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# Math 143 Calc III Study Guide Math 143

Cal Poly

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This 6 page Study Guide was uploaded by Ayan Notetaker on Wednesday January 20, 2016. The Study Guide belongs to Math 143 at California Polytechnic State University San Luis Obispo taught by Mark Stankus in Winter 2016. Since its upload, it has received 64 views. For similar materials see Calculus III in Math at California Polytechnic State University San Luis Obispo.

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Date Created: 01/20/16

Made by: Ayan Patel Math 143 – Calculus III – Study Guide Section 10.1 Sequences A sequence is a function whose domain is the set of positive integers. Ex: 3, 3.1, 3.14, 3.141 … Squeeze/Sandwich Theorem: If a < b n c fnr nn> 1 and the lim of a and c ns L, thnn the lim of b is also L. n Section 10.2 Infinite Series A sum of a sequence. 2 n Geometric Series: a + ar + ar + … + ar ????(1−???? ????+1) S n 1−???? ???? ???? If |r| < 1, then Sn= and lim ???? =???? and it converges. 1−???? ????→∞ 1−???? If |r| > = 1, then it diverges. Telescoping Series: ∑ (b n b )n+1 If ∑ anconverges and ∑ b converges, then ∑ (a + b )nconnerges to ∑ a + ∑ b n n If ∑ anconverges and ∑ b dinerges, then ∑ (a + b n dinerges. If both diverge, no conclusion. Test for Divergence: If ∑ anconverges, then lim ???? = 0???? ????→∞ If lim ???? ????s not 0 or doesn’t exist, ∑ a dnverges. ????→∞ Section 10.3 Integral Test Integral Test: If an= f(n) where integral of f(x) is easy to find. If f(x) is continuous, positive, and decreasing. If integral diverges, original series diverges, and if integral converges, original series converges. Made by: Ayan Patel Section 10.4 Comparison Test P-Series: ∑ 1/np If p > 1, it converges. If p <= 1, it diverges. Comparison Test: 0<=a <=n n If ∑ n diverges, then ∑ n diverges. If ∑ n converges, then ∑ anconverges. Else, no conclusion. Limit Comparison Test: ???????? If n > 0 and bn> 0 and ????→∞ ???????? exists and is not 0, then ∑ n and ∑ bnboth converge or diverge. Section 10.5 Ration and Root Tests Ratio Test: If lim???? ????+1= L; ????→∞ ???? ???? If L > 1, ∑na diverges and lim ???? = 0. ????→∞ ???? If L < 1, ∑ a converges. n If L = 1, no conclusion. Root Test: ???? If????→∞m √ ????????= L; If L > 1, ∑na diverges and lim ????????= 0. ????→∞ If L < 1, ∑na converges. If L = 1, no conclusion. Section 10.6 Alternating Series, Absolute/Conditional Convergence Alternating Series: ∑ (-1) un Alternating Series Test: If ∑ n is an alternating series, and lim |???? | = 0, and {|an|} is decreasing, then the ????→∞ ???? series converges. Made by: Ayan Patel Convergence: If ∑na is absolutely convergent, then n a converges. If ∑ na | converges, ∑na absolutely converges. If ∑na converges, and ∑ |n | diverges, n a conditionally converges. Section 10.7 Power Series Power Series: ∑ CnX = C0+ C1X + C2X … ???? ( ) ∑ ∞ ???? Ex: For ???? ???? = ????=0 ????+5 ???????? ????????+1 ????????= and ???? ????+1 = ????+5 ????+6 ????+1 ???? ????+5 ????+5 Ratio Test????→∞im ????+6 ∗ ???????? | = ????→∞ ???? ( ????+6) = |????| If |x| < 1, converges absolutely. If |x| > 1, diverges. Interval of Convergence (Domain): [-1, 1) Radius of Convergence: 1 Section 10.8 Taylor and McLaurin Series ???????? (????) Taylor Series: ????=0 (???? − ????)???? ????! ∞ ???????? (0) ???? McLaurin Series:∑ ????=0 ????! (????) 7x Ex: Find McLaurin Series for f(x) = e 7x f(x) = e f(0) = 1 f’(x) = 7e f’(0) = 7 f’’(x) = 7 e f’’(0) = 7 (n) n 7x (n) n f (x) = 7 e f (0) = 7 7x ∞ 7???? ???? e = ∑ ????=0 ???? ????! ???? 2????+1 sin ???? = ∑∞ (−1) ???? ????=0 (2????+1 ! (−1) ???????? cos ???? = ∑ ????=0 (2???? ! ???? ∞ ???????? ???? = ∑ ????=0 ????! Made by: Ayan Patel Section 10.9 Taylor Series Convergence and Estimates ????(????)???? ) ????(????+1)???? ) Taylor’s Formula: ???? ???? = ???? ???? + ???? ???? ???? − ???? + ⋯+ ) (???? − ????)???? + (???? − ????) ????+1 ????! (????+1 ! 3 ???? Ex: Estimation error for sin(x) with ???? −6 for 0 ≤ x ≤ ∏ ????(4(???? ????−0 )4 |sin(????)||???? | ????4 |f(x) - 3 (x)||= 4! |= 24 ≤ 24 Section 11.1 Parameterizations of Plane Curves Parametric Curve: x = f(t), y= g(t), with t being the parameter. Ex: x = 1 + t, y = t y = (x-1) 3 3 Ex: y = x x = t, y = t Section 11.2 Calculus with parametric curves ???????? ???????? ???????? ???????? = ???????? ???????? ???? ???????? ???? ???? ( ) = ???????? ???????? ???????? 2 ???????? ???????? Section 11.3 Polar Coordinates X = rcosΘ Y = rsinΘ x + y = r2 y/x = tanΘ Polar Coordinantes: (r, Θ) Polar Regions and Polar Curves Section 11.4 Graphing Polar Coordinates Find r and Θ values and graph on an r-Θ plane. Then graph on a x-y plane by figuring out what quadrants each value of Θ falls in. Made by: Ayan Patel Ex: Graph r = 1 – 2sinΘ Section 11.5 Areas and Lengths in Polar Coordinates ????√ ???????? 2 ???????? 2 ???? = ???? (???????? + ( )???????????? 2 ???? = ∫????√ ???? + ( ) ???????? ???? ???????? ???? 1 2 ???????????????? = ∫???? 2 (???? ????)) ???????? Section 12.1 Three dimensional coordinate system 3 R x-y-z plane 2 2 2 Distance =√∆???? + ∆???? + ∆???? Sphere: r = (x-h) + (y-k) + (z-l) Section 12.2 Vectors u = <3, 4> <3,0> + <0,4> = <3,4> 25<7,4> = <25(7),25(4)> ( ) ( )( ) |<4,-2>| = 4 4 + −2 −2 = √20 Made by: Ayan Patel Section 12.3 Dot Product ⃑⃑ ⃑⃑ ⃑ ∙ ???? = ???? ????1 1???? ???? = 2 2 |⃑ | |???????????? ???????????????? ???? = ???? ∙ ????⃑ ????⃑ |????|2 Section 12.4 Cross Product ???? ???? ???? ⃑ ???? ???? = |???? 1 ????2 ????3 | ???? 1 ????2 ????3 Area of Parallelogram = ????|⃑ ???? ????| Section 12.5 Lines and Planes in Space Equation of Line between two points Point P and Q, find vector PQ. Line r = point P + t(vector PQ) Equation of plane containing points Points P, Q, R, find vector PQ and PR. Find PQ X PR. (PQ X PR) • (<x,y,z> - point P) = 0 Equation of plane with a point and orthogonal line Point P and line L. Find normal vector n from line. n • (<x,y,z> - point P) = 0

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