12.6 Quadric surfaces Review: rv point on the line, direction abc, , Vector equation of a line L: r r0 t v 0 Parametric scalar equation of a line L: x x0 at, y y 0 bt, z z 0 ct r(t ) P0 t ( P 1 P 0 ) with 0 t 1 is the line which goes from P 0 to P 1 r0 on the plane Vector equation of a plane: n r r0 0 n abc, , normal to plane Scalar equation of the plane: ax by cz d nn12 Angle between two planes with normal nn12, : cos nn12 | (Qrn ) | Distance between a point Q and a plane: 0 n | (Qrv ) | Distance between a point Q and a line: 0 | v | Surfaces in 3-space: x2 y 2 z 2 Ellipsoid 1 a2 b 2 c 2 Hyperboloid of one sheet x2 y 2 k 2 x2 y 2 z 2 z k: 1 any k a2 b 2 c 2 2 2 2 1 abc horizontal traces are ellipses Hyperboloid of two sheet x2 y 2 k 2 z k: 2 2 2 1, | k | 1 x2 y 2 z 2 a b c 1 horizontal traces are ellipses a222 b c xy22 (Elliptical paraboloid) z ab22 Horizontal traces are ellipses (notice z 0) xy22 (Hyperbolic paraboloid) z ab22 Horizontal traces are hyperbolas Vertical traces are parabolas xy22 Cone z2 ab22 Horizontal traces are ellipses Vertical traces are hyperbolas y Notice: If x 0 z straight lines b Cylinders: xy22 yx22 y ax2 1 1 ab22 ab22 VII IV II III VI I VIII V Draw a picture of the following surfaces: x y22 z 22or a) x2 y 3 z 6 3 2 elliptic paraboloid opens in the positive x direction yz22 b) 4 x y22 4 z 0 or x 41 hyperbolic paraboloid Draw a picture of the surface : 4x2 y 2 4 z 2 4 y 24 z 36 0 4x2 y 2 4 y ___ 4 z 2 6 z ___9 36 ___4 ___36 4x2 y 222 4 z 3 4 22 x2 yz23 1 1 4 1 ellipsoid center: 0,2,3 notice e.g. y 0 Summary: x2 y2 z 2 1 Ellipsoid a2 b2 c2 x2 y 2 z 2 1 Hyperboloid of one sheet abc2 2 2 all variables present x2 y 2 z 2 1 Hyperboloid of two sheets a222 b c all variables squared z2 x 2 y 2 Cone elliptic c2 a 2 b 2 z x22 y Paraboloid elliptic c a22 b z x22 y one variable Paraboloid hyperbolic 22 c a b not squared one variable not present cylinder opening in the direction of the missing variable 22 22 xy xz 2 1 Elliptic cylinder 1 Hyperbolic cylinder z ax Parabolic cylinder ab22 ab22 13.1 Curves in Space A curve in space (or the plane): x f( t ), y g( t ), z h( t ) or in the plane: r(),,t f g h f i g j h k r()()()t f t j g t j This is a vector valued function (as opposed to scalar valued) Example: r(tt ) cos( ),sin(t), t Notice that x( t )22 y ( t ) 1 i.e. the curve lies on a cylinder A circular helix: A curve in the plane: r(tt ) 4cos(2 ) i 9sin(2 t) j xy22 Notice that 1 The particle travels along an ellipse 4922 r(0) 4 i = (4,0) Travels counter clockwise as 0t 2 the particle goes twice around the ellipse Trefoil knot: 3t 3 t 3 t rt 2 cos 2 cos t , 2 cos 2 sin t ,sin 2 For an animation go to : http://math.bu.edu/people/paul/225/trefoil_knot.html For a vector valued function: rt f t,, g t h t Take limits, derivatives and integrals component wise: limrt lim f t ,lim g t ,lim h t t a t a t a t a rt f t,, g t h t b b b b rtdt ftdt , gtdt , htdt a a a a sint Example: Let rrt , e2t ,ln 1 t . Find lim t t t0 sin t limr te lim ,0 ,ln 1 1,1,0 ij tt00t Examples: (a) Let rt sin12 t , 1 t ,ln 1 3 t . Find rt 1/2 1 1 2 1 13t rt , 12 tt , 3 ,, 1t 2 2 13 t 11tt2213 t 2 2 2 2 (b) t2i t t 1 j t sin t k dt tdttt2 , 1 dtt , sin tdt 1 1 1 1 2 3 2 t 81 7 t2 dt 33 3 1 3 1 2 1 1 1 3/2 1/2 225/2 3/2 u1 udu u u dt uu t t1 dt 53 0 0 0 1 ut1 du dt 22 6 10 16 tu10 53 15 15 15 tu21 2 tsin t dt ut dv sin t 1 1 udv uv vdu du dt vt cos 2 t 2 1 cost cos t dt 1 1 2 t 1 2 1 1 1 costt sin cos 2 22 sin 2 cos sin 2 1 21 3 1 0 1 0 2 7 16 3 t2i t t 1 j t sin t k dt , , 1 3 15 Differentiation Rules: Integration rules: d 1. ut v t u t v t (u(t ) v (t)) dt u ( t ) dt v ( t ) dt dt d 2. cuu t c t cuu()() t dt c t dt dt d 3. f tu t f t u t f t u t dt d 4. ut v t u t v t u t v t dt d 5. ut v t u t v t u t v t dt d 6. uu f t f t f t dt Geometry of the derivative vector : z secant vector P rrt h t Q r t rth y Tangent vector x tangent vector P rrt h t rt lim Q rrt h t r t h0 h h rth The tangent line to a smooth curve at tt 0 rt ftgtht ,, ft i gt j ht k It passes through the point r(t0 ) f t 0 , g t 0 , h t 0 , and has the same direction as the tangent vector at t0 r '(t0 ) f t 0 , g t 0 , h t 0 . So the equation of the tangent line is: s r(t00) s r '( t ) Or in parametric form: x f( t0 ) sf '( t 0 ), y g ( t 0 ) sg '( t 0 ), z h ( t 0 ) sh '( t 0 ) Problem : A drunken bee travels along the path r(tt ) cos(2 ),sin(2 t), t for 10 seconds. It then travels at constant speed in a straight line for 10 more seconds. At what point does the bee end up? Question: What is speed and velocity? Velocity is a vector: v(t ) r'(t) Speed is a scalar: | v(t ) | velocity: v(tt ) 2sin(2 ),2cos(2 t),1 Speed = | v(t ) | 9 3 At time t0 10, it travels along the tangent line with speed 3 ft/sec tangent line: s r(t00) s v ( t ) after 10 more seconds, it arrives at r(tt00) 10 v ( ) cos(20),sin(20),10 10 2sin(20),2cos(20),1 cos(20) 20sin(20),sin(20) 20cos(20),20 17.85, 9.07, 20 Question: What is distance traveled?.