3, 1, 3 3, 1, 1 3, 1, 1. x(t) = t cos(πt) y(t) = t sin(πt), z(t) = t 2 t 0

Transcription

1 Math 5 Final Eam olutions ecember 5, Problem. ( pts.) (a 5 pts.) Find the distance from the point P (,, 7) to the plane z +. olution. We can easil find a point P on the plane b choosing some values for and (for eample, we can take, then z and we get P (,, ), and then P P,, ). Rewriting the equation of the plane in the standard form, z +, we see than a normal vecor to the plane can be taken as n,,. Then istance(p (,, 7), z + ) comp n P P,,,,,, (b 5 pts.) Find parametric equations of the tangent line to the parametric curve at the point Q(, /, /). olution. We have (t) t cos(πt) (t) t sin(πt), z(t) t t r(t) t cos(πt), t sin(πt), t r(/), /, / OQ r (t) cos(πt) πt sin(πt), sin(πt) + πt cos(πt), t r (/) π/,, and therefore parametric equations of the tanget line are r(s) r(/) + r (/)s π/s, / + s, / + s or, in components, Problem. ( pts.) onsider the function f(, ) ( + ) around the point P (, ). (a pts.) Is f(, ) around P more sensitive to changes in, or changes in? Eplain (s) + ( π/)s (s) / + ()s z(s) / + ()s olution. ince the rate of change of f(, ) w.r.t. at P is f (, ) ()(+), 8 and the rate of change of f(, ) w.r.t. at P is f (, ),, we see that around P f(, ) is more sensitive to changes in. (b pts.) What is the rate of change of f at P in the direction of the vector v,? olution. We have f(, ) ( + ), and f(p ) 8,. Then v f(p ) comp v f(p ) f(p ) v v 8,,, 7. (c pts.) What is the maimum rate of change of f at P? In which direction does it occur? olution. The maimum rate of change is geven b the magnitude of the gradient vector, and so it is f(p ) 65 and it occurs in the direction of the gradient vector f(p ) 8, (or 8/ 65, / 65 if ou prefer to use the unit vector.)

2 Problem. ( pts.) Find and classif all critical points of the function f(, ) olution. First we have to find the critical points b computing the gradient of f(, ) and setting it to zero: { f 5 + 5, + /5 + /5 and we see that critical points are intersections of the following lines (see the graph): f : f : + / olving the first equation for, /5, and plugging it into the second equation we get /5 + + /5 ( and then ) or 5( and then ). To classif the critical points we will use the second derivative test. First, we compute the Hessian matri of second derivatives: [ ] [ ] f f H(, ) 5. f f /5 Then [ ] 5 H(, ) det H < and so it is the saddle. For the point (, 5) we have [ ] 5 H(, 5), det H >, f /5 (, 5) 5 > and so this point is a local minimum. Answer: The point (, ) is a saddle and the point (, 5) is a local minimum

3 Problem. (5 pts.) onsider the planar vector fields F(, ) ( )i + j and G(, ) ( + )e i + e j, and the curve from the point A(, ) to the point B(, ) that goes along the parabola (see the picture on the right). 5 B A (a 5 pts.) arefull eplain whether these vector fields are conservative or not. olution. First we check the necessar condition for a field to be conservative (i.e., gradient) b computing its curl: curl(f) Q P () ( ), so F can not be conservative. curl(g) Q P ( e + e ) ( e + ( + )e ), so G ma be conservative. In this case it is, since G is defined and is differentiable everwhere on the plane. Alternativel, we can eplicitl find the potential function. We have: f(, ) P d ( + )e d e + h() f(, ) Q d e d e + k(), and so G(, ) (e ) is a conservative vector field. (b 5 pts.) ompute the work done b the field F along the curve. olution. ince F is not conservative, we have to compute the line integral F dr using parametrization. Parametrizing b r(t) t, t, t, we get F dr ( ) d + d ( t t ) dt + d( t ) ( 8t t ) 5. (c 5 pts.) ompute the work done b the field G along the curve. olution. ince G is conservative, we can use the fundamental theorem: G dr df f(, ) (,) (,) (,) e (,) e +. Problem 5. (5 pts.) onsider the surface that is the part of the cone z + below the plane z (see the picture on the right). (a 5 pts.) Give a parametric representation of. Make sure to eplicitl describe or sketch the parametrization domain. 5 olution. For eample, we can parametrize as a graph of function f(, ) + using cartesian coordinates, r(, ),, +, with the parametrization domain the unit disk {(, ) + }.

4 (b 5 pts.) Find an equation of the tangent plane to at the point P (,, ). olution. First note that f + and f. Then, using the linearization, we have + z + f (, )( + ) + f (, )( ) ( + ) + ( ) or + z. (b 5 pts.) If the densit function σ(,, z) is equal to the distance to the plane, find the mass of the surface. olution. We have d + f + f da da and the densit finction is σ(,, z) z. Then π Mass σ d + da r rdr dθ 8 π. Problem 6. ( pts.) Find the circulation of the vector field F (cos + ln )i+(ln + sin )j over the first-quadrant petal of a four-petal rose given in polar coordinates b the equation r sin(θ) (see the picture) olution. B Green s Theorem, F dr ( (Q P ) da ( ) sin + sin da (ln + sin ) (cos + ln ) da switching to polar coordinates and using the equation r sin(θ) for the upper r-limit ) da π/ sin(θ) π/ sin θ cos θ cos θ rdr dθ r cos θ dθ π/ π/ sin(θ) cos(θ) dθ sin θ dθ.

5 5 Problem 7. ( pts.) Find the circulation of the field F (z )i + (z + z)j + ( + )k along the boundar of the Pringles r potato chip (i.e., the part of the surface z contained inside the clinder + ) oriented as shown. olution. B tokes Theorem, F dr curl(f) d. We have i j k curl(f) z,, z +,, z +. z z + z + z Parametrizing as a graph of the function f(, ) with the parametrization domain the unit disk {(, ) + } we get d f, f, da,, da and so F dr,, z + d,, +,, da ( + ) da switching to polar coordinates and noticing that π da b smmetr ( r ) rdr dθ π(/ /) π/. Problem 8. ( pts.) onsider the field F ( + z)i + ( + z)j + ( + z)k and the unit cube in the first octant (see the picture). z (a 5 pts.) Find the flu of F out of this cube. olution. B the divergence theorem, F d div(f) dv. ince div(f) P +Q +R z +, E F d div(f) dv dv Volume(unit cube). E cube (b 5 pts.) What is the flu of F through just the top surface of the cube? olution. The top surface of the cube can be parametrized b and, r(, ),,, where the parametrization domain is the unit square {(, ), }. Then d k da and we get F d F k da R(,, ) da ( + ) d d / / +.

6 6 Problem 9. ( pts.) The graph below is a plot of a gradient vector field F P (, )i + Q(, )j, together with the level curves of its potential function f(, ) (the function values on some level curves are given, and others can be determined from the fact that level separation is on this plot). Also on the plot ou can see three oriented curves,, and, and points A, B, M and N. Use this information to answer the following questions A 5 B M N 7 5 (a) Find the work done b F in moving an object from A to B along the path. olution. ince F f, using the Fundamental Theorem for line integrals of vector fields and the level curves information, we get F dr df f(, ) f(b) f(a) 5 7. (b) Is the circulation of F around the curve positive, negative, or zero? B A olution. irculation of gradient vector fields around closed curves is alwas zero. (c) Is the flu of F out of the region bounded b the curve positive, negative, or zero? olution. ince F clearl goes inside the region bounded b, its outward flu across is negative. (d) Is the divergence of F at N positive, negative, or zero? olution. ince N is a sink, div(f)(n) is negative. (e) What would a small paddle placed at M do? olution. The paddle will not rotate since curl(grad(f)).