOUT THE EXPONENTIAL FUNCTION CALCULUS PROPERTIES OF THE EXPONENTIAL THE NUMBER E EXPONENTIALS WITH ARBITRARY BASES ARBITRARY POWERS LOGARITHMS WITH ARBITRARY BASES CALCULUS WITH LOGS AND EXPONENTIALS TO ARBITRARY BASES DIFFERENTIATION AND INTEGRATION OF LOGA X AND AX GRAPHING OF LOGARITHMIC AND EXPONENTIAL FUNCTIONS LOGARITHMIC DIFFERENTIATION EXPONENTIAL GROWTH AND DECAY A DIFFERENTIAL EQUATION BACTERIAL GROWTH RADIOACTIVE DECAY COMPOUND INTEREST INVERSE TRIGONOMETRIC FUNCTIONS INVERSE SINE AND COSINE THE INVERSE TANGENT FUNCTION INTEGRALS IN WHICH INVERSE TRIGONOMETRIC FUNCTIONS ARISE OTHER INVERSE TRIGONOMETRIC FUNCTIONS AN EXAMPLE INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS METHODS OF INTEGRATION INTEGRATION BY PARTS PARTIAL FRACTIONS PRODUCTS OF LINEAR FACTORS QUADRATIC FACTORS SUBSTITUTION INTEGRALS OF TRIGONOMETRIC EXPRESSIONS APPLICATIONS OF THE INTEGRAL VOLUMES BY SLICING THE BASIC STRATEGY EXAMPLES VOLUMES OF SOLIDS OF REVOLUTION THE METHOD OF WASHERS THE METHOD OF CYLINDRICAL SHELLS DIFFERENT AXES WORK AVERAGES ARC LENGTH AND SURFACE AREA ARC LENGTH SURFACE AREA HYDROSTATIC PRESSURE NUMERICAL METHODS OF INTEGRATION THE TRAPEZOID RULE SIMPSON’S RULE BIBLIOGRAPHY SOLUTIONS TO EXERCISES FINAL EXAM TEAMFLY CHAPTER NUMBER SYSTEMS THE NUMBER SYSTEMS THAT WE USE IN CALCULUS ARE THE NATURAL NUMBERS, THE INTEGERS, THE RATIONAL NUMBERS, AND THE REAL NUMBERS LET US DESCRIBE EACH OF THESE: THE NATURAL NUMBERS ARE THE SYSTEM OF POSITIVE COUNTING NUMBERS 1, 2, 3, … WE DENOTE THE SET OF ALL NATURAL NUMBERS BY N THE INTEGERS ARE THE POSITIVE AND NEGATIVE WHOLE NUMBERS AND ZERO: , ?3, ?2, ?1, 0, 1, 2, 3, WE DENOTE THE SET OF ALL INTEGERS BY Z THE RATIONAL NUMBERS ARE QUOTIENTS OF INTEGERS ANY NUMBER OF THE FORM P/Q, WITH P, Q ?Z AND Q ?= 0, IS A RATIONAL NUMBER WE SAY THAT P/Q AND R/S REPRESENT THE SAME RATIONAL NUMBER PRECISELY WHEN PS = QR OF COURSE YOU KNOW THAT IN DISPLAYED MATHEMATICS WE WRITE FRACTIONS IN THIS WAY 2 + 3 = IS ACTUALLY A RATIONAL NUMBER, FOR IT REPRESENTS X = 3 16792 = OR M = 99000 SO, AS WE ASSERTED, M IS A RATIONAL NUMBER OR QUOTIENT OF INTEGERS THE THIRD KIND OF DECIMAL NUMBER IS ONE WHICH HAS A NON-TERMINATING DEC- IMAL EXPANSION THAT DOES NOT KEEP REPEATING AN EXAMPLE IS 3 14159265 THIS IS THE DECIMAL EXPANSION FOR THE NUMBER THAT WE ORDINARILY CALL ? SUCH A NUMBER IS IRRATIONAL, THAT IS, IT CANNOT BE EXPRESSED AS THE QUOTIENT OF TWO INTEGERS CHAPTER YOU TRY IT: SOLVE THE INEQUALITY |X?4| > 1 EXHIBIT YOUR ANSWER ON A NUMBER LINE YOU TRY IT: ON A REAL NUMBER LINE, SKETCH THE SET {X: X2 ?1 < 3} CHAPTER PLOT THE POINTS P = (3, ?2), Q = (?4, 6), R = (2, 5), S = (?5, ?3) SOLUTION THE ?RST COORDINATE 3 OF THE POINT P TELLS US THAT THE POINT IS LOCATED 3 UNITS TO THE RIGHT OF THE Y-AXIS (BECAUSE 3 IS POSITIVE) THE SECOND COORDINATE ?2 OF THE POINT P TELLS US THAT THE POINT IS LOCATED 2 UNITS BELOW THE X-AXIS (BECAUSE ?2 IS NEGATIVE) SEE FIG 1 7 THE ?RST COORDINATE ?4 OF THE POINT Q TELLS US THAT THE POINT IS LOCATED 4 UNITS TO THE LEFT OF THE Y-AXIS (BECAUSE ?4 IS NEGATIVE) THE SECOND COORDINATE 6 OF THE POINT Q TELLS US THAT THE POINT IS LOCATED 6 UNITS ABOVE THE X-AXIS (BECAUSE 6 IS POSITIVE) SEE FIG 1 7 THE ?RST COORDINATE 2 OF THE POINT R TELLS US THAT THE POINT IS LOCATED 2 UNITS TO THE RIGHT OF THE Y-AXIS (BECAUSE 2 IS POSITIVE) THE SECOND COORDINATE 5 OF THE POINT R TELLS US THAT THE POINT IS LOCATED 5 UNITS ABOVE THE X-AXIS (BECAUSE 5 IS POSITIVE) SEE FIG 1 7 THE ?RST COORDINATE ?5 OF THE POINT S TELLS US THAT THE POINT IS LOCATED 5 UNITS TO THE LEFT OF THE Y-AXIS (BECAUSE ?5 IS NEGATIVE) THE SECOND COORDINATE ?3 OF THE POINT S TELLS US THAT THE POINT IS LOCATED 3 UNITS BELOW THE X-AXIS (BECAUSE ?3 IS NEGATIVE) SEE FIG 1 7 CHAPTER X Q S P R SOLUTION THE POINT X IS 1 UNIT TO THE RIGHT OF THE Y-AXIS AND 3 UNITS BELOW THE X-AXIS THEREFORE ITS COORDINATES ARE (1, ?3) THE POINT Y IS 2 UNITS TO THE LEFT OF THE Y-AXIS AND 1 UNIT ABOVE THE X-AXIS THEREFORE ITS COORDINATES ARE (?2, 1) CHAPTER SKETCH THE SET OF POINTS S = {(X, Y): X > 2} ON A PAIR OF COORDINATE AXES SOLUTION NOTICE THAT THE SET S CONTAINS ALL POINTS WITH X-COORDINATE GREATER THAN 2 THESE WILL BE ALL POINTS TO THE RIGHT OF THE VERTICAL LINE X = 2 THAT SET IS EXHIBITED IN FIG 1 11 YOU TRY IT: SKETCH THE SET {(X, Y): X + Y < 4} YOU TRY IT: IDENTIFY THE SET (USING SET BUILDER NOTATION) THAT IS SHOWN IN FIG 1 12 CHAPTER Y X 1 ?(?1) = 2 CHAPTER 3 ?(?1) = 4 = 2 IF A LINE HAS SLOPE M, THEN, FOR EACH UNIT OF MOTION FROM LEFT TO RIGHT, THE LINE RISES M UNITS IN THE LAST EXAMPLE, THE LINE RISES 3/2 UNITS FOR EACH UNIT OF MOTION TO THE RIGHT OR ONE COULD SAY THAT THE LINE RISES 3 UNITS FOR EACH 2 UNITS OF MOTION TO THE RIGHT CHAPTER Y X R = (_2,10) S = (_1,5) T = (1,_5)