Derivatives (Zambak)

DDEERRIIVVAATTIIVVEESS
CHAPTER 11: DDIFFERENTIATION
1. INTRODUCTION TTO DDERIVATIVES .. .. .. .. .. 2
A. TANGENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
B. VELOCITIES . . . . . . . . . . . . . . . . . . . . . . . . . . 6
C. RATES OF CHANGE . . . . . . . . . . . . . . . . . . . . 8
D. DERIVATIVE OF A FUNCTION . . . . . . . . . . . .10
E. LEFT-HAND AND RIGHT-HAND DERIVATIVES . .13
F. DIFFERENTIABILITY AND CONTINUITY . . . . .14
Activity: Linear Approximation . . . . . . . . . . . .18
EXERCISES 1.1 . . . . . . . . . . . . . . . . . . . . . . . .20
2. TECHNIQUES OOF DDIFFERENTIATION .. .. ..22
A. BASIC DIFFERENTIATION RULES . . . . . . . . . .22
B. THE PRODUCT AND THE QUOTIENT RULES . . .28
C. THE CHAIN RULE . . . . . . . . . . . . . . . . . . . . . .33
D. HIGHER ORDER DERIVATIVES . . . . . . . . . . . .37
Activity: Linear Motion . . . . . . . . . . . . . . . . . . .40
EXERCISES 1.2 . . . . . . . . . . . . . . . . . . . . . . . .42
3. DERIVATIVES OOF EELEMENTARY
FUNCTIONS .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..45
A. DERIVATIVES OF EXPONENTIAL
AND LOGARITHMIC FUNCTIONS . . . . . . . . .45
1. Exponential Functions . . . . . . . . . .45
2. Logarithmic Functions . . . . . . . . . . .48
3. Logarithmic Differentiation . . . . . . .50
B. DERIVATIVES OF TRIGONOMETRIC
FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . .52
C. DERIVATIVES OF SPECIAL FUNCTIONS . . . . .56
1. Absolute Value Functions . . . . . . . .56
2. Sign Functions . . . . . . . . . . . . . . . . .58
3. Floor Functions . . . . . . . . . . . . . . . .59
D. IMPLICIT DIFFERENTIATION . . . . . . . . . . . . . .61
E. DERIVATIVES OF PARAMETRIC FUNCTIONS . . .64
F. DERIVATIVES OF INVERSE
TRIGONOMETRIC FUNCTIONS . . . . . . . . . . .67
EXERCISES 1.3 . . . . . . . . . . . . . . . . . . . . . . . .69
CHAPTER SUMMARY . . . . . . . . . . . . . .72
CONCEPT CHECK . . . . . . . . . . . . . . . . .73
CHAPTER REVIEW TEST 1A . . . . . . . . . . . . . . . .74
CHAPTER REVIEW TEST 1B . . . . . . . . . . . . . . . .76
CHAPTER REVIEW TEST 1C . . . . . . . . . . . . . . . .78
CHAPTER 22: APPLICATIONS OOF TTHE
DERIVATIVE
1. L’HOSPITAL’S RRULE .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..82
A. THE INDETERMINATE FORM 0/0 . . . . . . . . . .82
B. THE INDETERMINATE FORM ∞∞ /∞∞ . . . . . . . . .85
EXERCISES 2.1 . . . . . . . . . . . . . . . . . . . . . . . .87
2. APPLICATIONS OOF TTHE FFIRST
DERIVATIVES .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..88
A. INTERVALS OF INCREASE AND DECREASE . . .88
B. MAXIMUM AND MINIMUM VALUES . . . . . . . .92
1. Absolute and Local
Maximum and Minimum . . . . . . . . . .92
2. Finding the Local Extrema . . . . . . . .94
3. The First Derivative Test . . . . . . . . .97
4. Finding the Absolute Extrema . . . . .102
EXERCISES 2.2 . . . . . . . . . . . . . . . . . . . . . . . .105
3. APPLICATIONS OOF TTHE SSECOND
DERIVATIVES .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..108
A. CONCAVITY . . . . . . . . . . . . . . . . . . . . . . . . . .108
B. THE SECOND DERIVATIVE TEST . . . . . . . . . . .114
EXERCISES 2.3 . . . . . . . . . . . . . . . . . . . . . . . .116
4. OPTIMIZATION PPROBLEMS .. .. .. .. .. .. .. .. .. .. ..118
Activity: Refraction of Light . . . . . . . . . . . . . . .122
EXERCISES 2.4 . . . . . . . . . . . . . . . . . . . . . . . .124
5. PLOTTING GGRAPHS .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..126
A. ASYMPTOTES . . . . . . . . . . . . . . . . . . . . . . . . .126
B. CURVE PLOTTING . . . . . . . . . . . . . . . . . . . . .129
EXERCISES 2.5 . . . . . . . . . . . . . . . . . . . . . . . .137
CHAPTER SUMMARY . . . . . . . . . . . . . .138
CONCEPT CHECK . . . . . . . . . . . . . . . . .139
CHAPTER REVIEW TEST 2A . . . . . . . . .140
CHAPTER REVIEW TEST 2B . . . . . . . . .142
CHAPTER REVIEW TEST 2C . . . . . . . . . .144
ANSWERS . . . . . . . . . . . . . . . . . . . . . .146
GLOSSARY . . . . . . . . . . . . . . . . . . . . . .151