Applications in Differentiation Page 3

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1 Applications in Differentiation Page 3 Continuity and Differentiability Page 3 Gradients at Specific Points Page 5 Derivatives of Hybrid Functions Page 7 Derivatives of Composite Functions Page 8 Joining Functions Smoothly Page 9 Increasing, Decreasing and Constant Functions Page 10 Strictly Increasing and Decreasing Functions Page 14 Maxima, Minima and Stationary Points Page 18 Stationary Points Page 18 Locating Stationary Points Page 18 First Derivative Test (Sign Test) Page 19 Second Derivative Test Page 21 False Stationary Points Page 24 Global Maxima and Minima Page 26 Finding the Global Maximum/Minimum Values Page 27 Worded Applications Involving Maxima and Minima Page 29 The Derivative Function Page 31 Graphs of the Derivative Function Page 31 Sketching the Derivative Function Page 33 Rates of Change Page 34 Average and Instantaneous Rates of Change Page 34 Solving Worded Problems Involving Rates of Change Page 36 Vessels and Rates of Change Page 37 Tangents and Normals Page 40 Finding the Equation of a Tangent Page 41 Finding the Equation of a Normal Page 42 Proofs Involving Tangents and Normals Page 42

2 Integration and Its Applications Page 43 Integrating Algebraic Expressions Page 44 Integrating Expressions General Approach Page 45 Simplifying Expressions Page 46 Integrating ( ax b) n Page 48 Integrating Exponential Expressions Page 49 Integrating Trigonometric Expressions Page 50 Integrating 1 x Page 51 Integrating g'( x) gx ( ) Page 53 Integrating Hybrid Functions Page 55 Integrating Composite Functions Page 56 Applications in Integration Page 57 Solving for the Constant c Page 57 Integration by Recognition Page 58 Applications of Differentiation and Integration to Kinematics Page 59 Relationships Between Displacement, Velocity and Acceleration Page 61 Definite Integrals Page 65 Properties of the Definite Integral Page 68 Approximating the Area Under a Curve Page 69 The Left Endpoint Approximation Page 69 The Right Endpoint Approximation Page 70 Exact Areas Page 73 Areas Under Curves Page 73 Changing the Sign of an Area Page 74 Calculating Unsigned Areas or Areas Page 76 Areas Between Curves Page 78 Other Applications The Average Value of a Function Page 80 Graphs of Antiderivative Function Page 81 The Fundamental Theorem of Calculus Revisited Page 83

3 Sections 1: Introductory Probability Page 84 Probabilities of Simple Events Page 84 Complementary Events Page 85 Sets and Notations Page 86 Venn Diagrams Page 88 Probabilities of Compound Events Page 89 Choices and Combinations of Events Page 90 Combinations of Events The Multiplication Rule Page 93 Dependent Events Conditional Probability Page 94 Independent Events Page 96 Mutually Exclusive versus Independency of Events Page 97 Probability Diagrams Page 97 Karnaugh Maps Page 98 Tree Diagrams Page 99 Sections 2: Counting Techniques Page 100 Permutations Page 100 Combinations Page 100 Sections 3: Random Variables and their Distributions Page 101 Discrete Random Variables Page 101 Continuous Random Variables Page 102 Probability Distributions and Functions Page 103 Density and Mass Functions Page 105 Parameters Used to Describe Probability Functions Page 107 Measures of Central Tendency Page 107 A. The Mean Value (Expected Value) Page 107 B. The Median Page 108 C. The Mode Page 108

4 Measures of Spread Page 110 A. The Range Page 110 B. The Interquartile Range Page 111 C. The Variance and Standard Deviation Page 112 Confidence Intervals For Many Distributions Page 113 General Points Relating to all Probability Distributions Page 113 Inequalities Page 113 Profit and Loss Page 113 Percentiles Page 113 Section 4: General Discrete Probability Distributions Page 114 Calculating Probabilities Page 114 Calculating the Mean (Expected Value) Page 116 Properties of E( X) Page 117 Calculating the Mode Page 117 Calculating the Variance and Standard Deviation Page 118 Finding Probability Intervals/Confidence Intervals Page 120 Statistical Properties Involving Functions in Terms of X Page 121 The Expected Value of Any Function Page 121 The Expected Value of Linear Functions Page 122 Variance of Functions Page 123 Section 5: The Binomial Distribution (Bernoulli Sequences) Page 124 Calculating Probabilities Page 125 Calculating the Mean and Standard Deviation Page 127 Finding the Value of n (The Trial Size) Page 128 The Binomial Probability Distribution Graph Page 129

5 Section 6: The Continuous Probability Distribution Page 130 Conditions for the Existence of a Probability Density Function Page 131 Proving/Showing that a Probability Density Function Exists Page 131 Calculating Probabilities of Simple PDFS Page 133 Probabilities Involving Complex PDFS Page 134 The Mean of Simple PDFS Page 135 The Mean of Complex PDFS Page 136 The Mean of Functions in Terms of X Page 137 The Median of Simple PDFS Page 138 The Median of Complex PDFS Page 139 The Mode Page 141 The Variance and Standard Deviation of Simple PDFS Page 142 The Variance and Standard Deviation of Complex PDFS Page 143 The Variance of Functions in Terms of X Page 144 The Range Page 144 Percentiles and Quantiles Page 145 The Interquartile Range Page 146 Section 7: The Normal Distribution Page 147 Important Points Page 147 Calculating Probabilities (Given X Values) Page 148 Graphs of the Normal Distribution Page 149 Confidence Intervals Associated with a Normal Distribution Page 150 The Standard Normal Distribution Page 151 Calculating Probabilities (Given Z Values) Page 154 Symmetry Properties Page 154 The Inverse Normal Distribution Page 155 Finding the Value of X, Z, or Given a Probability or Area Page 155

6 Statistical Inference Page 159 Population and Samples Page 160 Parameters and Statistics Page 161 Random Sampling Page 162 What is Sampling? Page 164 The Sampling Process Page 164 Probability Sampling Techniques Page 165 Cluster Sampling Page 166 Simple Random Sampling Page 168 Stratified Random Sampling Page 169 Systematic Random Sampling Page 171 Non-Probability Sampling Page 173 Convenience Sampling Page 174 Judgement Sampling Page 175 Quota Sampling Page 176 Accuracy, Bias, Variability and Precision Page 178 Sampling Errors Page 183 Non-Sampling Errors Page 184 Populations and Samples Page 185 Common Notations Page 185 Population & Sample Proportions Page 185 Sample Proportions Page 186 Population Proportions Page 187 Sampling Distribution of the Sample Proportion Page 189 Sampling from Small Populations Page 190 Sampling from Large Populations Page 192 The Standard Error Page 194 Approximating the Sampling Distribution of the Sample Proportion Using the Normal Distribution Page 195 Large Sample Size Determination Page 196 Important Properties of the Normal Approximation for the Sampling Distribution of P Page 197 Calculating Probabilities Using the Normal Approximation Page 199

7 Confidence Intervals for the Population Proportion Page 200 Point Estimates Page 200 Confidence Intervals Page 201 Interpreting Confidence Intervals Page 203 Important Assumptions Page 204 Confidence Levels Page 205 Critical Values Page 206 A 95% Confidence Interval for the Population Proportion Page 207 Constructing Confidence Intervals Page 208 Confidence Levels and Multipliers Page 209 Cumulative Percentages and Critical Values Page 210 Finding a Confidence Interval for P Page 211 The Margin of Error Page 212 Calculations Involving the Margin of Error Page 214 Confidence Intervals in Terms of the Margin of Error Page 215 Finding the Point Estimate & the Margin of Error from a Confidence Interval Page 216 Factors That Affect the Width of Confidence Levels Page 217 Factors That Affect the Margin of Error Page 219 Reducing the Margin of Error Page 221 Certainty vs Precision Page 222 Calculating the Sample Sized Needed to Estimate the Population Proportion Page 224