6 - 2π/3 ( −1 / 2, 3 / 2) 3π/4 (− 2 / 2, 2 / 2) 5π/6...

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Unformatted text preview: 2π/3 ( −1 / 2, 3 / 2) 3π/4 (− 2 / 2, 2 / 2) 5π/6 (− 3 / 2, 1 / 2) π (−1,0) 7π/6 (− 3 / 2, − 1 / 2) 5π/4 (− 2 / 2 , − 2 / 2) ( −1 / 2, − π/2 (0,1) π/3 ( 1 / 2, 3 / 2) π/4 ( 2 / 2, 2 / 2) π/6 ( 3 / 2, 1 / 2) θ = 0 or 2 π (x,y) = (1,0) 11π/6 ( 3 / 2, − 1 / 2) 7π/4 ( 2 / 2 , − 2 / 2) tan θ = sin θ/ cos θ, sec θ = 1/ cos θ, sin2 θ + cos2 θ = 1, sin2 θ = 1 2 cot θ = cos θ/ sin θ, csc θ = 1/ sin θ, 1 + tan2 θ = sec2 θ, 1 2 cot θ = 1/ tan θ, 1 + cot2 θ = csc2 θ, − 1 2 cos(2θ), cos2 θ = + 1 2 cos(2θ), sin(2θ) = 2 sin θ cos θ, cos(2θ) = cos2 θ − sin2 θ 4π/3 3 / 2) x = cos θ 3π/2 (0,−1) 5π/3 ( 1 / 2, − 3 / 2) y = sin θ [sin x]′ = cos x [ln x]′ = b [cos x]′ = − sin x [ln g (x)]′ = g ′ (x) g (x) [tan x]′ = sec2 x [ex ]′ = ex [sec x]′ = sec x tan x [eg(x) ]′ = eg(x) g ′ (x) [csc x]′ = − csc x cot x [f g ]′ = f ′ g + f g ′ [cot x]′ = − csc2 x f g ′ 1 x = gf ′ − f g ′ g2 v du. f (x) dx = F (b) − F (a), a F ′ (x) = f (x). f (g (x)) g ′ (x) dx = f (u) du, u = g (x). u dv = uv − L’Hospital’s rule: Exponent rule: If 0 ∞ f (x) → or ± . g (x) 0 ∞ Then lim x→c f (x) f ′ (x) = lim ′ . x→c g (x) x→c g (x) x→c If f (x)g(x) → 00 , 1∞ , ∞0 or 0∞ . Then lim f (x)g(x) = eL where L = lim ln[f (x)g(x) ]. ∞ Geometric series rule: k=1 ∞ crk−1 = c/(1 − r), if |r| < 1 . divergent, if |r| ≥ 1 p-series rule: k =q 1 converges if p > 1, diverges if p ≤ 1. kp ∞ Divergence test: If lim ak = 0. k→∞ Then k =q ak is divergent. ∞ ∞ Integral test: If ak = f (k ), f (x) ≥ 0, f (x) continuous, f (x) decreasing. ∞ ∞ Then k =q ak behaves like q ∞ ∞ f (x) dx. Basic comp test: If 0 < ak ≤ bk . Then k =q bk conv ⇒ k =q ak conv, and k =q ak div ⇒ k =q bk div. Limit comp test: Given ak > 0, bk > 0 consider lim ak = L. k→∞ bk ∞ ∞ If 0 < L < ∞, then bk behave the same. ak and k =q k =q ∞ ∞ If L = 0, then ak conv if bk conv. k =q k =q ∞ ∞ If L = ∞, then ak div if bk div. k =q k =q ∞ ∞ Alt series test: If bk > 0, bk decreasing, lim bk = 0. k→∞ ∞ Then k =q ∞ (−1)k bk and k =q (−1)k−1 bk conv. Absolute conv thm: k =q |ak | conv ⇒ k =q ak conv. Ratio test: If lim k→∞ ak+1 = L. ak ∞ Then k =q ∞ ak is absolutely conv if L < 1, divergent if L > 1 or L = ∞. ak is absolutely conv if L < 1, divergent if L > 1 or L = ∞. k =q ∞ ∞ N ∞ Root test: If lim |ak |1/k = L. k→∞ ∞ Then Int estimate: If k =q ak satisfies conds of Int Test and is conv. ∞ Then N +1 f (x) dx ≤ k =q ∞ ak − k =q ak ≤ N N f (x) dx. Alt series estimate: If k =q (−1)k bk satisfies conds of Alt Ser Test and is conv. ∞ Then k =q (−1)k bk − k =q (−1)k bk ≤ bN +1 . Power series w/ref pt a: f (x) = k=0 ck (x − a)k . ∞ Taylor’s formula for power series w/ref pt a: nth -degree Taylor polynomial of f (x) w/ref pt a: n+1 f (x) = k=0 f (k) (a) (x − a)k . k! n Tn (x) = k=0 f (k) (a) (x − a)k . k! Taylor’s ineq: |f (x) − Tn (x)| ≤ Md for all x ∈ I , where M is max val of |f (n+1) (x)| in I = [a − d, a + d], [a − d, a] or [a, a + d]. (n + 1)! Classic Maclaurin series. 1 = 1−x ex = ∞ ∞ xk , k=0 ∞ −1 < x < 1. −∞ < x < ∞. sin(x) = k=0 ∞ (−1)k x2k+1 , (2k + 1)! (−1)k x2k , (2k )! ∞ −∞ < x < ∞. −∞ < x < ∞. arctan(x) = k=0 ∞ (−1)k x2k+1 , 2k + 1 pk x, k −1 ≤ x ≤ 1. k=0 xk , k! cos(x) = k=0 (1 + x)p = k=0 −1 < x < 1. Parametric curve Slope of C : Length of C : dy = dx dy dt dx dt C : (x, y ) = (f (t), g (t)), a ≤ t ≤ b, 2 w/surface of revolution S (about x-axis assuming y ≥ 0). d dy dt ( dx ) . dx dt . Concavity of C : dy = dx2 Net area b/w C and x-axis: Volume of S : y dx dt. dt π y 2 dx dt . dt ( dx )2 + ( dy )2 dt . dt dt Area of S : 2πy ( dx )2 + ( dy )2 dt . dt dt Simple parametric curves. Line: Circle w/radius r: x = x0 + (x1 − x0 )t, y = y0 + (y1 − y0 )t. Ellipse w/radii a, b in x, y : x = x0 + a cos(t), y = y0 + b sin(t). x = x0 + r cos(t), y = y0 + r sin(t). Polar coordinates (r, θ), Polar coords: polar graphs r = h(θ) in x, y -plane. Polar-Cartesian relation: x = r cos θ, y = r sin θ. θ=angle of radial axis, r=coord along radial axis. Polar graph: points in x, y -plane satisfying r = h(θ). Parametric eqs of polar graph: x = h(θ) cos θ, y = h(θ) sin θ. Polar area of polar graph: 12 2r dθ. ...
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This note was uploaded on 01/27/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas.

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