Calculus_Notes_Cheat_Sheet_2016 - Harolds Calculus Notes Cheat Sheet 24 February 2016 AP Calculus Limits Definition of Limit Let f be a function defined

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Unformatted text preview: Harold’s Calculus Notes Cheat Sheet 24 February 2016 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. =1 →0 Two Special Trig Limits 1 − =0 →0 Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor 1 Derivatives Definition of a Derivative of a Function Slope Function (See Larson’s 1-pager of common derivatives) ( + ℎ) − () ′ () = lim ℎ→0 ℎ Notation for Derivatives The Constant Rule The Power Rule The General Power Rule The Constant Multiple Rule The Sum and Difference Rule Position Function Velocity Function Acceleration Function Jerk Function The Product Rule The Quotient Rule The Chain Rule Exponentials ( , ) Logorithms ( , ) Sine Cosine Tangent Secent Cosecent Cotangent Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor 2 () − () → − ′ ′ (), () (), , , [()], [] [ ] = 0 [ ] = −1 [ ] = 1 (ℎ = 1 0 = 1) [ ] = −1 ′ ℎ = () [()] = ′ () [() ± ()] = ′ () ± ′ () 1 () = 2 + 0 + 0 2 () = ′ () = + 0 () = ′ () = ′′ () () = ′ () = ′′ () = (3) () [] = ′ + ′ ′ − ′ [ ]= 2 [(())] = ′ (())′ () = · [ ] = , [ ] = (ln ) 1 1 [ln ] = , [log ] = (ln ) [()] = cos() [()] = −() [()] = 2 () [()] = () () [()] = − () () [()] = − 2 () ′ () = lim 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). If f(a) = f(b), then there exists at least one number c in (a,b) such that f’(c) = 0. () − () − () = () + ( − )′() Find ‘c’. () lim () = lim = → → () Mean Value Theorem If f meets the conditions of Rolle’s Theorem, then 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 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 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. (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) Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor 3 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. ( = 2 ; = −3 ; ′ = 4 ; ′ = ? ) 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. 2 8 ( ′ = − ) ⦁4= −3 3 Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor 4 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 ∑ 6 = =1 ∑ 7 = =1 2 ( + 1)2 (22 + 2 − 1) 6 5 54 2 = + + − 12 6 2 12 12 ( + 1)(2 + 1)(34 + 63 − 3 + 1) 42 2 ( + 1)2 (34 + 63 − 2 − 4 + 2) 24 −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 © 2015-2016 by Harold Toomey, WyzAnt Tutor 5 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 1 and ̅ = (−1 + ) = [−1 , ] 2 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 Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor 6 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] Integration ∫ ′ () = () + Basic Integration Rules Integration is the “inverse” of differentiation, and vice versa. ∫ () = () () = 0 ∫ 0 = () = = 0 ∫ = + The Constant Multiple Rule 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() (See Harold’s Fundamental Theorem of Calculus Cheat Sheet) Mean Value Theorem for Integrals ∫ () = (ℎ())ℎ′ () − (())′() () ∫ () = ()( − ) Find ‘’. The Average Value for a Function Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor 7 1 ∫ () − 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 © 2015-2016 by Harold Toomey, WyzAnt Tutor 8 Partial Fractions Condition Case I: Simple linear ( degree) Case II: Multiple degree linear ( degree) Case III: Simple quadratic ( degree) Case IV: Multiple degree quadratic ( degree) Example Expansion Typical Solution for Cases I & II Typical Solution for Cases III & IV Series ( mposition) () () = () where () () are polynomials and degree of () < () ( + ) + + 2 ( + ) ( + ) ( + )3 + 2 ( + + ) + + + + + ( 2 + + ) ( 2 + + )2 ( 2 + + )3 () ( + )( + )2 ( 2 + + ) + = + + + ( + ) ( + ) ( + )2 ( 2 + + ) ∫ = | + | + + ∫ 2 = −1 ( ) + 2 + Arithmetic Geometric lim = (Limit) →∞ Sequence Example: ( , +1 , +2 , …) = ∑ Summation Notation =1 Summation Expanded Sum of n Terms (finite series) = 1 + 2 + ⋯ + −1 + (Partial Sum) −1 = ∑ 0 = ∑ 0 −1 =0 = 0 + 0 + 0 2 + ⋯ + 0 −1 1 + ) = ( 2 = (21 + ( − 1)) 2 =1 = 0 1 − 1− (1 − ) = →∞ 1− 1− = lim Sum of ∞ Terms (infinite series) Recursive nth Term Explicit nth Term →∞ = −1 + = 1 + ( − 1) Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor 9 only if || < 1 where is the radius of convergence and (−, ) is the interval of convergence = −1 = 0 −1 Convergence Tests (See Harold’s Series Convergence Tests Cheat Sheet) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Convergence Tests ℎ Term Geometric Series p-Series Alternating Series Integral Ratio Root Direct Comparison Limit Comparison Telescoping ∞ ∑( − +1 ) =1 Telescoping Series Converges if lim = → ∞ Diverges if N/A Sum: = 1 − Taylor Series +∞ Power Series ∑ ( − ) = 0 + 1 ( − ) + 2 ( − )2 + ⋯ =0 +∞ Power Series About Zero ∑ = 0 + 1 + 2 2 + ⋯ =0 +∞ Maclaurin Series Taylor series about zero () ≈ () = ∑ = () () ! () = () + () +∞ Maclaurin Series with Remainder =∑ =0 () (0) (+1) ( ∗ ) +1 + ! ( + 1)! where ≤ ∗ ≤ and lim () = 0 →+∞ +∞ Taylor Series () ≈ () = ∑ =0 () () ( − ) ! () = () + () +∞ Taylor Series with Remainder () () (+1) ( ∗ ) =∑ ( − ) + ( − )+1 ! ( + 1)! =0 where ≤ ∗ ≤ and lim () = 0 →+∞ Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor 10 Common Series Exponential Functions ∞ ! =∑ =0 ∞ = ln() = ∑ =0 1++ ( ln()) ! 1 + ln() + 2 3 4 + + +⋯ 2! 3! 4! ( ln())2 ( ln())3 + +⋯ 2! 3! Natural Logarithms ∞ ln (1 − ) = ∑ =1 ∞ || < 1 + (−1) ( − 1) ln () = ∑ || < 1 =1 ∞ ln (1 + ) = ∑ =1 ( − 1) + (−1)−1 || < 1 2 3 4 5 + + + +⋯ 2 3 4 5 ( − 1)2 ( − 1)3 ( − 1)4 + + +⋯ 2 3 4 − 2 3 4 5 + − + −⋯ 2 3 4 5 Geometric Series ∞ 1 = ∑(−1) ( − 1) 0 < < 2 =0 1 − ( − 1) + ( − 1)2 − ( − 1)3 + ( − 1)4 + ⋯ ∞ 1 = ∑(−1) || < 1 1+ =0 1 − + 2 − 3 + 4 − ⋯ ∞ 1 = ∑ || < 1 1− 1 + + 2 + 3 + 4 + ⋯ 1 = ∑ −1 || < 1 (1 − )2 1 + 2 + 3 2 + 4 3 + 5 4 + ⋯ =0 ∞ =1 ∞ 1 ( − 1) −2 =∑ || < 1 3 (1 − ) 2 1 + 3 + 6 2 + 10 3 + 15 4 + ⋯ =2 Binomial Series +∞ (1 + ) = ∑ ( ) =0 || < 1 and all complex r where −+1 ( )=∏ =1 ( − 1)( − 2) … ( − + 1) = ! Trigonometric Functions ∞ sin () = ∑ =0 1 + + (−1) 2+1 (2 + 1)! ∞ cos () = ∑ =0 (−1) 2 (2)! Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor 11 ( − 1) 2 ( − 1)( − 2) 3 + +⋯ 2! 3! − 3 5 7 9 + − + −⋯ 3! 5! 7! 9! 1− 2 4 6 8 + − + −⋯ 2! 4! 6! 8! ∞ 2 (−4) (1 − 4 ) 2−1 (2)! =1 || < 2 Bernoulli Numbers: 1 1 1 1 0 = 1, 1 = , 2 = , 4 = , 6 = , 2 6 30 42 1 5 691 7 8 = , 10 = , 12 = , 14 = 30 66 2730 6 ∞ (−1) 2 2 sec () = ∑ (2)! =0 || < 2 Euler Numbers: 0 = 1, 2 = −1, 4 = 5, 6 = −61, 8 = 1,385, 10 = −50,521, 12 = 2,702,765 ∞ (2)! arcsin () = ∑ 2 2+1 (2 !) (2 + 1) tan () = ∑ +2 3 5 7 9 + 16 + 272 + 7936 − ⋯ 3! 5! 7! 9! 1 2 5 17 7 2 9 = + 3 + + + −⋯ 3 15 315 945 1+ 2 4 6 8 10 + 5 + 61 + 1385 + 50,521 +⋯ 2! 4! 6! 8! 10! + =0 || ≤ 1 arccos () = − arcsin () 2 || ≤ 1 ∞ (−1) 2+1 arctan () = ∑ (2 + 1) 3 1 ∙ 3 5 1 ∙ 3 ∙ 5 7 + + +⋯ 2∙3 2∙4∙5 2∙4∙6∙7 3 1 ∙ 3 5 1 ∙ 3 ∙ 5 7 −− − − −⋯ 2 2∙3 2∙4∙5 2∙4∙6∙7 − 3 5 7 9 + − + −⋯ 3 5 7 9 − − 2+1 sinh () = =∑ (2 + 1)! 2 + 3 5 7 9 + + + +⋯ 3! 5! 7! 9! + − 2 cosh () = =∑ (2)! 2 1+ 2 4 6 8 + + + +⋯ 2! 4! 6! 8! =0 || < 1, ≠ ± Hyperbolic Functions ∞ =0 ∞ ∞ =0 2 4 (4 − 1) 2−1 tanh () = ∑ (2)! =1 || < 2 ∞ (−1) (2)! arcsinh () = ∑ 2 2+1 (2 !) (2 + 1) 3 5 7 9 − 2 + 16 − 272 + 7936 − ⋯ 3! 5! 7! 9! 1 3 2 5 17 7 2 9 − + − + −⋯ 3 15 315 945 − =0 || ≤ 1 3 1 ∙ 3 5 1 ∙ 3 ∙ 5 7 + − +⋯ 2∙3 2∙4∙5 2∙4∙6∙7 ∞ 2− arccosh () = − ∑ 2+1 2 ! (2 + 1) =0 || ≤ 1 ∞ arctanh () = ∑ =0 3 ∙ 1 ∙ 3 5 ∙ 1 ∙ 3 ∙ 5 7 −− − − −⋯ 2 2∙3 2∙4∙5 2∙4∙6∙7 2+1 (2 + 1) || < 1, ≠ ±1 Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor 12 + 3 5 7 9 + + + +⋯ 3 5 7 9 ...
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