INDEX UNIT 4 TSFX REFERENCE MATERIALS 2013 APPLICATIONS IN DIFFERENTIATION

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1 INDEX UNIT 4 TSFX REFERENCE MATERIALS 2013 APPLICATIONS IN DIFFERENTIATION Applications in Differentiation Page 1 Conditions For Differentiability Page 1 Gradients at Specific Points Page 3 Derivatives of Hybrid Functions Page 5 Derivatives of Composite Functions Page 6 Joining Functions Smoothly Page 7 Increasing and Decreasing Functions Page 8 Maxima and Minima Page 11 Maximum and Minimum Values Page 12 Finding Maximum and Minimum Values Page 12 Stationary Points Page 14 Locating Stationary Points Page 14 First Derivative Test (Sign Test) Page 15 Second Derivative Test Page 17 False Stationary Points Page 20 Worded Applications Involving Maxima and Minima Page 21 Graphs of the Derivative Function Page 23 Sketching the Derivative Function Page 25 Tangents and Normals Page 26 Finding the Equation of a Tangent Page 27 Finding the Equation of a Normal Page 28 Proofs Involving Tangents and Normals Page 28 Rates of Change Page 29 Average and Instantaneous Rates of Change Page 29 Solving Worded Rate Problems Page 31 Relationships Between Displacement, Velocity and Acceleration Page 32 Vessels and Rates of Change Page 33 Related Rates of Change Page 36 Relative Rates of Change Page 37 Approximations Page 38

2 INTEGRATION AND ITS APPLICATIONS Integration and Its Applications Page 40 Integrating Algebraic Expressions Page 40 Integrating Expressions General Approach Page 41 Simplifying Expressions Page 42 n Integrating ( ax b) Page 44 Integrating Exponential Expressions Page 45 Integrating Trigonometric Expressions Page 46 1 Integrating x Page 47 Integrating Hybrid Functions Page 50 Integrating Composite Functions Page 51 Applications in Integration Page 52 Solving For the Constant c Page 52 Integration by Recognition Page 53 Definite Integrals Page 54 Important Properties of the Definite Integral Page 56 Approximating Areas Using Left Rectangles Page 57 Approximating Areas Using Right Rectangles Page 58 Areas Under Curves Page 59 Signed and Unsigned Areas Page 59 Area Notations Page 60 Areas Between Curves Page 62 Sketching f ( x ) Given the Graph of f '( x ) Page 64 The Average Value of a Function Page 66 The Fundamental Theorem of Calculus Revisited Page 67

3 PROBABILITY Introductory Probability Page 68 Calculating Probabilities Page 68 Complementary Events Page 69 Choices and Combinations of Events Page 70 Karnaugh Maps Page 72 Tree Diagrams Page 73 Conditional Probability Page 74 Sets and Notations Page 76 Venn Diagrams Page 77 Selections and Arrangements (Permutations and Combinations) Page 79 Measures of Central Tendency Page 80 The Mean Value (Expectation) Page 80 The Median Page 81 The Mode Page 82 Measures of Spread Page 83 The Range Page 83 The Interquartile Range Page 84 The Variance and Standard Deviation Page 85 Confidence Intervals ( % Rule) Page 85 General Points Relating to all Probability Distributions Page 86 Inequalities Page 86 Profit and Loss Page 86 Percentiles Page 86 Random Variables and Their Distributions Page 87 Discrete and Continuous Random Variables Page 87 Probability Distributions Page 88 Graphical Representations of Probability Distributions Page 88 How to Choose the Appropriate Technique in Probability Page 90

4 Discrete Probability Distributions Page 91 Calculating Probabilities Page 91 Calculating the Mean Value (Expectation) Page 93 Properties of E(X) Page 94 Calculating the Median Page 95 Calculating the Mode Page 96 Calculating the Variance and Standard Deviation Page 97 Finding Probability/Confidence Intervals Page 99 Statistical Properties Involving Functions in Terms of X Page 100 The Binomial (Bernoulli) Distribution Page 103 Calculating Probabilities Page 104 Calculating the Mean and Standard Deviation Page 105 Finding the Value of n (the trial size) Page 106 The Binomial Probability Distribution Graph Page 107 Markov Sequences Page 108 Calculating Probabilities Using Tree Diagrams Page 180 Transition Probability Tables Page 111 Using Matrices to Solve Markov Applications Page 112 The Steady State Page 114

5 The Continuous Probability Distribution Page 116 Conditions For the Existence of a Probability Density Function Page 117 Proving/Showing that a Probability Density Function Exists Page 117 Calculating Probabilities of Simple PDFS Page 119 Probabilities Involving Complex PDFS Page 120 The Mean of Simple PDFS Page 121 The Mean of Complex PDFS Page 122 The Mean of a Function in Terms of X Page 123 The Median of Simple PDFS Page 124 The Median of Complex PDFS Page 125 The Mode Page 127 The Variance and Standard Deviation of Simple PDFS Page 128 The Variance and Standard Deviation of Complex PDFS Page 129 The Variance of Functions in Terms of X Page 130 The Range Page 130 Percentiles and Quartiles Page 131 The Interquartile Range Page 132 The Normal Distribution Page 133 Important Points Page 133 Confidence Intervals Page 134 Graphs of the Normal Distribution Page 134 The Standard Normal Distribution Page 135 Calculating Probabilities Page 136 The Inverse Normal Distribution Page 138