Rectangular Polar Parametric

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Hrold s AP Clculus BC Rectngulr Polr Prmetric Chet Sheet 15 Octoer 2017 Point Line Rectngulr Polr Prmetric f(x) = y (x, y) (, ) Slope-Intercept Form: y = mx + Point-Slope Form: y y 0 = m (x x 0 )

1 Hrold s AP Clculus BC Rectngulr Polr Prmetric Chet Sheet 15 Octoer 2017 Point Line Rectngulr Polr Prmetric f(x) = y (x, y) (, ) Slope-Intercept Form: y = mx + Point-Slope Form: y y 0 = m (x x 0 ) Generl Form: Ax + By + C = 0 Clculus Form: f(x) = f () x + f(0) (r, θ) or r θ Polr Rect. Rect. Polr x = r cos θ y = r sin θ tn θ = y x r 2 = x 2 + y 2 r = ± x 2 + y 2 θ = tn 1 ( y x ) Point (,) in Rectngulr: x(t) = y(t) = <, > t = 3 rd vrile, usully time, with 1 degree of freedom (df) < x, y > = < x 0, y 0 > + t <, > < x, y > = < x 0 + t, y 0 + t > where <, > = < x 2 x 1, y 2 y 1 > x(t) = x 0 + t y(t) = y 0 + t m = y x = y 2 y 1 x 2 x 1 = r = r 0 + sv + tw Plne n x (x x 0 ) +n y (y y 0 ) + n z (z z 0 ) = 0 Vector Form: n (r r 0 ) = 0 where: v nd w re given vectors defining the plne r 0 is the vector of fixed point on the plne Copyright y Hrold Toomey, WyzAnt Tutor 1

0 Circle: A = C nd B = 0 Ellipse: AC > 0 or B 2 4AC < 0 Prol: AC = 0 or B 2 4AC = 0 Hyperol: AC < 0 or B 2 4AC > 0 Note: If A +

= (x, y ), Old = (x, y) rottes through ngle θ from x-xis r = p 1 e cos θ (1 e 2 ) 0 e < 1 where p = { 2d for { e = 1 (e 2 1) e >

2 Rectngulr Polr Prmetric. Generl Eqution for All Conics: Generl Eqution for All Conics: Conics Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 where Line: A = B = C = 0 Circle: A = C nd B = 0 Ellipse: AC > 0 or B 2 4AC < 0 Prol: AC = 0 or B 2 4AC = 0 Hyperol: AC < 0 or B 2 4AC > 0 Note: If A + C = 0, squre hyperol Rottion: If B 0, then rotte coordinte system: A C cot 2θ = B x = x cos θ y sin θ y = y cos θ + x sin θ New = (x, y ), Old = (x, y) rottes through ngle θ from x-xis r = p 1 e cos θ (1 e 2 ) 0 e < 1 where p = { 2d for { e = 1 (e 2 1) e > 1 p = semi-ltus rectum or the line segment running from the focus to the curve in direction prllel to the directrix Eccentricity: Circle e = 0 Ellipse 0 e < 1 Prol e = 1 Hyperol e > 1 Copyright y Hrold Toomey, WyzAnt Tutor 2

3 Rectngulr Polr Prmetric. x 2 + y 2 = r 2 (x h) 2 + (y k) 2 = r 2 Centered t Origin: r = (constnt) θ = θ [0, 2π] or [0, 360 ] Circle Center: (h, k) Vertices: NA Focus: (h, k) Centered t (r 0, φ): r 2 + r 0 2 2rr 0 cos(θ φ) = R 2 Hint: Lw of Cosines or r = r 0 cos(θ φ) + 2 r 0 2 sin 2 (θ φ) x(t) = r cos(t) + h y(t) = r sin(t) + k [t min, t mx ] = [0, 2π) (h, k) = center of circle (h, k) Copyright y Hrold Toomey, WyzAnt Tutor 3

1 + e cos θ where e = c = 2 2 reltive to center (h,k) x(t) = cos(t) + h y(t) = sin(t) + k [t min, t mx ] = [0, 2π] (h, k) = center of ellipse Rotted Ellipse:

4 Rectngulr Polr Prmetric. (x h) 2 2 (y k)2 + 2 = 1 Center: (h, k) Vertices: (h ±, k) nd (h, k ± ) Foci: (h ± c, k) Focus length, c, from center: c = 2 2 Ellipse: r = (1 e2 ) for 0 e < e cos θ where e = c = 2 2 reltive to center (h,k) x(t) = cos(t) + h y(t) = sin(t) + k [t min, t mx ] = [0, 2π] (h, k) = center of ellipse Rotted Ellipse: x(t) = cos t cos θ sin t sin θ + h y(t) = cos t sin θ + sin t cos θ + k Ellipse θ = the ngle etween the x-xis nd the mjor xis of the ellipse Interesting Note: The sum of the distnces from ech focus to point on the curve is constnt. d 1 + d 2 = k Copyright y Hrold Toomey, WyzAnt Tutor 4

4p(x h) Vertex: (h, k) Focus: (h + p, k) Directrix: x = h p Prol: 2d r = for e = 1 1 + e cos θ where d = 2p Trigonometric Form: y = x 2 r sin θ = r 2 cos 2 θ

c] (h, k) = vertex of prol Horizontl Axis of Symmetry: y(t) = 2pt + k x(t) = pt 2 + h (opens to the right) or x(t) = pt 2 h (opens to the left) [t min, t mx ]

5 Prol Rectngulr Polr Prmetric. Verticl Axis of Symmetry: x 2 = 4 py (x h) 2 = 4p(y k) Vertex: (h, k) Focus: (h, k + p) Directrix: y = k p Horizontl Axis of Symmetry: y 2 = 4 px (y k) 2 = 4p(x h) Vertex: (h, k) Focus: (h + p, k) Directrix: x = h p Prol: 2d r = for e = e cos θ where d = 2p Trigonometric Form: y = x 2 r sin θ = r 2 cos 2 θ sin θ r = cos 2 = tn θ sec θ θ Verticl Axis of Symmetry: x(t) = 2pt + h y(t) = pt 2 + k (opens upwrds) or y(t) = pt 2 k (opens downwrds) [t min, t mx ] = [ c, c] (h, k) = vertex of prol Horizontl Axis of Symmetry: y(t) = 2pt + k x(t) = pt 2 + h (opens to the right) or x(t) = pt 2 h (opens to the left) [t min, t mx ] = [ c, c] (h, k) = vertex of prol Projectile Motion: x(t) = x 0 + v x t y(t) = y 0 + v y t 16t 2 feet y(t) = y 0 + v y t 4.9t 2 meters v x = v cos θ v y = v sin θ Generl Form: x = At 2 + Bt + C y = Dt 2 + Et + F where A nd D hve the sme sign Copyright y Hrold Toomey, WyzAnt Tutor 5

1 reltive to center (h,k) Left-Right Opening Hyperol: x(t) = sec( t) + h y(t) = tn( t) + k [t min, t mx ] = [ c, c] (h, k) = vertex of hyperol Up-Down Opening Hyperol: x(t) = tn(t) + h y(t) = sec(t)

6 Hyperol Rectngulr Polr Prmetric. (x h) 2 2 (y k)2 2 = 1 Center: (h, k) Vertices: (h ±, k) Foci: (h ± c, k) Focus length, c, from center: c = Hyperol: r = (e2 1) for e > e cos θ Eccentricity: where e = c = = sec θ > 1 reltive to center (h,k) Left-Right Opening Hyperol: x(t) = sec( t) + h y(t) = tn( t) + k [t min, t mx ] = [ c, c] (h, k) = vertex of hyperol Up-Down Opening Hyperol: x(t) = tn(t) + h y(t) = sec(t) + k [t min, t mx ] = [ c, c] (h, k) = vertex of hyperol p = semi-ltus rectum or the line segment running from the focus to the curve in the directions θ = ± π 2 Interesting Note: The difference etween the distnces from ech focus to point on the curve is constnt. d 1 d 2 = k Generl Form: x(t) = At 2 + Bt + C y(t) = Dt 2 + Et + F where A nd D hve different signs Copyright y Hrold Toomey, WyzAnt Tutor 6

7 Rectngulr Polr Prmetric. 1 st Derivtive 2 nd Derivtive f f(x + h) f(x) (x) = lim h 0 h f f(x) f(c) (c) = lim x c x c f (x) = dy = y = D x f (x) = d (dy ) = d2 y 2 = y dy = dy = dr sin θ + r cos θ dr cos θ r sin θ Hint: Use Product Rule for y = r sin θ x = r cos θ d 2 y 2 = d (dy ) = d (dy ) dy = dy, provided 0 d 2 y 2 = d (dy ) = Riemnn Sum: n d (dy ) S = f(y i )(x i x i 1 ) i 1 Left Sum: S = ( 1 n ) [f() + f ( + 1 n ) + f ( + 2 n ) + + f( 1 n )] Integrl F(x) = f(x) = F() F() Middle Sum: S = ( ) [f ( + ) + f ( + n 2n 2n ) + + f( 1 2n )] Right Sum: S = ( 1 n ) [f ( + 1 n ) + f ( + 2 n ) + + f()] Copyright y Hrold Toomey, WyzAnt Tutor 7

8 Inverse Functions Arc Length Rectngulr Polr Prmetric. f(f 1 (x)) = f 1 (f(x)) = x Inverse Function Theorem: f 1 1 () = f () where = f () L = 1 + [f (x)] 2 Proof: s = (x x 0 ) 2 + (y y 0 ) 2 s = ( x) 2 + ( y) 2 ds = 2 + dy 2 ds = 2 + dy 2 ( 2 2) ds = 2 + ( dy ) 2 2 ds = 2 (1 + ( dy ) 2 ) if y = sin θ if y = cos θ if y = tn θ if y = csc θ if y = sec θ if y = cot θ L = ds ds = r 2 + ( dr ) 2 Circle: L = s = rθ then θ = sin 1 y then θ = cos 1 y then θ = tn 1 y then θ = csc 1 y then θ = sec 1 y then θ = cot 1 y Proof: L = (frction of circumference) π (dimeter) L = ( θ ) π (2r) = rθ 2π θ = rcsin y θ = rccos y θ = rctn y θ = rccsc y θ = rcsec y θ = rccot y L = ( L = ( + ( dy + ( dy + ( dz ds = 1 + ( dy ) 2 L = ds Perimeter Squre: P = 4s Rectngle: P = 2l + 2w Tringle: P = + + c Circle: C = πd = 2πr Ellipse: C π( + ) π Ellipse: C = ( c )2 sin 2 θ 0 Copyright y Hrold Toomey, WyzAnt Tutor 8

9 Are Lterl Surfce Are Rectngulr Polr Prmetric. Squre: A = s² Rectngle: A = lw Rhomus: A = ½ Prllelogrm: A = h Trpezoid: A = ( 1+ 2 ) h 2 Kite: A = d 1 d 2 2 Tringle: A = ½ h Tringle: A = ½ sin(c) Tringle: A = s(s )(s )(s c), where s = ++c 2 Equilterl Tringle: A = ¼ 3s 2 Frustum: A = 1 3 ( 1+ 2 ) h 2 Circle: A = πr² Circulr Sector: A = ½ r²θ Ellipse: A = π Cylinder: S = 2πrh Cone: S = πrl S = 2π f(x) 1 + [f (x)] 2 A = 1 2 [f(θ)]2 where r = f(θ) Proof: Are of sector: A = s dr = r θ dr = 1 2 r2 θ where rc length s = r θ For rottion out the x-xis: S = 2πy ds For rottion out the y-xis: S = 2πx ds A = g(t) f (t) where f(t) = x nd g(t) = y or x(t) = f(t) nd y(t) = g(t) Simplified: A = y(t) (t) Proof: f(x) y = f(x) = g(t) = df(t) = f (t) For rottion out the x-xis: S = 2πy ds For rottion out the y-xis: S = 2πx ds ds = r 2 + ( dr 2 ) r = f(θ), θ ds = ( 2 ) + ( dy 2 ) if x = f(t), y = g(t), t Copyright y Hrold Toomey, WyzAnt Tutor 9

10 Totl Surfce Are Surfce of Revolution Volume Rectngulr Polr Prmetric. Cue: S = 6s² Rectngulr Box: S = 2lw + 2wh + 2hl Regulr Tetrhedron: S = 2h Cylinder: S = 2πr (r + h) Cone: S = πr² + πrl = πr (r + l) For revolution out the x-xis: A = 2π f(x) 1 + ( dy ) 2 For revolution out the y-xis: A = 2π x 1 + ( dy ) 2 dy Cue: V = s³ Rectngulr Prism: V = lwh Cylinder: V = πr²h Tringulr Prism: V= Bh Tetrhedron: V= ⅓ h Pyrmid: V = ⅓ Bh Cone: V = ⅓ h = ⅓ πr²h Sphere: S = 4πr² Ellipsoid: S = (too complex) For revolution out the x-xis: A = 2π r cos θ r 2 + ( dr ) 2 For revolution out the y-xis: A = 2π r sin θ Sphere: V = 4 3 πr3 r 2 + ( dr ) 2 Ellipsoid: V = 4 3 πc For revolution out the x-xis: A = 2π y(t) ( + ( dy For revolution out the y-xis: A = 2π x(t) ( + ( dy Copyright y Hrold Toomey, WyzAnt Tutor 10

11 Rectngulr Polr Prmetric. Disc Method - Rottion out the x- xis: V = π [f(x)] 2 Cylindricl Shell Method: = (re of circle) d(thickness) Disc Method: Volume of Revolution Wsher Method - Rottion out the x-xis: V = π { [f(x)] 2 [g(x)] 2 } Cylinder Method - Rottion out the y-xis: V = 2πx f(x) = (circumference) (hight) Copyright y Hrold Toomey, WyzAnt Tutor 11

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