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AP Calculus AB Notes 2018‐2019 Arbor View HS Name: Period: 1 Table of Contents 1.2 Finding Limits Graphically and Numerically (48)................................................................................... 4 1.3 Evaluating Limits Analytically (57) ........................................................................................................ 7 1.4 Continuity and One-Sided Limits (68) ................................................................................................... 10 1.5 Infinite Limits (80) ................................................................................................................................. 14 3.5 Limits at Infinity (192) ........................................................................................................................... 18 2.1 The Derivative and the Tangent Line Problem (94) .............................................................................. 20 2.2 Basic Differentiation Rules (105) .......................................................................................................... 23 2.2 Day 2 Rates of Change (109) ................................................................................................................. 26 2.3 Product and Quotient Rules (117) .......................................................................................................... 29 2.3 Day 2 Trigonometric & Higher-Order Derivatives (121) ...................................................................... 32 2.4 Day 1 Chain Rule (127) ......................................................................................................................... 36 2.4 Day 2 Chain Rule (132) ......................................................................................................................... 39 2.5 Day 1 Implicit Differentiation (132) ...................................................................................................... 42 2.5 Day 2 Implicit Differentiation (132) ...................................................................................................... 45 2.6 Day 1 Related Rates (144) ..................................................................................................................... 47 2.6 Day 2 Related Rates (144) ..................................................................................................................... 50 3.1 Extrema on an Interval (160) ................................................................................................................. 53 3.2 Rolle’s & Mean Value Theorems (168) ................................................................................................. 55 3.3 Increasing/Decreasing f(x)s and the 1st Derivative Test (174) .............................................................. 58 3.4 Concavity and the 2nd Derivative Test (184) ........................................................................................ 61 3.6 Summary of Curve Sketching (202) ...................................................................................................... 64 3.7 Optimization (211) ................................................................................................................................. 68 2 3.8 Newton’s Method (222) ......................................................................................................................... 72 3.9 Differentials (228) .................................................................................................................................. 74 4.1 Antiderivatives and Indefinite Integration (242).................................................................................... 79 4.2 Area (242) .............................................................................................................................................. 83 4.3 Riemann Sums and Definite Integrals (265) .......................................................................................... 87 4.4 The Fundamental Theorem of Calculus (275) ....................................................................................... 90 Slope Fields (Appendix pg. A6)................................................................................................................... 94 4.5 Integration by Substitution (288) ........................................................................................................... 97 4.6 Numerical Integration (300) ................................................................................................................. 100 5.1 The Natural Log Function: Differentiation (314) ................................................................................ 103 5.2 The Natural Log Function: Integration (324)....................................................................................... 106 5.3 Inverse Functions (332) ....................................................................................................................... 109 5.4 Exponential f(x)s: Differentiation & Integration (341) ........................................................................ 112 5.5 Bases Other than e and Applications (351) .......................................................................................... 115 5.6 Differential Equations: Grow and Decay (361) ................................................................................... 118 5.7 Differential Eqs: Separation of Variables (369) ................................................................................... 120 5.8 Inverse Trig Functions - Differentiation (380)..................................................................................... 123 5.9 Inverse Trig Functions - Integration (388) ........................................................................................... 126 7.7 Indeterminate Forms and L'Hopital's Rule (530) ................................................................................. 128 6.1 Area of Region Between Two Curves (412) ........................................................................................ 130 6.2 Volume the Disk Method (421) ........................................................................................................... 133 6.3 Volume the Shell Method (432) ........................................................................................................... 137 3 Notes #1-1 Date: ______ 1.2 Finding Limits Graphically and Numerically (48) Letter of recommendation: lim f(x) = L * The limit (L) of f(x) as x approaches c. participate in class, stand xc out – in a good way! 2 1) A penny: .01 = x 32x lim x2 = ? 2) Go ½ the distance x 2 each time over 10 ft. x 1.75 1.9 1.999 2 2.001 2.1 2.25 y ? Ex.1 Find 1c osx numerically and graphically. lim x x 0 3 x -2 Ex.2 fx() ? lim 1 x -2 x 2 “hiccup” function * Existence at the point is irrelevant. Limits that fail to exist: 1. f(x) approaches different values from the left and right f ()xf (x) sides of c. lim lim xc xc from the left from the right Diving board f(x) 2 x 1 Ex.3 lim f ()x if fx() Exists everywhere else. x 1 0 x 1 f ()x lim x0 limf ()x x3 4
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