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Precalculus: Real Mathematics, Real People
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Chapter 2 / Exercise 3
Precalculus: Real Mathematics, Real People
Larson
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Unformatted text preview: Harold’s Calculus Notes Cheat Sheet 15 October 2017 AP Calculus Limits Definition of Limit Let f be a function defined on an open interval containing c and let L be a real number. The statement: lim () = → means that for each > 0 there exists a > 0 such that if 0 < | − | < , then |() − | < Tip : Direct substitution: Plug in () and see if it provides a legal answer. If so then L = (). The Existence of a Limit The limit of () as approaches a is L if and only if: lim () = → − lim () = → + Prove that () = − is a continuous function. Definition of Continuity A function f is continuous at c if for every > 0 there exists a > 0 such that | − | < and |() − ()| < . Tip: Rearrange |() − ()| to have |( − )| as a factor. Since | − | < we can find an equation that relates both and together. |() − ()| = |( 2 − 1) − ( 2 − 1)| = | 2 − 1 − 2 + 1| = | 2 − 2 | = |( + )( − )| = |( + )| |( − )| Since |( + )| ≤ |2| |() − ()| ≤ |2||( − )| < So given > 0, we can choose = || > in the Definition of Continuity. So substituting the chosen for |( − )| we get: 1 |() − ()| ≤ |2| (| | ) = 2 Since both conditions are met, () is continuous. Two Special Trig Limits →0 =1 1 − =0 →0 Copyright © 2015-2017 by Harold Toomey, WyzAnt Tutor 1 Derivatives Definition of a Derivative of a Function Slope Function Notation for Derivatives 0. The Chain Rule 1. The Constant Multiple Rule 2. The Sum and Difference Rule 3. The Product Rule 4. The Quotient Rule 5. The Constant Rule 6a. The Power Rule 6b. The General Power Rule 7. The Power Rule for x 8. Absolute Value 9. Natural Logorithm 10. Natural Exponential 11. Logorithm 12. Exponential 13. Sine 14. Cosine 15. Tangent 16. Cotangent 17. Secant (See Larson’s 1-pager of common derivatives) ( + ℎ) − () ′ () = lim ℎ→0 ℎ () − () ′ () = lim → − ′ (), () (), , ′ , [()], [] [(())] = ′ (())′ () = · [()] = ′ () [() ± ()] = ′ () ± ′ () [] = ′ + ′ ′ − ′ [ ]= 2 [] = 0 [ ] = −1 [ ] = −1 ′ ℎ = () [] = 1 (ℎ = 1 0 = 1) [||] = || 1 [ln ] = [ ] = 1 [log ] = (ln ) [ ] = (ln ) [()] = cos() [()] = −() [()] = 2() [()] = − 2 () [()] = () () Copyright © 2015-2017 by Harold Toomey, WyzAnt Tutor 2 Derivatives 18. Cosecant 19. Arcsine 20. Arccosine 21. Arctangent 22. Arccotangent 23. Arcsecant 24. Arccosecant 25. Hyperbolic Sine 26. Hyperbolic Cosine 27. Hyperbolic Tangent 28. Hyperbolic Cotangent 29. Hyperbolic Secant 30. Hyperbolic Cosecant 31. Hyperbolic Arcsine 32. Hyperbolic Arccosine 33. Hyperbolic Arctangent 34. Hyperbolic Arccotangent 35. Hyperbolic Arcsecant 36. Hyperbolic Arccosecant Position Function Velocity Function Acceleration Function Jerk Function (See Larson’s 1-pager of common derivatives) [()] = − () () 1 [sin−1 ()] = √1 − 2 −1 [cos −1()] = √1 − 2 1 [tan−1()] = 1 + 2 −1 [cot −1 ()] = 1 + 2 1 [sec −1()] = || √ 2 − 1 −1 [csc −1 ()] = || √ 2 − 1 [ℎ()] = cosh() [ℎ()] = ℎ() [ℎ()] = ℎ2() [ℎ()] = −ℎ2 () [ℎ()] = − ℎ() ℎ() [ℎ()] = − ℎ() ℎ() 1 [sinh−1()] = √ 2 + 1 1 [cosh−1()] = 2 √ − 1 1 [tanh−1 ()] = 1 − 2 1 [coth−1()] = 1 − 2 −1 [sech−1()] = √1 − 2 −1 [csch−1()] = || √1 + 2 1 () = 2 + 0 + 0 2 () = ′ () = + 0 () = ′ () = ′′ () () = ′ () = ′′ () = (3) () Copyright © 2015-2017 by Harold Toomey, WyzAnt Tutor 3 Applications of Differentiation Rolle’s Theorem f is continuous on the closed interval [a,b], and f is differentiable on the open interval (a,b). Mean Value Theorem If f meets the conditions of Rolle’s Theorem, then If f(a) = f(b), then there exists at least one number c in (a,b) such that f’(c) = 0. () − () − () = () + ( − )′() Find ‘c’. If f takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval. ′ () = Intermediate Value Therem f is a continuous function with an interval, [a, b], as its domain. () = → () lim () = lim → L’Hôpital’s Rule 0 ∞ { , , 0 • ∞, 1∞ , 00 , ∞0 , ∞ − ∞} , {0∞ }, 0 ∞ () ′ () ′′ () ℎ lim = lim ′ = lim ′′ =⋯ → () → () → () Graphing with Derivatives Test for Increasing and Decreasing Functions The First Derivative Test The Second Deriviative Test Let f ’(c)=0, and f ”(x) exists, then Test for Concavity Points of Inflection Change in concavity 1. If f ’(x) > 0, then f is increasing (slope up) ↗ 2. If f ’(x) < 0, then f is decreasing (slope down) ↘ 3. If f ’(x) = 0, then f is constant (zero slope) → 1. If f ’(x) changes from – to + at c, then f has a relative minimum at (c, f(c)) 2. If f ’(x) changes from + to - at c, then f has a relative maximum at (c, f(c)) 3. If f ’(x), is + c + or - c -, then f(c) is neither 1. If f ”(x) > 0, then f has a relative minimum at (c, f(c)) 2. If f ”(x) < 0, then f has a relative maximum at (c, f(c)) 3. If f ’(x) = 0, then the test fails (See 1 derivative test) 1. If f ”(x) > 0 for all x, then the graph is concave up ⋃ 2. If f ”(x) < 0 for all x, then the graph is concave down ⋂ If (c, f(c)) is a point of inflection of f, then either 1. f ”(c) = 0 or 2. f ” does not exist at x = c. Copyright © 2015-2017 by Harold Toomey, WyzAnt Tutor 4 Analyzing the Graph of a Function x-Intercepts (Zeros or Roots) y-Intercept Domain Range Continuity Vertical Asymptotes (VA) Horizontal Asymptotes (HA) Infinite Limits at Infinity Differentiability Relative Extrema Concavity Points of Inflection (See Harold’s Illegals and Graphing Rationals Cheat Sheet) f(x) = 0 f(0) = y Valid x values Valid y values No division by 0, no negative square roots or logs x = division by 0 or undefined lim− () → and lim+ () → →∞ →∞ lim− () → ∞ and lim+ () → ∞ →∞ →∞ Limit from both directions arrives at the same slope Create a table with domains, f(x), f ’(x), and f ”(x) If ”() → +, then cup up ⋃ If ”() → −, then cup down ⋂ f ”(x) = 0 (concavity changes) Approximating with Differentials ( ) ′( ) = + = ′ ()( − ) + () Newton’s Method Finds zeros of f, or finds c if f(c) = 0. +1 = − Tangent Line Approximations Function Approximations with Differentials Related Rates ( + ∆) ≈ () + = () + ′ () Steps to solve: 1. Identify the known variables and rates of change. ( = 15 ; = 20 ; ′ = 2 ; ′ = ? ) 2. Construct an equation relating these quantities. ( 2 + 2 = 2 ) 3. Differentiate both sides of the equation. (2 ′ + 2 ′ = 0) 4. Solve for the desired rate of change. ( ′ = − ′ ) 5. Substitute the known rates of change and quantities into the equation. 15 3 ( ′ = − ⦁ 2 = ) 20 2 Copyright © 2015-2017 by Harold Toomey, WyzAnt Tutor 5 Summation Formulas ∑ = =1 ∑ = =1 ( + 1) 2 = + 2 2 2 ∑ 2 = =1 ( + 1)(2 + 1) 3 2 = + + 6 3 2 6 2 3 ∑ = (∑ ) = =1 Sum of Powers =1 2 ( + 1)2 4 3 2 = + + 4 4 2 4 ( + 1)(2 + 1)(32 + 3 − 1) 5 4 3 ∑ 4 = = + + − 30 5 2 3 30 =1 ∑ 5 = =1 2 ( + 1)2 (22 + 2 − 1) 6 5 54 2 = + + − 12 6 2 12 12 ( + 1)(2 + 1)(34 + 63 − 3 + 1) ∑ = 42 6 =1 2 ( + 1)2 (34 + 63 − 2 − 4 + 2) ∑ = 24 7 =1 −1 ( + 1)+1 1 +1 () = ∑ = − ∑( ) () +1 +1 =1 =0 2 ∑ ( + 1) = ∑ + ∑ = =1 Misc. Summation Formulas =1 =1 ( + 1)( + 2) 3 1 ∑ = ( + 1) + 1 =1 ∑ =1 1 ( + 3) = ( + 1)( + 2) 4( + 1)( + 2) Copyright © 2015-2017 by Harold Toomey, WyzAnt Tutor 6 Numerical Methods 0 () = ∫ () = lim ∑ (∗ ) ∆ ‖‖→0 =1 where = 0 < 1 < 2 < ⋯ < = and ∆ = − −1 and ‖‖ = {∆ } Types: • Left Sum (LHS) • Middle Sum (MHS) • Right Sum (RHS) Riemann Sum 0 () = ∫ () ≈ ∑ (̅ ) ∆ = =1 ∆[(̅1 ) + (̅2 ) + (̅3 ) + ⋯ + (̅ )] − where ∆ = Midpoint Rule (Middle Sum) 1 2 and ̅ = (−1 + ) = [−1 , ] Error Bounds: | | ≤ (−)3 242 1 () = ∫ () ≈ ∆ [(0 ) + 2(1 ) + 2(3 ) + ⋯ + 2(−1 ) 2 + ( )] − where ∆ = and = + ∆ Trapezoidal Rule Error Bounds: | | ≤ (−)3 122 2 () = ∫ () ≈ Simpson’s Rule ∆ [(0 ) + 4(1 ) + 2(2 ) + 4(3 ) + ⋯ 3 + 2(−2 ) + 4(−1 ) + ( )] Where n is even − and ∆ = and = + ∆ Error Bounds: | | ≤ (−)5 1804 [MATH] fnInt(f(x),x,a,b), [MATH] [1] [ENTER] TI-84 Plus TI-Nspire CAS Example: [MATH] fnInt(x^2,x,0,1) 1 1 ∫ 2 = 3 0 [MENU] [4] Calculus [3] Integral [TAB] [TAB] [X] [^] [2] [TAB] [TAB] [X] [ENTER] Shortcut: [ALPHA] [WINDOWS] [4] Copyright © 2015-2017 by Harold Toomey, WyzAnt Tutor 7 Integration (See Harold’s Fundamental Theorem of Calculus Cheat Sheet) ∫ ′ () = () + Basic Integration Rules Integration is the “inverse” of differentiation, and vice versa. ∫ () = () () = 0 ∫ 0 = () = = 0 ∫ = + 1. The Constant Multiple Rule 2. The Sum and Difference Rule The Power Rule () = ∫ () = ∫ () ∫[() ± ()] = ∫ () ± ∫ () ∫ = +1 + , ℎ ≠ −1 +1 = −1, ℎ ∫ −1 = ln|| + The General Power Rule If = (), ′ = () then +1 ∫ ′ = + , ℎ ≠ −1 +1 Reimann Sum ∑ ( ) ∆ , ℎ −1 ≤ ≤ =1 ‖∆‖ = ∆ = Definition of a Definite Integral Area under curve lim ∑ ( ) ∆ = ∫ () ‖∆‖→0 Swap Bounds Additive Interval Property − =1 ∫ () = − ∫ () ∫ () = ∫ () + ∫ () The Fundamental Theorem of Calculus ∫ () = () − () ∫ () = () () ∫ () = (())′ () The Second Fundamental Theorem of Calculus ℎ() ∫ () = (ℎ())ℎ′ () − (())′() () Mean Value Theorem for Integrals The Average Value for a Function Copyright © 2015-2017 by Harold Toomey, WyzAnt Tutor ∫ () = ()( − ) Find ‘’. 1 ∫ () − 8 Integration Methods 1. Memorized See Larson’s 1-pager of common integrals ∫ (())′ () = (()) + Set = (), then = ′ () 2. U-Substitution ∫ () = () + = _____ = _____ ∫ = − ∫ = _____ = _____ 3. Integration by Parts = _____ = _____ Pick ‘’ using the LIATED Rule: L – Logarithmic : ln , log , . I – Inverse Trig.: tan−1 , sec −1 , . A – Algebraic: 2 , 3 60 , . T – Trigonometric: sin , tan , . E – Exponential: , 19 , . D – Derivative of: ⁄ () ∫ () where () () are polynomials 4. Partial Fractions Case 1: If degree of () ≥ () then do long division first Case 2: If degree of () < () then do partial fraction expansion ∫ √2 − 2 Substutution: = sin Identity: 1 − 2 = 2 5a. Trig Substitution for √ − ∫ √ 2 − 2 Substutution: = sec Identity: 2 − 1 = 2 5b. Trig Substitution for √ − ∫ √ 2 + 2 5c. Trig Substitution for √ + 6. Table of Integrals 7. Computer Algebra Systems (CAS) 8. Numerical Methods 9. WolframAlpha Substutution: = tan Identity: 2 + 1 = 2 CRC Standard Mathematical Tables book TI-Nspire CX CAS Graphing Calculator TI –Nspire CAS iPad app Riemann Sum, Midpoint Rule, Trapezoidal Rule, Simpson’s Rule, TI-84 Google of mathematics. Shows steps. Free. Copyright © 2015-2017 by Harold Toomey, WyzAnt Tutor 9 Partial Fractions Condition Example Expansion Typical Solution (See Harold’s Partial Fractions Cheat Sheet) () () = () where () () are polynomials and degree of () < () If degree of () ≥ () ℎ () ( + )( + )2 ( 2 + + ) + = + + + 2 2 ( + ) ( + ) ( + ) ( + + ) ∫ = | + | + + Sequences & Series (See Harold’s Series Cheat Sheet) lim = (Limit) →∞ Sequence Example: ( , +1 , +2 , …) (1 − ) = →∞ 1− 1− = lim Geometric Series Convergence Tests Series Convergence Tests Taylor Series only if || < 1 where is the radius of convergence and (−, ) is the interval of convergence (See Harold’s Series Convergence Tests Cheat Sheet) 1. Divergence or ℎ Term 6. Ratio 2. Geometric Series 7. Root 3. p-Series 8. Direct Comparison 4. Alternating Series 9. Limit Comparison 5. Integral 10. Telescoping (See Harold’s Taylor Series Cheat Sheet) () = () + () +∞ Taylor Series =∑ =0 () () (+1) ( ∗ ) ( − ) + ( − )+1 ! ( + 1)! where ≤ ∗ ≤ (worst case scenario ∗ ) and lim () = 0 →+∞ Copyright © 2015-2017 by Harold Toomey, WyzAnt Tutor 10 ...
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