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Intermediate Algebra
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Unformatted text preview: Harold’s Calculus Notes “Cheat Sheet” 26 January 2015 AP Calculus AB & BC 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 f(a) and see if it provides a legal answer. If so then L = f(a). The Existence of a Limit The limit of f(x) as x approaches a is L if and only if: lim () = → − and lim+ () = → Definition of Continuity Prove that () = − is a continuous function. |() − ()| = |( 2 − 1) − ( 2 − 1)| = | 2 − 1 − 2 + 1| = | 2 − 2 | = |( + )( − )| = |( + )| |( − )| This can be used to prove that f(x) is a Since |( + )| ≤ |2| |() − ()| ≤ |2||( − )| < Continuous Function. For −∞ < < ∞, |() substitute into − ()| < . So given > 0, we can choose = | | > in the Definition of Continuity. So substituting the Note: The trick is to rearrange |() − ()| chosen for |( − )| we get: to have |( − )| as a factor. Since | − | < 1 |() − ()| ≤ |2| (| | ) = we can find an equation that relates both 2 and together. Since both conditions are met, the function is continuous. =1 →0 The Definition of Continuity states that a function f is continuous at c if for every > 0 there exists a > 0 such that | − | < and |() − ()| < . Two Special Trig Limits 1 − =0 →0 Copyright © 2013 by Harold Toomey, WyzAnt Tutor 1 Derivatives Definition of a Derivative of a Function Slope 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 The Product Rule The Quotient Rule The Chain Rule Exponentials ( , ) Logorithms ( , ) Sine Cosine Tangent Secent Cosecent Cotangent (See Larson’s 1-pager of common derivatives) ( + ∆) − () ′ () = lim ∆→0 ∆ () − () ′ () = lim → − ′ ′ (), () (), , , [()], [] [] = 0 [ ] = −1 [] = 1 (think = 1 0 = 1) [ ] = −1 ′ [()] = [()]−1 [()] = ′ () [() ± ()] = ′ () ± ′ () 1 () = 2 + 0 + 0 2 () = ′ () = + 0 () = ′ () = ′′ () [] = ′ + ′ ′ − ′ [ ]= 2 = (), = () ≠ 0 = · [(())] = ′ (())′ () [ ] = , [ ] = (ln ) 1 1 [ln ] = , [log ] = (ln ) [()] = cos() [()] = −() [()] = 2() [()] = () () [()] = − () () [()] = − 2 () Copyright © 2013-2015 by Harold A. Toomey, WyzAnt Tutor 2 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 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 1st derivative test) 1. If f”(x) > 0 for all x, then the graph is concave upward 2. If f”(x) < 0 for all x, then the graph is concave downward 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 Copyright © 2013-2015 by Harold A. Toomey, WyzAnt Tutor 3 Relative Extrema Concavity Points of Inflection 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 ( + ∆) ≈ () + = () + ′ () 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 © 2013-2015 by Harold A. Toomey, WyzAnt Tutor 4 Integration Basic Integration Rules Integration is the “inverse” of differentiation. Differentiation is the “inverse” of integration. ∫ ′ () = () + () = 0 ∫ 0 = () = = 0 ∫ = + [∫ () ] = () The Constant Multiple Rule The Sum and Difference Rule ∫ () = ∫ () ∫[() ± ()] = ∫ () ± ∫ () ∫ = The Power Rule () = +1 + , ℎ ≠ −1 +1 If = −1 then ∫ −1 = ln|| + If = (), ′ = () then +1 ∫ ′ = + , ℎ ≠ −1 +1 The General Power Rule Reimann Sum ∑ ( ) ∆ , ℎ −1 ≤ ≤ =1 ‖∆‖ = ∆ = Definition of a Definite Integral Area under curve ‖∆‖→0 =1 ∫ () = − ∫ () Additive Interval Property ∫ () = ∫ () + ∫ () The Second Fundamental Theorem of Calculus lim ∑ ( ) ∆ = ∫ () Swap Bounds The Fundamental Theorem of Calculus − ∫ () = () − () (See Harold’s Fundamental Theorem of Calculus “Cheat Sheet”) = [∫ () ] = () () [∫ () ] = () ′() ′ () = (See Harold’s Fundamental Theorem of Calculus “Cheat Sheet”) Copyright © 2013-2015 by Harold A. Toomey, WyzAnt Tutor 5 Mean Value Theorem for Integrals ∫ () = ()( − ) The Average Value for a Function Find ‘c’. 1 ∫ () − ∑ = =1 ( + 1) 2 ∑ = = + 2 2 2 =1 ( + 1)(2 + 1) 3 2 ∑ = = + + 6 3 2 6 2 Summation Formulas =1 2 ( + 1)2 4 3 2 ∑ = = + + 4 4 2 4 3 =1 ∑ 4 = =1 ( + 1)(2 + 1)(32 + 3 − 1) 30 = Copyright © 2013-2015 by Harold A. Toomey, WyzAnt Tutor 5 4 3 + + − 5 2 3 30 6 Integration Methods 1. Memorized See Larson’s 1-pager of common integrals ∫ (())′ () = (()) + Set = (), then = ′ () 2. U-Substitution ∫ () = () + = _____ = _____ ∫ = − ∫ = ____ = _____ 3. Integration by Parts = _____ = _____ Pick ‘’ using the LIATE Rule: L - Logarithmic : ln , log , . I - Inverse Trig.: tan−1 , sec −1 , . A - Algebraic: 2 , 3 60 , . T - Trigonometric: sin , tan , . E - Exponential: , 19 , . () ∫ () 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 5. Trig Substitution for √ − Substutution: = sin Identity: 1 − 2 = 2 ∫ √ 2 − 2 6. Trig Substitution for √ − Substutution: = sec Identity: 2 − 1 = 2 ∫ √ 2 + 2 7. Trig Substitution for √ + 8. Table of Integrals 9. Computer Algebra Systems (CAS) 10. Numerical Methods 11. 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 Google of mathematics. Shows steps. Free. WolframAlpha iPhone/iPad app Copyright © 2013-2015 by Harold A. Toomey, WyzAnt Tutor 7 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) Typical Solution for Cases I & II Typical Solution for Cases III & IV Numerical Methods ( mposition) () () = () where () () are polynomials and degree of () < () ( + ) + + 1 2 ( + ) ( + ) ( + )3 + 2 ( + + ) + + + + + 2 2 2 2 ( + + )3 ( + + ) ( + + ) ∫ = | + | + + ∫ 2 = −1 ( ) + 2 + ∫ () = lim ∑ (∗ ) ∆ ‖‖→0 Riemann Sum =1 where = 0 < 1 < 2 < ⋯ < = and ∆ = − −1 and ‖‖ = {∆ } Types: Left Sum, Middle Sum, Right Sum ∫ () ≈ ∑ (̅ ) ∆ = =1 ∆[(̅1 ) + (̅2 ) + (̅3 ) + ⋯ + (̅ )] − where ∆ = Midpoint Rule and ̅ = 1 (−1 2 + ) = [−1 , ] Error Bounds: | | ≤ (−)3 242 ∫ () ≈ Trapezoidal Rule ∆ [(0 ) + 2(1 ) + 2(3 ) + ⋯ + 2(−1 ) 2 + ( )] − where ∆ = and = + ∆ Error Bounds: | | ≤ Copyright © 2013-2015 by Harold A. Toomey, WyzAnt Tutor (−)3 122 8 ∫ () ≈ Simpson’s Rule ∆ [(0 ) + 4(1 ) + 2(2 ) + 4(3 ) + ⋯ 3 + 2(−2 ) + 4(−1 ) + ( )] Where n is even − and ∆ = and = + ∆ Error Bounds: | | ≤ Infinite Sequences and Series (See Harold’s Series “Cheat Sheet”) lim = (Limit) Sequence →∞ Example: ( , +1 , +2 , …) ∞ Arithmetic Series (infinite) (−)5 1804 ∑ = = 1 + 2 + 3 + ⋯ + + ⋯ =1 Geometric Series (finite) ∑ −1 = = + + 2 + ⋯ + −1 =1 (1 − ) () 1− (1 − ) = lim = →∞ 1− 1− only if || < 1 where is the radius or interval of convergence = Geometric Series (infinite) Partial Sum = ∑ = 1 + 2 + 3 + ⋯ + Similar to an arithmetic series Copyright © 2013-2015 by Harold A. Toomey, WyzAnt Tutor 9 Convergence Tests (See Harold’s Series Convergence Tests “Cheat Sheet”) 1. Term Test 2. Geometric Series Test ∞ ∑( − +1 ) =1 3. Telescoping Series Converges if lim = → ∞ Diverges if N/A Sum: = 1 − 4. p-Series Test 5. Alternating Series Test 6. Integral Test 7. Ratio Test 8. Root Test 9. Direct Comparison Test 10. Limit Comparison Test Taylor Series +∞ Power Series ∑ ( − ) = 0 + 1 ( − ) + 2 ( − )2 + ⋯ =0 Power Series About Zero +∞ ∑ = 0 + 1 + 2 2 + ⋯ =0 Maclaurin Series Taylor series about zero +∞ () ≈ () = ∑ () (0) ! =0 +∞ () Taylor Series () ≈ () = ∑ =0 () ( − ) ! () = () + () +∞ =∑ =0 Taylor Series with Remainder () () ( − ) + () ! (+1) ( ∗ ) ( − )+1 ( + 1)! where ≤ ∗ ≤ and lim () = 0 () = +∞ →+∞ ( − 1)( − 2) … ( − + 1) (1 + ) = 1 + ∑ ! =1 Binomial Series (1 + ) = 1 + + || < 1 ( − 1) 2 ( − 1)( − 2) 3 + 2! 3! +⋯ (See Harold’s Illegals and Graphing Rationals “Cheat Sheet”) Copyright © 2013-2015 by Harold A. Toomey, WyzAnt Tutor 10 Common Series ∞ =∑ =0 ∞ ! 1++ 1 = ∑(−1) ( − 1) 0 < < 2 =0 2 3 4 + + +⋯ 2! 3! 4! 1 − ( − 1) + ( − 1)2 − ( − 1)3 + ( − 1)4 + ⋯ ∞ 1 = ∑(−1) || < 1 1+ =0 1 − + 2 − 3 + 4 − ⋯ ∞ 1 = ∑ || < 1 1− 1 + + 2 + 3 + 4 + ⋯ =0 ∞ ln () = ∑(−1)−1 =0 ( − 1) 0 < ≤ 2 ∞ ln (1 + ) = ∑(−1)−1 =0 ∞ sin () = ∑(−1) =0 ∞ − ( − 1)2 ( − 1)3 ( − 1)4 ( − 1)5 + − + −⋯ 2 3 4 5 || < 1 2 3 4 5 − + − + −⋯ 2 3 4 5 2+1 (2 + 1)! − 3 5 7 9 + − + −⋯ 3! 5! 7! 9! 2 (2)! 1− 2 4 6 8 + − + −⋯ 2! 4! 6! 8! + 3 5 7 9 + + + +⋯ 3! 5! 7! 9! 1+ 2 4 6 8 + + + +⋯ 2! 4! 6! 8! cos () = ∑(−1) =0 ∞ − − 2+1 sinh () = =∑ (2 + 1)! 2 + cosh () = 2 − =0 ∞ 2 =∑ (2)! =0 2+1 ∞ arcsin () = ∑(2)! =0 ∞ (2 !)2 (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∙3 2∙4∙5 2∙4∙6∙7 || ≤ 1 arcsinh () = ∑(−1) (2)! =0 2+1 (2 !)2 (2 + 1) || ≤ 1 2+1 arctan () = ∑(−1) || < 1 (2 + 1) ∞ =0 ∞ 2+1 arctanh () = ∑ || < 1 (2 + 1) =0 Euler’s Equation Copyright © 2013-2015 by Harold A. Toomey, WyzAnt Tutor − 3 5 7 9 + − + −⋯ 3 5 7 9 3 5 7 9 + + + +⋯ 3 5 7 9 +1=0 = cos() + sin() + 11 ...
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