166 Calculus, Practice Exam 4. (* The actual Exam 4 is Friday, April 25, 2003 7:30 to 9:00 AM *) Instructions: You are welcome to check your answers with your calculators. But you will not be given full credit unless your work is clear, complete, and correct. In particular when you use the cross product, write out the full 3x3 matrix with i,j,k in the top row. 1. Choose the appropriate name for the quadric surfaces. The possible choices are from Sphere, Ellipsoid, Hyperboloid of one Sheet, Hyperboloid of two sheets, Cone, hyperbolic paraboloid, Cylindrical Surface Paraboloid (a) x^2 + y + z^2 = 0 ____________________________________ Paraboloid (b) x^2 - y + z^2 = 0 ____________________________________ Hyperbolic Paraboloid (c) x^2 + y - z^2 = 0 ____________________________________ Hyperbolic Paraboloic (d) -x^2 - y + z^2 = 0 ____________________________________ Cone (e) x^2 - y^2 - z^2 = 0 ____________________________________ Sphere (f) x^2 + y^2 + z^2 = 1 ____________________________________ Hyperboloid of one sheet (g) x^2 + y^2 - z^2 = 1 ____________________________________ Hyperboloid of two sheets (h) x^2 - y^2 - z^2 = 1 ____________________________________ ParametricPlot3D[{ r Cos[t], -r^2, r Sin[t]},{r,0,-5},{t,0,2 Pi}]; ParametricPlot3D[{ r Cos[t], r^2, r Sin[t]},{r,0,-5},{t,0,2 Pi}]; ParametricPlot3D[ {x,-x^2+z^2,z},{x,-2,2},{z,-2,2}]; ParametricPlot3D[ {x, -x^2+z^2,z},{x,-2,2},{z,-2,2}]; ParametricPlot3D[ { r, r Cos[t], r Sin[t]}, {r,-10,10},{t,0,2 Pi}]; ParametricPlot3D[ { Sin[f] Cos[t], Sin[f] Sin[t], Cos[f]},{t,0,2 Pi},{f,0,Pi}]; ParametricPlot3D[ { r Cos[t], r Sin[t], Sqrt[r^2-1] }, {r, 1, 3},{t,0,2 Pi}]; ParametricPlot3D[ { r Cos[t], r Sin[t],-Sqrt[r^2-1] }, {r, 1, 3},{t,0,2 Pi}]; ParametricPlot3D[ {Sqrt[1+r^2], r Cos[t], r Sin[t]}, {r,0,5},{t,0,2 Pi}]; ParametricPlot3D[ {-Sqrt[1+r^2], r Cos[t], r Sin[t]}, {r,0,5},{t,0,2 Pi}]; 2. Find the length of the curve r(t) = (2 t, 3 Sin[t], 3 Cos[t]), from t=0 to t=Pi. r(t) = ( 2 t, 3 Sin[t], 3 Cos[t]); v(t) = (2, 3 Cos[t], -3 Sin[t]); s(t) = Sqrt[4 + 9 Cos[t]^2 + 9 Sin[t]^2] = Sqrt[13] t=Pi length = INT Sqrt[13] t dt = Pi Sqrt[13]. <================ t=0 r[t_] := { 2 t, 3 Sin[t], 3 Cos[t]}; a = ParametricPlot3D[r[t],{t,0,Pi}]; b = Table[ Graphics3D[ Point[r[t/Sqrt[13]]]],{t,1,10}]; c = Show[a,b,PlotLabel->" r(t) = (2 t, 3 Sin[t], 3 Cos[t]) "]; Display["2.ps",c]; 3. At a time in space V(t) = (1,6,5) and A(t) = (2,3,1). (a) Find T (* Unit Tangent Vector *) (b) Find N (* Unit Normal Vector *) (c) Find At (* Component of Acceleration in the Direction T *) (d) Find An (* Component of Acceleration in the Direction N *) (e) Find k (* The curvature *) V = (1,6,5) A = (2,3,1) s = Sqrt[1+36+25] = Sqrt[62] T = (1.6,5)/Sqrt[62] <=================== (a) AoV 2+18+5 25 At = ----- = ---------- = ----------- <=============== (c) ||V|| Sqrt[62] Sqrt[62] | i j k | | 2 3 1 | |AxV| | 1 6 5 | |( 9, -9, 9 )| 9 Sqrt[3] An = --------- = ---------------- = ------------ = -------- <======== (d) ||V|| Sqrt[62] Sqrt[62] Sqrt[62] (2,3,1)-25/Sqrt[62] (1,6,5)/Sqrt[62] N = ( A - At T)/An = ------------------------------------ 9 Sqrt[3]/Sqrt[62] (2,3,1)-25/62 (1,6,5) N = ------------------------------ 9 Sqrt[3]/Sqrt[62] 62 (2,3,1)-25 (1,6,5) (124-25, 186-150,62-125) N = ------------------------------ = ----------------------- 9 62 Sqrt[3]/Sqrt[62] 9 Sqrt[186] (99, 36, 63) (11,4,7) N = ----------------------- = ------------ <=================== (b) 9 Sqrt[186] Sqrt[186] An 9 Sqrt[3] k = --------- = ------------ <======================= (e) ||V||^2 62 Sqrt[62] V = {1,6,5}; A = {2,3,1}; T = {1,6,5}/Sqrt[62]; NN = 1/Sqrt[186] {11,4,7}; k = 9 Sqrt[3]/(62 Sqrt[62]); red[{a_,b_,c_}] := {a,b,c,RGBColor[1,0,0]}; green[{a_,b_,c_}] := {a,b,c,RGBColor[0,1,0]}; blue[{a_,b_,c_}] := {a,b,c,RGBColor[0,0,1]}; a1 = ParametricPlot3D[ red[V x],{x,0,1}]; a2 = ParametricPlot3D[ green[A x],{x,0,1}]; a3 = ParametricPlot3D[ T x,{x,0,1}]; a4 = ParametricPlot3D[ NN x,{x,0,1}]; a5 = ParametricPlot3D[ 1/k (Cos[t] T + Sin[t] NN) + 1/k NN,{t,0,2 Pi}]; a6 = Show[a1,a2,a3,a4,a5,PlotLabel->" V = {1,6,5}, A = {2,3,1}"]; Display["3.ps",%]; 4. For the curve (e^t, Sqrt[2] t, e^(-t) ), find the Normal and Tangential components of Acceleration. r(t) = (e^t, Sqrt[2] t, e^(-t) ) v(t) = (e^t, Sqrt[2], -e^(-t) ) s(t) = Sqrt[ e^(2t) + 2 + e^(-2t)] = e^t + e^(-t) a(t) = (e^t, 0, e^(-t) ) AoV e^(2t)-e^(-2t) At = -------- = ---------------- = e^t - e^(-t) <================= ||V|| e^t + e^(-t) | i j k | | e^t 0 e^(-t) | |AxV| | e^t Sqrt[2] -e^(-t) | An = -------- = -------------------------- ||V|| || V || | ( -Sqrt[2] e^(-t), 2, e^t Sqrt[2] ) | An = ------------------------------------------ e^t + e^(-t) Sqrt[ 2 e^(-2t) + 4 + 2 e^(2t) ] An = -------------------------------- e^t + e^(-t) Sqrt[2] Sqrt[ e^(-2t) + 2 + e^(2t) ] An = ------------------------------------ e^t + e^(-t) Sqrt[2] ( e^(-t)+ e^t) An = ------------------------ = Sqrt[2]. <======================== e^t + e^(-t) r[t_] := { E^t, Sqrt[2] t, E^(-t) }; v[t_] = D[r[t],t]; a[t_] = D[v[t],t]; s[t_] := Sqrt[v[t].v[t]]; At[t_] := a[t].v[t]/s[t]; An[t_] := Sqrt[Cross[a[t],v[t]].Cross[a[t],v[t]]]/s[t]; TT[t_] := v[t]/s[t]; NN[t_] := (a[t]-At[t] TT[t])/An[t]; red[{a_,b_,c_}] := {a,b,c,RGBColor[1,0,0]}; green[{a_,b_,c_}] := {a,b,c,RGBColor[0,1,0]}; blue[{a_,b_,c_}] := {a,b,c,RGBColor[0,0,1]}; a1 = ParametricPlot3D[r[t],{t,-2,2}]; a2 = Table[ ParametricPlot3D[ red[r[i/5]+x TT[i/5]],{x,0,1}],{i,-10,10}]; a3 = Table[ ParametricPlot3D[ green[r[i/5]+x NN[i/5]],{x,0,1}],{i,-10,10}]; a4 = Show[a1,a2,a3,PlotLabel->"r(t) = (e^t, Sqrt[2] t, e^(-t) )"]; Display["4.ps",a4]; 5. For the curve ( Sin[t], t, Cos[t] ) find the curvature at time t. r(t) = (Sin[t],t Cos[t] ) v(t) = (Cos[t],1, -Sin[t]) s(t) = Sqrt[2] a(t) = ( -Sin[t],0, -Cos[t]); | i j k | | -Sin[t] 0 -Cos[t] | |AxV| | Cos[t] 1 -Sin[t] | An = ------ = ------------------------------ ||V|| Sqrt[2] | ( Cos[t], -1 , -Sin[t] ) | An = -------------------------------------------- Sqrt[2] Sqrt[2] An = --------- = 1 Sqrt[2] An 1 k = --------- = --- ||V||^2 2 r[t_] := {Sin[t], t, Cos[t] }; v[t_] = D[r[t],t]; a[t_] = D[v[t],t]; s[t_] := Sqrt[v[t].v[t]]; At[t_] := a[t].v[t]/s[t]; An[t_] := Sqrt[Cross[a[t],v[t]].Cross[a[t],v[t]]]/s[t]; TT[t_] := v[t]/s[t]; NN[t_] := (a[t]-At[t] TT[t])/An[t]; red[{a_,b_,c_}] := {a,b,c,RGBColor[1,0,0]}; green[{a_,b_,c_}] := {a,b,c,RGBColor[0,1,0]}; blue[{a_,b_,c_}] := {a,b,c,RGBColor[0,0,1]}; a1 = ParametricPlot3D[r[t],{t,-2,2}]; a2 = Table[ ParametricPlot3D[ red[r[i/5]+x TT[i/5]],{x,0,1}],{i,-10,10}]; a3 = Table[ ParametricPlot3D[ green[r[i/5]+x NN[i/5]],{x,0,1}],{i,-10,10}]; a4 = Show[a1,a2,a3,PlotLabel->"r(t) = ( Sin[t], t, Cos[t] )"]; Display["5.ps",a4]; 6. For the function f(x,y,z) = x^2 + 3y^2 - 5 z^2 (a) What are the level surfaces? (b) What is the Normal to the surface? (c) What is the tangent plane to the level surface at (1,1,2)? (d) What is the normal line to the surface at (1,1,2). (a) Level surfaces are Hyperboloids of One sheet or two sheets. (b) Normal = ( 2x, 6y, -10z ) (c) Normal(1,1,2) = (2,6,-20) Plane is 2x + 6y -20z = -32 (d) Normal line is x = 1 + 2 t y = 1 + 6 t z = 2 - 20 t --------------------------------------------------------- x^2 + 3 y^2 - 5 z^2 = -16 -5 z^2 = -16 - x^2 - 3 y^2 z^2 = (16 + x^2 + 3 y^2)/5 f[x_,y_] := Sqrt[ (16 + x^2 + 3 y^2)/5] top = ParametricPlot3D[{ r Cos[t], r Sin[t], f[r Cos[t], r Sin[t]]}, {r,0,10},{t,Pi/4, 5 Pi/4},PlotPoints->30]; plane = Plot3D[ (-32-2x-6y)/(-20), {x,-10,10},{y,-10,10}]; normal = ParametricPlot3D[{1+2t,1+6t,2-20t},{t,-10,10}]; x = Show[top,plane,normal,PlotLabel->"x^2 + 3 y^2 -5z^2 = -16"]; Display["6.ps",x]; ---------------------------------------------------------- 7. Derive the formula A = At T + An N and explain why An = |v|^2/r. V = v T A = v' T + v T' Since T is a unit vector, T' is perpendicular to T and so T' is in the direction of N. T' = |T'| N. and A = v' T + v |T'| N At = v' An = v |T'|. Since |T'|/v = k = 1/r An = v^2 k = v^2/r 8. For the function z = x^2 + y^2. (a) What is the normal to the surface at (3,4,25). (b) What is the tangent plane to the surface at (3,4,25). x^2 + y^2 - z = 0 Normal = (2x, 2y, -1) (a) Normal(3,4,25) = (6,8,-1) (b) Plane 6x + 8 y -z = 18+32-25 = 25 f[x_,y_] := x^2 + y^2; top = ParametricPlot3D[{ r Cos[t], r Sin[t], f[r Cos[t], r Sin[t]]}, {r,0,6},{t,0, 2 Pi},PlotPoints->30]; plane = Plot3D[ 6x+8 y -25, {x,2.5,3.5},{y,3.5,4.5}]; normal = ParametricPlot3D[{3+6t,4+8t,25-t},{t,-10,10}]; x = Show[top,plane,normal,PlotLabel->"z = x^2 + y^2" ]; Display["7.ps",x]; 9. For the curve y = Sin[x] find the radius of curvature at any point x. y = Sin[x] y' = Cos[x] y" = -Sin[x]; | Sin[x] | k = --------------- (1+Cos[x]^2)^(3/2) (1+Cos[x]^2)^(3/2) r = 1/k = ----------------------- | Sin[x] | ========================================================== r(t) = {t, Sin[t],0}; v(t) = {1, Cos[t],0}; a(t) = {0,-Sin[t],0}]; T = {1, Cos[t], 0 } ------------------ Sqrt[1+Cos[t]^2]; AoV -Sin[t] Cos[t] At = --------- = ------------------- ||V|| Sqrt[1+Cos[t]^2]; | i j k | | 0 -Sin[t] 0 | |AxV| | 1 Cos[t] 0 | An = --------- = -------------------------------- ||V|| Sqrt[1+Cos[t]^2] |( 0, 0, Sin[t] ) } | Sin[t] | An = --------------------- = ------------------- Sqrt[1+Cos[t]^2] Sqrt[1+Cos[t]^2] N = (A - At T)/An -Sin[t] Cos[t] (1, Cos[t],0) (0, -Sin[t],0) - ----------------- -------------------- Sqrt[1+Cos[t]^2] Sqrt[1+Cos[t]^2] = ------------------------------------------------------------------------------- |Sin[t]|/ Sqrt[1+Cos[t]^2] Sin[t] Cos[t] (0, -Sin[t],0) + --------------- (1,Cos[t],0) 1+Cos[t]^2 = ------------------------------------------------- |Sin[t]|/ Sqrt[1+Cos[t]^2] Sin[t] Cos[t] +Sin[t] Cos[t]^2 ( ---------------, -Sin[t] + --------------------, 0 ) 1+Cos[t]^2 1+Cos[t]^2 -------------------------------------------------------- |Sin[t]|/ Sqrt[1+Cos[t]^2] ( Sin[t] Cos[t], -Sin[t] , 0 ) -------------------------------------------------------- |Sin[t]| Sqrt[1+Cos[t]^2] Sin[t] N = --------------------------- ( Cos[t], -1 , 0 ) |Sin[t]| Sqrt[1+Cos[t]^2] r[t_] := {t,Sin[t]}; s[t_] := Sqrt[1+Cos[t]^2]; TT[t_] := {1, Cos[t] }/ Sqrt[1+Cos[t]^2]; NN[t_] := Sin[t]/(Abs[Sin[t]] Sqrt[1+Cos[t]^2]) {Cos[t],-1}; An[t_] := Abs[ Sin[t]]/Sqrt[1+Cos[t]^2]; k[t_] := An[t]/s[t]^2; a1 = Plot[Sin[t],{t,-2 Pi,2 Pi},PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; a2 = Table[ ParametricPlot[ 1/k[2 t Pi/3 ]( Cos[x] TT[2 t Pi/3] + Sin[x] NN[2 t Pi/3]) + r[2 t Pi/3] + 1/k[2 t Pi/3] NN[2 t Pi/3], {x,0,2 Pi}], {t,-3,3}]; a3 = Table[ Graphics[Point[ r[2 t Pi/3] ]],{t,-3,3}]; a4 = Show[a1,a2,a3,PlotLabel->" y = Sin[x] ",AspectRatio->Automatic]; Display["9.ps",a3]; =================================================================== 10. A particle starts at the origin with initial velocity (1,2,1). Its acceleration is (t,1,t^2). Find its position function. a = (t, 1, t^2 ) v = (t^2/2, t, t^3/3) + (1,2,1) r = (t^3/6, t^2/2, t^4/12) + (1,2,1)t. a = ParametricPlot3D[ {t^3/6, t^2/2, t^4/12} + {1,2,1}t, {t,-1,1}]; b = Show[a,PlotLabel->"a = (t, 1, t^2}, v(0) = (1,2,1)"]; Display["10.ps",b];