Math 21a. CurvesVector-Valued Functions and Curves in SpaceDerivatives and Integrals of Vector-Valued FunctionsT. JudsonHarvard UniversitySpring 2008Learning Objectives1• To understand and be able to the concept of a vector-valued function,r(t) = hf(t), g(t), h(t)i = f(t)i + g(t)j + h(t)k.• To be able to apply the concepts of limits and continuity to vector-valued functions.• To understand that a curve in R3can be represented parametricallyx = f(t) y = g(t) z = h(t)or by a vector-valued functionr(t) = hf(t), g(t), h(t)i.• To understand and be able to calculate the derivative of a vector-valued function.• To understand and be able to apply the basic rules of differentiation for vector-valued func-tions.• To understand and be able to find the definite integral of a vector-valued function.Testing Your Knowledge1. Two particles travel along the space curvesr1(t) = h3t, 7t − 12, t2i r2(t) = h4t − 3, t2, 5t − 6iDo the particles collide?1Sections 10.1 and 10.2 in J. Stewart. Multivariable Calculus: Concepts and Contexts, third edition. Brooks/Cole,Belmont CA, 2005.12. Given the plane curve described by the vector equation r(t) = sin ti + 2 cos tj:(a) Sketch the plane curve. (b) Find r0(t)(c) Sketch the position vector r(π/4) and the tangent vector r0(π/4).3. Given r(t) = ht, cos 2t, sin 2ti, find(a) r0(t)(b)Zπ0r(t) dt4. Match the parametric equations with the graph below.(a) x = cos 4t, y = t, z = sin 4t(b) x = t, y = t2, z = e−t(c) x = t, y = 1/(1 + t2), z = t2(d) x = cos t, y = sin t, z = sin 5t(e) x = cos t, y = sin t, z = ln t(f) x = e−tcos 5t, y = e−tsin 5t, z = e−te modei ofy of a posi-ls E and B.of the pat-, cycloid we;hoid inves-x : t , y : U 0 + t , ) , z : t 2x : e-' cos 10r, y - e-' sin l0r, z : e-'l l . " : . o t t , ) : s i n t , z : s i n 5 t.22. x: cos t, y : sin t. z : lntf ,13" Strow that the curve with parametric equations-r: : t cos t,'t y: t sin t, z : r lies on the cone z' : x' * y2, and use this 'fact to help sketch the curve.stcTlot{ l0.l vt(ToR t|Jl,lffl0l,|s A1'lD SPA(t (|JRvts s 70Iffi 30. Craptr the curve with parametric equations'r : ",/1 - 0)5 co* lOt cos rv : .[ : o'2sioitTdT sin rz : 0.5 cos l0rExplain the appearance ofthe graph by showing that it lieson a sphere.31. Show that the curve with parametric equations x : 12,y : I - 3 t , z : 1 + l r p as se s t h r ou gh t h e po in ts ( 1 ' 4' 0)and (9, -8, 28) but not through the point (4"7, -6).32-34 s Find a vector function that represents the curve ofintersection of the two surfaces.3?. The cylinder x2 + y': 4 and the surface z : xy33. The cone z : tfplV and the plane z : I * y34. The paraboloid z : 4x2 + y2 and the parabolic cylinderJ : x 2ffi SS. fty to sketch by hand the curve of intersection of the circu-lar cylinder x' i y': 4 and the parabolic cylinder z : xt'Then find parametric equations for this curve and use theseequations and a computer to graph the curve.ffi St. fry to sketch by hand the curve ofintersection oftheparabolic cylinder y : .r2 and the top half of the ellipsoidx' + 4y' I 4zz : 16. Then find parametric equations forthis curve and use these equations and a computer to graphthe curve.37. If two objects travel through space along two differentcurves, it's often important to know whether they will col-lide. (Will a missile hit its moving target? Will two aircraftcollide?) The curves might intersect, but we need to knowwhether the objects are in the same position at the samerlne. Suppose the trajectories of two particles are given bythe vector functionsnU) : Q' ,lt - 12,t2) rz(t) : (4t - 3,t2'5t - 6)for r > 0. Do the Particles collide?38. Two particles travel along the space curvesr , ( t ) : \ t , t 2 , t 3 ) r r ( r ) : ( l + 2 t , I + 6 t , l + l 4 t >Do the particles collide? Do their paths intersect?39. Suppose u and v are'/ector functions that possess limits ast '--> a and let c be a constant. Prove the following prop-erties of limits.(u) lgt"(r) + v(r)l: lg"ttl + limv(t)(b) lim cu(r) : c lim u(r)(c) lim [u(r) ' v(t)] : lg "ttl ' Jg "ttt(0) l{: t"(t) x v(r)l : lg utO x lim v(r)rf the trajecto'the graPhs; 24. Show that the curve with parametric equations l: sin t,i=. y: cos /, z: sin2t is the curve ofintersection ofthe. + f i + 2 k L s u r f a c e s z : x 2 a n d x ' + y ' : 1 . U s e t h i s f a c t t o h e l p; sketch the curve.i. tS. At what point does the curve r(r) : t i + (2t - t' ) k inter-1 sect the Paraboloid z: x2 + Y'?, ,i:e-) equations for F 26-28 q Use a computer to graph the curve with the given vec-! tor equation. Make sure you choose a parameter domain and7, viewpoints that reveal the true nature of the curve.r n / t 1 l \ lt 1 u \ 1 1 J . ' , 1 6 . r ( l ) : ( t o _ t , + l , i , , 2 )i u. rlry : \,',r/, - t.J5 - t)" I tU r(1) : (sin r, sin 2r, sin 3r)fr 21. Craptr the curve with parametric equationst x: (1 + cos 16r) cos /, ) : (l * cos 16r) sin t,ll t : I * cos 16r. Explain the appearance ofthe graph byi: showing that it lies on a cone.!;5. Which of the following curves are smooth? That is, which curves satisfy the property thatr0(t) 6= 0 for all t?(a) r(t) = ht3, t4, t5i (b) r(t) = ht3+ t, t4, t5i (c) r(t) = hcos3t,
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