Denis Auroux -MIT多变量微积分重点试卷汇编.pdf

18 02 Practice Exam 1 A Problem 1 15 points A unit cube lies in the fi rst octant with a vertex at the origin see fi gure a Express the vectors OQ a diagonal of the cube and OR joining O to the center of a face in terms of k O Q R z y x b Find the cosine of the angle between OQ and OR Problem 2 10 points The motion of a point P is given by the position vector R 3 cos t t 3sin t k Compute the velocity and the speed of P Problem 3 15 points 10 5 a Let A 1 2 1 3 0 1 2 1 0 then det A 2 and A 1 1 2 1 1 2 a 2 2 b 5 6 fi nd a and b x 1 b Solve the system AX B where X y and B 2 z 1 c In the matrix A replace the entry 2 in the upper right corner by c Find a value of c for which the resulting matrix M is not invertible For this value of c the system M X 0 has other solutions than the obvious one X 0 fi nd such a solution by using vector operations Hint call U V and W the three rows of M and observe that M X 0 if and only if X is orthogonal to the vectors U V and W Problem 4 15 points The top extremity of a ladder of length L rests against a vertical wall while the bottom is being pulled away Find parametric equations for the midpoint P of the ladder using as parameter the angle between the ladder and the P horizontal ground Problem 5 25 points 10 5 10 a Find the area of the space triangle with vertices P0 2 1 0 P1 1 0 1 P2 2 1 1 b Find the equation of the plane containing the three points P0 P1 P2 c Find the intersection of this plane with the line parallel to the vector V 1 1 1 and passing through the point S 1 0 0 Problem 6 20 points 5 5 10 a Let R x t z t y t k be the position vector of a path Give a simple intrinsic formula for d R R in vector notation not using coordinates dt b Show that if R has constant length then R and V are perpendicular c let A be the acceleration still assuming that R has constant length and using vector diff er entiation express the quantity R A in terms of the velocity vector only 18 02 Practice Exam 1B Problem 1 Let P Q and R be the points at 1 on the x axis 2 on the y axis and 3 on the z axis respectively a 6 Express QP and QR in terms of i j and k b 9 Find the cosine of the angle PQR Problem 2 Let P 1 1 1 Q 0 3 1 and R 0 1 4 a 10 Find the area of the triangle PQR b 5 Find the plane through P Q and R expressed in the form ax by cz d c 5 Is the line through 1 2 3 and 2 2 0 parallel to the plane in part b Explain why or why not Problem 3 A ladybug is climbing on a Volkswagen Bug VW In its starting position the the surface of the VW is represented by the unit semicircle x2 y2 1 y 0 in the xy plane The road is represented as the x axis At time t 0 the ladybug starts at the front bumper 1 0 and walks counterclockwise around the VW at unit speed relative to the VW At the same time the VW moves to the right at speed 10 a 15 Find the parametric formula for the trajectory of the ladybug and fi nd its position when it reaches the rear bumper At t 0 the rear bumper is at 1 0 b 10 Compute the speed of the bug and fi nd where it is largest and smallest Hint It is easier to work with the square of the speed Problem 4 1 2 3 1 1 4 1 1 M 3 2 1 M a 7 8 12 2 1 1 b 5 4 a 5 Compute the determinant of M b 10 Find the numbers a and b in the formula for the matrix M 1 x 2y 3z 0 c 10 Find the solution r x y z to 3x 2y z t as a function of t 2x y z 3 d r d 5 Compute dt Problem 5 a 5 Let P t be a point with position vector r t Express the property that P t lies on the plane 4x 3y 2z 6 in vector notation as an equation involving r and the normal vector to the plane d r b 5 By diff erentiating your answer to a show that is perpendicular to the normal vector dt to the plane 18 02 Practice Exam 2 A Problem 1 10 points 5 5 Let f x y xy x4 a Find the gradient of f at P 1 1 b Give an approximate formula telling how small changes x and y produce a small change w in the value of w f x y at the point x y 1 1 Problem 2 20 points On the topographical map below the level curves for the height function h x y are marked in feet adjacent level curves represent a diff erence of 100 feet in height A scale is given dh a Estimate to the nearest 1 the value at the point P of the directional derivative where ds u u is the unit vector in the direction of h h b Mark on the map a point Q at which h 2200 0 and 0 y 0 c Determine the nature of this critical point by using the second derivative test d Find the maximum of f in the fi rst quadrant justify your answer Problem 5 15 points In Problem 4 above instead of substituting for z one could also use Lagrange multipliers to maximize the volume V xyz with the same constraint x2 y2 z 1 a Write down the Lagrange multiplier equations for this problem b Solve the equations still assuming x 0 y 0 Problem 6 10 points w w Let w f u v where u xy and v x y Using the chain rule express and in terms of x y x y fu and fv Problem 7 15 points w Suppose that x2y xz2 5 and let w x3y Express as a function of x y z and evaluate z y it numerically when x y z 1 1 2 18 02 Practice Exam 2 B Problem 1 Let f x y x2y2 x a 5 Find f at 2 1 b 5 Write the equation for the tangent plane to the graph of f at 2 1 2 c 5 Use a linear approximation to fi nd the approximate value of f 1 9 1 1 d 5 Find the directional derivative of f at 2 1 in the direction of Problem 2 10 On the contour plot below mark the portion of the level curve f 2000 on f which 0 y Problem 3 a 10 Find the critical points of w 3x 2 4xy y 2 12y 16x and say what type each critical point is b 10 Find the point of the fi rst quadrant x 0 y 0 at which w is largest Justify your answer Problem 4 Let u y x v x2 y2 w w u v a 10 Express the partial derivatives wx and wy in terms of wu and wv and x and y b 7 Express xwx ywy in terms of wu and wv Write the coeffi cients as functions of u and v c 3 Find xwx ywy in case w v5 Problem 5 a 10 Find the Lagrange multiplier equations for the point of the surface x 4 y 4 z 4 xy yz zx 6 at which x is largest Do not solve b 5 Given that x is largest at the point x0 y0 z0 fi nd the equation for the tangent plane to the surface at that point Problem 6 Suppose that x 2 y 3 z 4 1 and z 3 zx xy 3 a 8 Take the total diff erential of each of these equations b 7 The two surfaces in part a intersect in a curve along which y is a function of x Find dy dx at x y z 1 1 1 18 02 Practice Exam 3 A 1 Let x y be the center of mass of the triangle with vertices at 2 0 0 1 2 0 and uniform density 1 a 10 Write an integral formula for y Do not evaluate the integral s but write explicitly the integrand and limits of integration b 5 Find x 2 15 Find the polar moment of inertia of the unit disk with density equal to the distance from the y axis 23 3 Let F ax y y3 1 2x bxy2 2 be a vector fi eld where a and b are constants a 5 Find the values of a and b for which F is conservative b 5 For these values of a and b fi nd f x y such that F f c 5 Still using the values of a and b from part a compute F d r along the curve C such C that x et cost y et sint 0 t 4 10 For F yx3 y2 fi nd C F d r on the portion of the curve y x2 from 0 0 to 1 1 5 Consider the region R in the fi rst quadrant bounded by the curves y x2 y x2 5 xy 2 and xy 4 a 10 Compute dxdy in terms of dudv if u x2 y and v xy b 10 Find a double integral for the area of R in uv coordinates and evaluate it 6 a 5 Let C be a simple closed curve going counterclockwise around a region R Let M M x y Express Mdx as a double integral over R C b 5 Find M so that Mdx is the mass of R with density x y x y 2 C 7 Consider the region R enclosed by the x axis x 1 and y x3 a 5 Use the normal form of Green s theorem to fi nd the fl ux of F 1 y2 out of R b 5 Find the fl ux out of R through the two sides C1 the horizontal segment and C2 the vertical segment c 5 Use parts a and b to fi nd the fl ux out of the third side C3 18 02 Practice Exam 3 B 1 2x Problem 1 a Draw a picture of the region of integration of dydx 0 x b Exchange the order of integration to express the integral in part a in terms of integration in the order dxdy Warning your answer will have two pieces Problem 2 a Find the mass M of the upper half of the annulus 1 x2 y2 1 which is to say outside the cylinder of radius one with axis the z axis a Compute the fl ux outward through S of the vector fi eld F yi xj zk b Show that the fl ux of this vector fi eld through any part of the cylindrical surface is zero c Using the divergence theorem applied to F compute the volume of the region between S and the cylinder Problem 3 20 Let S be the part of the spherical surface x2 y2 z2 2 lying in z 1 Orient S upwards and give its bounding circle C lying in z 1 the compatible orientation a Parametrize C and use the parametrization to evaluate the line integral I xzdx ydy ydz C b Compute the curl of the vector fi eld F xzi yj yk c Write down a fl ux integral through S which can be computed using the value of I Problem 4 15 Use the divergence theorem to compute the fl ux of F i j k outwards across the closed surface x4 y4 z4 1 Problem 5 15 Consider the surface S given by the equation z x 2 y 2 z 2 2 a Show that S lies in the upper half space z 0 b Write out the equation for the surface in spherical polar coordinates c Using the equation obtained in part b give an iterated integral with explicit integrand and limits of integration which gives the volume of the region inside this surface Do not evaluate the integral Problem 6 15 Let S be the part of the surface z xy where x2 y2 1 Compute the fl ux of F yi xj zk upward across S by reducing the surface integral to a double integral over the disk x2 y2 1 1 18 02 Practice Exam 48 Problem 1 10 poir ts Let C 1 e the portion of the cylinder z2 y2 21lying in the f i t octant L 20 y 20 z 20 and 1 elow t he plane z 1 Set 1111 a triple integral in cglCri i rlr ctr l coo r rlCri i ates which gives the monierlt of inertia of C a1 olit the z axis assnine the density to 1 e 15 1 Give integrancl and limits of integration l lit do not ln1 i ate Problem 2 20 poir ts 5 5 10 a A solid sphere S of radins a is place 1 a1 ove the cy plane so it is tangent at the origin and its diameter lies along the z axis Give its eqiiation in sl he ical coor iinntrs 1 Give the equation of the horizontal plan