3. [805/22] Let a = [8,1, 4] and b = [5, 2,1]. Find a + b,

Transcription

1 MATH 251: Calculus 3, SET8 EXAMPLES [Belmonte, 2018] 12 Vectors; Geometry of Space 12.1 Three-Dimensional Coordinate Systems 1. [796/6] What does the equation y = 3 represent in R 3? What does z = 5 represent? What does the pair of equations y = 3, z = 5 represent? In other words, describe the set of points (x,y,z) such that y = 3 and z = 5. Illustrate with a sketch. 3. [805/22] Let a = [8,1, 4] and b = [5, 2,1]. Find a + b, 4a + 2b, a, and a b. 4. [805/26] Find the vector that has the same direction as [6,2, 3] but has length [805/28] What is the angle between the vector 8i + 6j and the positive direction of the x-axis? 6. [805/30] A child pulls a sled through the snow on a level path with a force of 50N exerted 38 above the horizontal. Find the horizontal and vertical components of the force. 7. [806/32] Find the magnitude of the resultant force and the angle it makes with the positive x-axis. 2. [796/8] Describe and sketch the surface in R 3 represented by the equation x 2 + z 2 = [797/14] Find an equation of the sphere with center (2, 6,4) and radius 5. Describe its intersection with each of the three coordinate planes. 4. [797/16] Find an equation of the sphere that passes through the origin and whose center is (1,2,3). 8. [806/38] The tension T at each end of a chain has magnitude 25N. What is the weight of the chain? 5. [797/20] Show the equation 3x 2 + 3y 2 + 3z 2 = y + 12z represents a sphere, and find its center and radius. 6. [797/34] Describe in words the region of R 3 represented by x 2 + y 2 + z [797/38] Describe in words the region of R 3 represented by x 2 + y 2 + z 2 > 2z. 8. [797/42] Write inequalities to describe the solid upper hemisphere of the sphere of radius 2 centered at the origin. 9. [797/44] Consider the points P such that the distance from P to A( 1,5,3) is twice the distance from P to B(6,2, 2). Show that the set of all such points is a sphere, and find its center and radius. 10. [797/46] Find the volume of the solid that lies inside of both of the spheres x 2 + y 2 + z 2 + 4x 2y + 4z + 5 = 0 and x 2 + y 2 + z 2 = Vectors 1. [805/8] If the vectors in the figure satisfy u = v = 1 and u + v + w = 0, what is w? 9. [806/40] Three forces act on an object. Two of the forces are at an angle of 100 to each other and have magnitudes 25N and 12N. The third is perpendicular to the plane of these two forces and has magnitude 4N. Calculate the magnitude of the force that would exactly counterbalance these three forces. 10. [806/42] Consider the curve y = 2sinx. (a) Find the unit vectors that are parallel to the tangent line to the curve at the point (π/6,1). (b) Find the unit vectors that are perpendicular to the tangent line. (c) Sketch the curve y = 2sinx and the vectors in parts (a) and (b), all starting at (π/6,1) The Dot Product 1. [812/8] Find a b for a = [3,2, 1] and b = [4,0,5]. 2. [812/10] Find a b given a = 80, b = 50, and (a,b) = 3 4 π. 3. [812/12] If u is a unit vector, find u v and u w. 2. [805/14] Given points A(0,6, 1) and B(3,4,4), find a vector a with representation given by the directed line segment AB. Draw AB and the equivalent representation starting at the origin. 1

2 4. [813/22] Find the angles of the triangle with vertices A(1,0, 1), B(3, 2,0), C (1,3,3), to the nearest degree. 5. [813/24] In each case, determine whether the vectors are orthogonal (perpendicular), parallel, or neither. (a) u = [ 5,4, 2], v = [3,4, 1] (b) u = 9i 6j + 3k, v = 6i + 4j 2k (c) u = [c,c,c], v = [c,0, c] where c is a constant 6. [813/28] Find two unit vectors that make an angle of 60 with v = [3,4]. 7. [813/30] Find the acute angle between the lines x + 2y = 7 and 5x y = [813/32] Find the acute angle between the curves y = sinx, y = cos x, 0 x π/2 at their point of intersection. (Recall the angle between two curves is the angle between their tangent lines at the point of intersection.) 9. [813/44] Find the scalar and vector projections of b = [5,0, 1] onto a = [1,2,3]. 10. [813/52] A boat sails south with the help of a wind blowing in the direction S36 E with magnitude 400lb. Find the work done by the wind as the boat moves 120ft The Cross Product 1. [821/2] Let a = [4,3, 2] and b = [2, 1,1]. Find the cross product a b and verify that it is orthogonal to both a and b. 2. [821/6] Find c = a b for vectors a = [t, cost, sint] and b = [1, sint,cost]. Show c a and c b. 3. [821/8] If a = i 2k and b = j + k, find c = a b. Sketch them as position vectors. 4. [821/14] Find u v and determine whether u v is directed into the page or out of the page. 6. [821/20] Find two unit vectors orthogonal to both u = j k and v = i + j. 7. [821/28] Find the area of the parallelogram with vertices P(1,0,2), Q(3,3,3), R(7,5,8), and S(5,2,7). 8. [822/32] Let P(2, 3, 4), Q( 1, 2, 2), and R(3, 1, 3). (a) Find a nonzero vector orthogonal to the plane through the points. (b) Find the area of PQR, the triangle through the points. 9. [822/36] Given points P(3, 0, 1), Q( 1, 2, 5), R(5, 1, 1), and S(0,4,2), find the volume of the parallelepiped (sheared box) with adjacent edges PQ, PR, and PS. 10. [822/42] Let v = 5j and let u be a vector with length 3 that starts at the origin and rotates in the xy-plane. Find both the maximum and minimum values of the length of the vector u v. In what direction does u v point? 12.5 Equations of Lines and Planes 1. [831/4] Find a vector equation and parametric equations for the line through the point (0,14, 10) and parallel to the line x = 1 + 2t, y = 6 3t, z = 3 + 9t. 2. [831/10] Find parametric equations and symmetric equations for the line through (2,1,0) and perpendicular to both i + j and j + k. 3. [831/18] Find parametric equations for the line segment from ( 2, 18, 31) to (11, 4, 48). 4. [831/20] Determine whether the lines L 1 and L 2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. L 1 : x = 5 12t, y = 3 + 9t, z = 1 3t L 2 : x = 3 + 8s, y = 6s, z = 7 + 2s 5. [832/34] Find an equation of the plane through the points (3, 0, 1), ( 2, 2, 3), and (7, 1, 4). 5. [821/16] The figure shows a vector a in the xy-plane and a vector b in the direction of k. Here a = 3 and b = [832/46] Find the point where the line x = t 1, y = 1 + 2t, z = 3 t, intersects the plane 3x y + 2z = [832/62] Find an equation for the plane consisting of all points equidistant from the points (2,5,5) and ( 6,3,1). 8. [832/70] Find the distance from the point (0,1,3) to the line x = 2t, y = 6 2t, z = 3 +t. 9. [833/74] Find the distance between the parallel planes 6z = 4y 2x and 9z = 1 3x + 6y. (a) Find a b. (b) Use the right-hand rule to decide whether each component of a b is positive, negative, or zero. 10. [833/80] Let L 1 be the line through (1,2,6) and (2,4,8). Let L 2 be the line of intersection of the plane x y + 2z + 1 = 0 and the plane through (3,2, 1), (0,0,1), and (1,2,1). Find the distance between the lines L 1 and L 2. 2

3 12.6 Cylinders and Quadric Surfaces 1. [839/4] Describe and sketch the surface 4x 2 + y 2 = [839/8] Describe and sketch the surface z = siny. 3. [840/28] Describe and sketch the surface y = x 2 z [840/36] Reduce the equation x 2 y 2 z 2 4x 2z + 3 = 0 to a standard form, classify the surface, then sketch it. 5. [840/38] Reduce 4x 2 + y 2 + z 2 24x 8y + 4z + 55 = 0 to one of the standard forms, classify the surface, and sketch it. 6. [840/42] Graph the surface x 2 6x + 4y 2 z = [840/44] Sketch the region bounded by the paraboloids z = x 2 + y 2 and z = 2 x 2 y [841/46] Find an equation of the surface obtained by rotating the line z = 2y about the z-axis. 9. [841/48] Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is twice the distance from P to the yz-plane. Identify and sketch the surface. 10. [841/52] Show that the curve of intersection of the surfaces x 2 + 2y 2 z 2 + 3x = 1 and 2x 2 + 4y 2 2z 2 5y = 0 lies in a plane. Illustrate graphically. 13 Vector Functions 13.1 Vector Functions and Space Curves [ t 2 t 1. [853/4] Find the limit lim t 1 t 1, t + 8, sinπt ]. lnt 2. [854/10] Sketch the curve r(t) = [sin πt, t, cos πt], indicating the direction in which t increases. 3. [854/22] Sketch the curve x = cost, y = sint, z = 1/ ( 1 +t 2). 4. [854/28] Show that the curve with parametric equations x = sint, y = cost, z = sin 2 t is the curve of intersection of the surfaces z = x 2 and x 2 + y 2 = 1. Sketch the curve and surfaces on the same plot. 5. [854/32] At what points does the helix r(t) = [sint,cost,t] intersect the sphere x 2 + y 2 + z 2 = 5? Illustrate. 6. [854/36] Graph the curve r(t) = [cos(8 cost) sint, sin(8 cost) sint, cost]. 7. [854/40] Graph the curve x = cos2 10t cost, x = cos2 10t sint, z = 1 2 cos10t. 8. [855/46] Find the curve of intersection of the cylinder x 2 + z 2 = 1 and the semiellipsoid x 2 + y 2 + 4z 2 = 4, y [855/48] Find the curve of intersection of the top half of the ellipsoid x 2 + 4y 2 + 4z 2 = 16 and the parabolic cylinder y = x 2. Illustrate. 10. [855/50] Two particles travel along the space curves r 1 (t) = [ t,t 2,t 3] and r 2 (s) = [1 + 2s, 1 + 6s, s]. Do they collide? Do their paths intersect? Illustrate Derivatives and Integrals of Vector Functions 1. [860/4] Consider the plane curve r(t) = [ t 2,t 3]. (a) Graph the curve in the xy-plane. (b) Compute the derivative r (t). (c) Sketch the position vector r(1) and the tangent vector r (1) on the same plot as the curve. 2. [860/10] Find the derivative of r(t) = [ e t, t t 3, lnt ]. 3. [860/14] Find r (t) for r(t) = sin 2 at i +te bt j + cos 2 ct k. 4. [860/20] Let r(t) = [ sin 2 t, cos 2 t, tan 2 t ]. Find the unit tangent vector T(π/4). 5. [860/26] Find the tangent line to the curve x = t 2 + 3, y = ln ( t ), z = t at the point P(2, ln4, 1). Illustrate. 6. [860/30] Find the tangent line to the curve x = 2cost, y = 2sint, z = 4cos2t at the point P ( 3, 1, 2 ). Illustrate. 7. [861/34] At what point do curves r 1 (t) = [ t, 1 t, 3 +t 2] and r 2 (s) = [ 3 s, s 2, s 2] intersect? Find their angle of intersection to the nearest degree. 8. [861/36] Evaluate the integral 4 1 [ 2t 3/2, 0, (t + 1) t ] dt. 9. [861/42] Find r(t) given that r (t) = [t, e t, te t ] and r(0) = [1,1,1]. 10. [861/50] If r(t) = u(t) v(t) where u(2) = [1,2, 1], u (2) = [3,0,4], and v(t) = [ t,t 2,t 3],find r (2) Arc Length and Curvature 1. [868/4] Find the length of the curve r(t) = [cost,sint,lncost], 0 t π/4. 2. [868/12] Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x 2 + y 2 = 4 and the plane x + y + z = [868/18] Let r(t) = [ t 2, sint t cost, cost +t sint ], t > 0. (a) Find unit tangent and unit normal vectors T(t) & N(t). (b) Use Formula 9 (on page 864) to find the curvature κ (t). 4. [868/24] Find the curvature of r(t) = [ t 2, lnt, t lnt ] at the point (1, 0, 0). 3

4 5. [868/30] Where does y = lnx have maximum curvature? What happens to the curvature as x? 6. [869/36] Plot the curve r(t) = [t sint, 1 cost, 4cos(t/2)] and its curvature function κ (t) for 0 t 8π. Comment on how the curvature reflects the shape of the curve. 7. [869/48] Find T, N, B, the unit tangent, normal, and binormal vectors, for r(t) = [cost, sint ln cost] at the point (1, 0, 0). 8. [869/53] At what point on the curve x = t 3, y = 3t, z = t 4 is the normal plane parallel to the plane 6x + 6y 8z = 1? 9. [870/64] Show the circular helix r(t) = [a cost, a sint, bt], where a and b are positive constants, has constant curvature and constant torsion. 10. [870/66] Find the curvature and torsion of the curve x = sinht, y = cosht, z = t, at the point (0,1,0) Motion in Space: Velocity and Acceleration 1. [878/4] For r(t) = [ t 2,1/t 2], find the velocity, acceleration, and speed of a particle with this position function. Graph the path of the particle along with the velocity and acceleration vectors for t = [878/8] For r(t) = [t, 2 cost, sint], find the velocity, acceleration, and speed of a particle with this position function. Graph the path of the particle along with the velocity and acceleration vectors for t = [878/12] Find the velocity, acceleration, and speed of a particle with position function r(t) = [ t 2,2t,lnt ]. 4. [878/16] Find the velocity and position vectors of a particle that has acceleration a(t) = sint i + 2cost j + 6t k, initial velocity v(0) = k, and initial position r(0) = j 4k. 5. [878/18] Let a(t) = [t,e t,e t ], v(0) = [0,0,1], and r(0) = [0,1,1]. (a) Find the position vector r(t). (b) Graph the path of the particle. 6. [878/26] A projectile is fired from a tank with initial speed 400m/s. Find two angles of elevation that can be used to hit a target 3000m away. 7. [879/36] A particle moves in space. (a) If it moves along a straight line, what can you say about its acceleration vector? (b) If it moves with constant speed along a curve, what can you say about its acceleration vector? 8. [879/38] Let r(t) = [ 2t 2, 2 3 t3 2t ]. Find the tangential and normal components of the acceleration vector. 9. [879/40] Let r(t) = [ t,2e t,e 2t]. Find the tangential and normal components of the acceleration vector. 10. [879/42] Let r(t) = 1 t i + 1 j + 1 k. Find the tangential and t 2 t normal components of the acceleration 3 vector at (1,1,1). 14 Partial Derivatives 14.1 Functions of Several Variables 1. [900/10] Let F (x,y) = y 2. (a) Evaluate F (3, 1). (b) Find and sketch the domain of F. (c) Find the range of F. 2. [900/12] Let g(x,y,z) = x 3 y 2 z 10 x y z. (a) Evaluate g(1,2,3). (b) Find and describe the domain of g. 3. [900/18] Sketch the domain of g(x,y) = ln(2 x) 1 x 2 y [900/28] Sketch the graph of f (x,y) = 2 x 2 y [901/54] Sketch both a contour map and a graph of the function f (x,y) = 36 9x 2 4y [902/58] Graph the dog saddle f (x,y) = xy 3 yx [903/68] Describe the level surfaces of the function f (x,y,z) = x 2 + 3y 2 + 5z [903/74] Plot f (x,y) = xye x2 y 2 as both a contour map and a graph. Try to identify any maximum or minimum points on the graph. 9. [903/76] Graph f (x,y) = xy x 2. Comment on the limiting + y2 behaviors of the function as (x,y) approaches the origin and as both x and y become large. 10. [903/80] Graph f (x,y) = x 2 + y 2, f (x,y) = e x 2 +y 2, f (x,y) = ln x 2 + y 2, f (x,y) = sin x 2 + y 2, and finally f (x,y) = 1/ x 2 + y 2. In general, if g(t) is a function of ( ) one variable, how is the graph of f (x,y) = g x 2 + y 2 obtained from the graph of g? 14.2 Limits and Continuity 1. [910/4] Find the limit of f (x,y) = xy x 2 as (x,y) (0,0) + 2y2 or state that it does not exist. Explain your conclusion. 2. [910/8] Find the limit lim (x,y) (3,2) e 2x y or show that it does not exist (DNE). 3. [910/12] Find lim (x,y) (1,0) 4. [910/16] Find lim (x,y) (0,0) xy y (x 1) 2 or show that it DNE. + y2 xy 4 x 4 or show that it DNE. + y4 4

5 5. [910/17] Determine the limit lim 6. [911/22] Find lim (x,y) (0,0,0) (x,y) (0,0) x 2 + y 2 x 2 + y x 2 y 2 z 2 x 2 + y 2 or show that it DNE. + z [911/28] Graph the function f (x,y) = 1 x 2 y 2. Observe where it is discontinuous. Use the formula to explain what you have observed. 8. [911/36] Determine the set of points at which the function f (x,y,z) = y x 2 lnz is continuous. 9. [911/40] Find lim ( x 2 + y 2) ln ( x 2 + y 2) by using polar (x,y) (0,0) coordinates. sin ( x 2 + y 2) 10. [911/42] Find lim (x,y) (0,0) x 2 + y 2 by using polar coordinates Partial Derivatives 1. [924/10] Use the contour map below to estimate the partial derivatives f x (2,1) and f y (2,1). 10. [927/98] The paraboloid z = 6 x x 2 2y 2 intersects the plane x = 1 in a parabola. Find parametric equations for the tangent line to this parabola at the point (1,2, 4). Graph the paraboloid, the parabola, and the tangent line on one plot Tangent Planes and Linear Approximation 1. [934/2] Find an equation of the tangent plane to the surface z = (x + 2) 2 2(y 1) 2 5 at the point (2,3,3). 2. [934/6] Find an equation of the tangent plane to the surface z = ln(x 2y) at the point (3,1,0). 3. [934/8] Graph the surface z = 9 + x 2 y 2 and its tangent plane at the point (2,2,5). 4. [934/14] Explain why f (x,y) = 1 + y is differentiable at 1 + x (1,3). Then find the linearization L(x,y) of the function at the point. 5. [935/20] Find the linear approximation of the function f (x,y) = 1 xy cosπy at (1,1) and use it to approximate f (1.02,0.97). 6. [935/30] Find the differential of the function f = xze y2 z [935/32] If z = x 2 xy + 3y 2 and (x,y) changes from (3, 1) to (2.96, 0.95), compare the values of z and dz. 2. [924/16] Find the first partial derivatives of the function f (x,y) = x 2 y 3y [924/30] Find the first partial derivatives of the function F (α,β) = β α t 3 + 1dt. 4. [924/40] Find the first partial derivatives of the function u = sin(x 1 + 2x nx n ). 5. [925/44] Let f (x,y,z) = x yz. Find f z (e,1,0). 6. [925/50] Given yz + xlny = z 2, use implicit differentiation to find z/ x and z/ y. 7. [925/66] Let g(r,s,t) = e r sin(st). Find g rst. 8. [925/76] Determine whether each of the following functions is a solution of Laplace s equation u xx + u yy = 0. (a) u = x 2 + y 2 (d) u = ln x 2 + y 2 (b) u = x 2 y 2 (e) u = sinxcoshy + cosxsinhy (c) u = x 3 + 3xy 2 (f) u = e x cosy e y cosx 9. [926/82] The temperature at a point (x,y) on a flat metal plate is give by T (x,y) = 60/ ( 1 + x 2 + y 2), where T is measured in C and x,y in meters. Determine the rate of change of the temperature with respect to distance at the point (2,1) in (a) the x-direction and (b) the y-direction. 8. [935/34] Use differentials to estimate the amount of metal in a closed cylindrical can that is 10cm high and 4cm in diameter if the metal in the top and bottom is 0.1cm thick and the metal in the sides is 0.05cm thick. 9. [935/38] The pressure, volume, and temperature of a mole of an ideal gas are related by the equation PV = 8.31T, where P is measured in kilopascals, V in liters, and T in kelvins. Use differentials to find the approximate change in the pressure if the volume increases from 12L to 12.3L and the temperature decreases from 310K to 305K. 10. [936/42] Suppose you need to know an equation of the tangent plane to a surface S at the point P(2,1,3). You don t have an equation for S but you know that the curves r 1 (t) = [ 2 + 3t, 1 t 2, 3 4t +t 2] r 2 (u) = [ 1 + u 2, 2u 3 1, 2u + 1 ] both lie on S. Find an equation of the tangent plane at P The Chain Rule 1. [943/4] Let z = 1 + xy, x = tan t, y = arctan t. Use the Chain Rule to find dz/dt. 2. [943/6] Let w = ln x 2 + y 2 + z 2, x = sin t, y = cos t, z = tan t. Use the Chain Rule to find dw/dt. 5

6 3. [943/8] Let z = tan 1 ( x 2 + y 2), x = slnt, y = te s. Use the Chain Rule to find z/ s and z/ t. 4. [944/16] Suppose f is a differentiable function of x and y, and g(r,s) = f ( 2r s, s 2 4r ). Use the table of values below to calculate g r (1,2) and g s (1,2). 5. [957/30] Near a buoy, the depth of a lake at the point with coordinates (x,y) is z = x y 3, where x, y, and z are measured in meters. A fisherman in a small boat starts at the point (80,60) and moves toward the buoy, which is located at (0,0). Is the water under the boat getting deeper or shallower when he departs? Explain. 6. [958/46] Find equations of (a) the tangent plane and (b) the normal line to the surface x 4 + y 4 + z 4 = 3x 2 y 2 z 2 at (1,1,1). 5. [944/22] Let T = v 2u + v, u = pq r, v = p qr. Use the Chain Rule to find the first order partial derivatives of T with respect to p, q, and r, when p = 2, q = 1, r = [944/24] Let P = u 2 + v 2 + w 2, u = xe y, v = ye x, w = e xy. Use the Chain Rule to find the first order partial derivatives of P with respect to x and y when x = 0 and y = [944/32] Let x 2 y 2 + z 2 2z = 4. Find z/ x and z/ y via implicit differentiation. 8. [945/38] The radius of a right circular cone is increasing at a rate of 1.8cm/s while its height is decreasing at a rate of 2.5cm/s. At what rate is the volume of the cone changing when the radius is 120cm and the height is 140cm? 9. [945/42] A manufacturer has modeled its yearly production function P (the value of its production, in millions of dollars) as a Cobb-Douglas function P(L,K) = 1.47L 0.65 K 0.35 where L is the number of labor hours (in thousands) and K is the invested capital (in millions of dollars). Suppose that when L = 30 and K = 8, the labor force is decreasing at a rate of 2000 labor hours per year and capital is increasing at a rate of $500,000 per year. Find the rate of change of production. 10. [945/46] If u = f (x,y) where x = e s cost and y = e s sint, ( ) 2 ( ) ( 2 ( ) 2 ( ) ) 2 show that u x + u y = e 2s u s + u t. 7. [959/54] At what points on the ellipsoid x 2 + y 2 + 2z 2 = 1 is the tangent plane parallel to the plane x + 2y + z = 1? 8. [959/60] At what points does the normal line through the point (1,2,1) on the ellipsoid 4x 2 + y 2 + 4z 2 = 12 intersect the sphere x 2 + y 2 + z 2 = 102? 9. [959/64] The plane y + z = 3 intersects the cylinder x 2 + y 2 = 5 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1,2,1). 10. [959/66] The helix r(t) = [cos(πt/2), sin(πt/2), t] intersects the sphere x 2 + y 2 + z 2 = 2 in two points. (a) Find the angle of intersection at each point. (b) Graph the sphere, helix, and points of intersection on the same plot Maximum and Minimum Values 1. [968/4] Use level curves in the figure below to ascertain the location of the critical points of f (x,y) = 3x x 3 2y 2 + y 4 and whether f has a saddle point, a local maximum, or a local minimum at each critial point. Explain your reasoning. Then use the Second Derivatives Test to confirm your assertions Directional Derivatives and the Gradient Vector 1. [956/4] Let f (x,y) = xy 3 x 2. Find the directional derivative of f at P(1,2) in the direction given by θ = π/3. 2. [957/10] Let f (x,y,z) = y 2 e xyz. (a) Find the gradient of f. (b) Evaluate the gradient at the point P(0,1, 1). (c) Find the rate of change of f at P in the direction of the unit vector û = [ 3 13, 4 13, 12 13]. 3. [957/20] Find the directional derivative of f (x,y,z) = xy 2 z 3 at P(2,1,1) in the direction toward Q(0, 3,5). 4. [957/26] Find the maximum rate of change of the function f (p, q, r) = arctan(pqr) at P(1, 2, 1) and the direction in which it occurs. 2. [968/12] Find and analyze the critical points of f (x,y) = x 3 + y 3 3x 3 3y 2 9x. Illustrate. 3. [968/16] Find and analyze the critical points of f (x,y) = xye (x2 +y 2 )/2. Illustrate. 4. [968/24] Find and analyze the critical points of the function f (x,y) = (x y)e x2 y 2. Illustrate. 6

7 5. [968/35] Find the absolute maximum and minimum values of the function f (x,y) = x 2 + 2y 2 2x 4y + 1 on the domain D = {(x,y) : 0 x 2,0 y 3} and where they occur. 6. [968/36] Find extreme values and where they occur for f (x,y) = xy 2 on D = { (x,y) : x 0,y 0,x 2 + y 2 3 }. 7. [969/42] Find the point on the plane x 2y + 3z = 6 that is closest to the point (0,1,1). 8. [969/48] Find the dimensions of the box with volume 1000cm 3 that has minimal surface area. 9. [969/50] Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64cm [970/60] Find an equation of the plane that passes through the point (1,2,3) and cuts off the smallest volume in the first octant Lagrange Multipliers 1. [977/2] For this problem, use MATLAB or a calculator. (a) Graph the circle x 2 + y 2 = 1. On the same plot, graph several curves of the form x 2 + y = c until you find two that just touch the circle. What is the significance of the values of c for these two curves? (b) Use Lagrange multipliers to find the extreme values of f (x,y) = x 2 + y subject to the constraint x 2 + y 2 = 1. Compare answers with part (a). 2. [977/4] Use Lagrange multipliers to find the extreme values of f (x,y) = 3x + y subject to the constraint x 2 + y 2 = [977/16] Find the minimum of f (x,y,z) = x 2 + 2y 2 + 3z 2 subject to the constraint x + 2y + 3z = 10. Show that f has no maximum value with this constraint. 4. [977/20] Find the extreme values of f (x,y,z) = x 2 + y 2 + z 2 subject to the constraints x y = 1 and y 2 z 2 = [977/22] Find extreme values of f (x,y) = 2x 2 + 3y 2 4x 5 subject to x 2 + y [978/26] Consider the function f (x,y) = x 3 + y 3 + 3xy subject to the constraint (x 3) 2 + (y 3) 2 = 9. (a) Estimate extreme values of f via graphical methods. (b) Use Lagrange multipliers to find extreme values exactly. 7. [978/32] Use Lagrange multipliers to find the point on the plane x 2y + 3z = 6 that is closest to the point (0,1,1). 8. [978/38] Use Lagrange multipliers to find the dimensions of the box with volume 1000cm 3 that has minimal surface area. 9. [978/40] Use Lagrange multipliers to find the dimensions of the rectangular box with largest volume if the total surface area is given as 64cm [978/48] Find the extreme values of f (x,y,z) = x + y + z subject to the constraints x 2 y 2 = z and x 2 + z 2 = Multiple Integrals 15.1 Double Integrals Over Rectangles 1. [999/10] Evaluate the double integral R 2x + 1dA where R = {(x,y) : 0 x 2,0 y 4}. 2. [999/14] Let f (x,y) = y x + 2. Find 2 0 f (x,y) dx and f (x,y) dy [1000/18] Calculate π/6 π/2 0 0 sinx + sinydydx. 4. [1000/22] Calculate yex y dxdy. 5. [1000/26] Calculate s +t dsdt. 6. [1000/30] Calculate R tanθ da where 1 t 2 R = { (θ,t) : 0 θ π/3, 0 t 1 2}. 7. [1000/34] Calculate 1 R 1+x+y da where R = [1,3] [1,2], a Cartesian set product. 8. [1000/40] Find volume of solid enclosed by surface z = x 2 + xy 2 and planes z = 0, x = 0, x = 5, and y = ±2. 9. [1000/44] Graph the solid that lies between the surface z = 2xy/ ( x ) and the plane z = x + 2y and is bounded by planes x = 0, x = 2, y = 0, and y = 4. Find its volume. 10. [1000/48] Find the average value of f (x,y) = e y x + e y over the rectangle R = [0,4] [0,1] Double Integrals Over General Regions 1. [1008/6] Evaluate the integral 1 0 e v e v dwdv. 2. [1008/14] Evaluate the integral D xyda where D is enclosed by the curves y = x 2 and y = 3x. Show that you get the same answer using either order of integration. 3. [1008/20] Evaluate D xyda where D is enclosed by the quarter-circle y = 1 x 2,x 0, and the coordinate axes. 4. [1008/28] Find the volume of the solid bounded by the planes z = x, y = x, x + y = 2, and z = [1009/36] Find the volume of the solid that is enclosed by the parabolic cylinder y = x 2 and the planes z = 3y and z = 2 + y. Do this by subtracting two volumes. 6. [1009/42] Find the volume between the paraboloids z = 2x 2 + y 2 and z = 8 x 2 2y 2 and inside the cylinder x 2 + y 2 = [1009/50] For 1 π/4 0 arctanx f (x,y) dydx, sketch the region of integration, then change the order of integration. 7

8 8. [1009/56] Evaluate the integral y e x4 dxdy by reversing the order of integration. 9. [1009/62] Find the average value of f (x,y) = xsiny over the region D enclosed by the curves y = 0, y = x 2, and x = [1010/70] Graph the solid bounded by the plane x + y + z = 1 and the paraboloid z = 4 x 2 y 2 and find its exact volume Double Integrals in Polar Coordinates 1. [1014/4] A region R is shown below. Decide whether to use polar coordinates or rectangular coordinates and write R f (x,y) da as an iterated integral, where f is an arbitrary continuous function on R. 10. [1015/39] Use polar coordinates to combine the sum 1 1/ 2 x 1 x 2 xydydx+ 2 x xydydx+ into one double integral, then evaluate it Applicatons of Double Integrals 2 4 x 2 0 xydydx 1. [1024/2] Electric charge is distributed over the unit disk x 2 + y 2 1 so that the charge density is σ (x,y) = x 2 + y 2 (measured in coulombs per square meter). Find the total charge on the disk. 2. [1025/6] Find the mass and center of mass of the lamina that occupies the triangular region D enclosed by the lines y = 0, y = 2x, and x + 2y = 1 and has density δ = x. 3. [1025/10] Find the mass and center of mass of the lamina that occupies the region D enclosed by y = 0 and y = cosx, π/2 x π/2 and has density δ = y. 2. [1014/8] Use polar coordinates to evaluate R 2x yda, where R is the region in the first quadrant enclosed by the circle x 2 + y 2 = 4 and the lines x = 0 and y = x. 3. [1015/12] Use polar coordinates to find D cos x 2 + y 2 da, where D is the disk with center the origin and radius [1015/16] Use a double integral to find the area of the region common to both cardioids r = 1 + cosθ and r = 1 cosθ. 5. [1015/20] Use polar coordinates to find the volume below the cone z = x 2 + y 2 and above the ring 1 x 2 + y [1015/24] Find the volume bounded by the paraboloid z = 1 + 2x 2 + 2y 2 and the plane z = 7 in the first octant. 7. [1015/30] Evaluate a 0 polar coordinates. a 2 y 2 2x + y dx dy by switching to a 2 y2 8. [1015/32] Evaluate 2 2x x x 2 + y 2 dy dx by switching to polar coordinates. 9. [1015/36] An agricultural sprinkler distributes water in a circular pattern of radius 100ft. It supplies water to a depth of e r feet per hour at a distance r feet from the sprinkler. (a) If 0 < R 100, what is the total amount of water supplied per hour to the region inside the circle of radius R centered at the sprinkler? (b) Determine an expression for the average amount of water per hour per square foot supplied to the region inside the circle of radius R. 4. [1025/12] A lamina occupies the part of the disk x 2 + y 2 1 in the first quadrant. Find its mass and center of mass if the density at any point is proportional to the square of its distance from the origin. 5. [1025/14] The boundary of a lamina consists of the two semicircles y = 1 x 2 and y = 4 x 2 together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density at any point is inversely proportional to its distance from the origin. 6. [1025/18] Find the moments of inertia I x, I y, I 0 for the lamina specified in #2 [1025/6]. 7. [1025/24] A lamina with constant density δ = k occupies the region under the curve y = sinx from x = 0 to x = π. Find the moments of inertia I x and I y and radii of gyration = x and = y. 8. [1025/26] Find the mass, center of mass, and moments of inertia for the lamina D = {(x,y) : 0 y xe x,0 x 2} with density δ = x 2 y 2. { 4xy if 0 x 1, 0 y 1 9. [1025/28] Let f (x, y) = 0 otherwise. (a) Verify that f is a joint density function. (b) If X and Y are random variables with joint density function f, find P ( X 1 2) and P ( X 1 2,Y 1 2). (c) Find the expected values of X and Y. { [1025/29] Let f (x, y) = 10 e x/2 y/5 if x 0, y 0 0 otherwise. (a) Verify that f is a joint density function. (b) If X and Y are random variables with joint density function f, find P(Y 1) and P(X 2,Y 4). (c) Find the expected values of X and Y. 8

9 15.5 Surface Area 1. [1028/2] Find the surface area of the part of the plane 6x + 4y + 2z = 1 that lies inside the cylinder x 2 + y 2 = [1028/4] Find the surface area of the part of the surface 2y + 4z x 2 = 5 that lies above the triangle with vertices (0,0), (2,0), and (2,4). 3. [1028/6] Find the area of the part of the cylinder x 2 + z 2 = 4 that lies above the square with vertices (0,0), (1,0), (1,1), and (0,1). 4. [1028/8] Find the area of the surface z = 3 2 ( x 3/2 + y 3/2), 0 x 1, 0 y [1028/10] Find the area of the part of the sphere x 2 + y 2 + z 2 = 4 that lies above the plane z = [1028/12] Find the area of the part of the sphere x 2 + y 2 + z 2 = 4z that lies inside the paraboloid z = x 2 + y [1028/14] Find the area of the surface z = cos ( x 2 + y 2) lying inside the cylinder x 2 + y 2 = 1 (correct to 4 decimal places). 8. [1029/16] Consider z = xy + x 2 + y 2, 0 x 2, 0 y 2. (a) Estimate its surface area by using the Midpoint Rule for double integrals with m = n = 2. (b) Compute said area to four decimal places using a CAS. 9. [1029/18] Graph the surface z = 1 + x + y + x 2, 2 x 1, 1 y 1. Then compute its surface area. 10. [1029/20] Graph the surface z = ( 1 + x 2) / ( 1 + y 2) lying above the rotated square x + y 1. Compute its surface area to 4 decimal places Triple Integrals in Rectangular Coordinates 1. [1037/4] Evaluate the integral 1 0 2y x+y y 0 6xydzdxdy. 2. [1037/8] Evaluate x 2 y xye z dzdydx. 3. [1037/12] Evaluate E sinydv, where E lies below the plane z = x and above the triangular region with vertices (0,0,0), (π,0,0), and (0,π,0). 4. [1037/14] Evaluate E x ydv, where E is enclosed by the surfaces z = x 2 1, z = 1 x 2, y = 0, and y = [1038/20] Use a triple integral to find the volume of the solid enclosed by the paraboloids y = x 2 + z 2 and y = 8 x 2 z [1038/26] Consider the integral B xe xyz dv, where B = {(x,y,z) : 0 x 4, 0 y 1, 0 z 2}. Compute its value to four decimal places using a CAS. 7. [1039/42] Find the mass and center of mass of the solid tetrahedron E bounded by the planes x = 0, y = 0, z = 0, x + y + z = 1 with density δ = y. 8. [1039/46] Find I z, the moment of inertia about the z-axis, for the solid cone x 2 + y 2 z h with constant density δ = k. 9. [1039/48] Compute (a) the mass, (b) center of mass, and (c) moment of inertia about the z-axis for the solid hemisphere x 2 + y 2 + z 2 1, z 0 with density δ = x 2 + y 2 + z 2. Use a CAS and spherical coordinates. (Refer to Section 15.8.) 10. [1039/50] Let E be the solid in the first octant bounded by the cylinder y 2 + z 2 = 9 and the planes x = 0, y = 3x, and z = 0, and having density δ = x 2 + y 2. Use a CAS to find its (a) mass, (b) center of mass, and (c) moment of inertia about the z-axis Triple Integrals in Cylindrical Coordinates 1. [1043/4] Change from rectangular to cylindrical coordinates. ( (a) 2, ) 2,1 (b) (2,2,2) 2. [1043/8] Identify the surface r = 2sinθ. 3. [1043/10] Write the equations in cylindrical coordinates. (a) 2x 2 + 2y 2 z 2 = 4 (b) 2x y + z = 1 4. [1043/12] Sketch the solid 0 θ π/2, r z [1043/16] Sketch the solid whose volume is given by the integral 2π 2 r r dzdr dθ then evaluate the integral. 6. [1043/18] Evaluate E zdv, where E is enclosed by the paraboloid z = x 2 + y 2 and the plane z = [1044/20] Evaluate E x ydv, where E is the solid that lies between the cylinders x 2 + y 2 = 1 and x 2 + y 2 = 16, above the xy-plane, and below the plane z = y [1044/24] Find the volume of the solid that lies between the paraboloid z = x 2 + y 2 and the sphere x 2 + y 2 + z 2 = [1044/28] Find the mass of the ball x 2 + y 2 + z 2 a 2 if the density at any point is proportional distance from z-axis. 9 x 2 y [1044/30] Evaluate 3 9 x by changing to cylindrical coordinates. x 2 + y 2 dzdydx 15.8 Triple Integrals in Spherical Coordinates 1. [1049/4] Change from rectangular to spherical coordinates. (a) ( 1,0, 3 ) (b) ( 3, 1,2 3 ) 2. [1050/10] Write the equations in spherical coordinates. (a) z = x 2 + y 2 (b) z = x 2 y 2 3. [1050/14] Sketch the solid ρ 2, ρ csc φ. 9

10 4. [1050/22] Evaluate E y2 z 2 dv, where E lies above the cone φ = π/3 and below the sphere ρ = [1050/26] Evaluate E x 2 + y 2 + z 2 dv, where E lies above the cone z = x 2 + y 2 and between the spheres x 2 + y 2 + z 2 = 1 and x 2 + y 2 + z 2 = [1050/28] Find the average distance from a point in a ball of radius a to its center. 7. [1050/32] Let H be a solid hemisphere of radius a whose density at any point is proportional to its distance from the center of the base. (a) Find the mass of H. (b) Find the center of mass of H. (c) Find the moment of inertia of H about its axis. 8. [1050/34] Find the mass and center of mass of a solid hemisphere of radius a if the density at any point is proportional to its distance from the base. 9. [1050/35] Find the volume and centroid of the solid E that lies above the cone z = x 2 + y 2 and below the sphere x 2 + y 2 + z 2 = 1. Use your choice of coordinates. 10. [1051/43] Change to spherical coordinates, then compute. 2 4 x x 2 y 2 ( x 2 + y 2 + z 2) 3/2 dzdydx 4 x x 2 y Change of Variables in Multiple Integrals 1. [1060/2] Find the Jacobian of the transformation x = u 2 + uv, y = uv [1060/6] Find the Jacobian of the transformation x = u + vw, y = v + wu, z = w + uv. 3. [1060/8] Let S be the square region bounded by the lines u = 0, u = 1, v = 0, v = 1 in the uv-plane. Find its image under the transformation x = v, y = u ( 1 + v 2). 4. [1060/14] Let R be the region in the xy-plane bounded by the hyperbolas y = 1/x, y = 4/x and the lines y = x, y = 4x, in the first quadrant. Find a transformation T (u,v) that maps a rectangular region in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes. 5. [1060/16] Let R be the parallelogram with vertices ( 1,3), (1, 3), (3, 1), (1,5). Use the transformation x = 1 4 (u + v), y = 1 4 (v 3u) to evaluate the integral R 4x + 8ydA. 6. [1060/18] Let R be the region that is bounded by the ellipse x 2 xy + y 2 = 2. Use the transformation x = 2u 2/3v, y = 2u + 2/3v, to evaluate R x2 xy + y 2 da. 7. [1060/20] Let R be the region in the first quadrant bounded by the curves xy = 1, xy = 2, xy 2 = 1, xy 2 = 2. Use inverse transformation u = xy, v = xy 2, to compute R y2 da. 8. [1060/22] An important problem in thermodynamics is to find the work done by an ideal Carnot engine. A cycle consists of alternating expansion and compression of gas in a piston. The work done by the engine is equal to the area of the region R enclosed by two isothermal curves xy = a, xy = b, and two adiabatic curves xy 1.4 = c, xy 1.4 = d, where 0 < a < b and 0 < c < d. Compute the work by determining the area of R. 9. [1060/24] Evaluate the integral R (x + y)ex2 y 2 da by making an appropriate change of variables. Here R is the rectangle enclosed by lines x y = 0, x y = 2, x + y = 0, x + y = [1063/56] Use the transformation x = u 2, y = v 2, z = w 2 to find the volume of the region bounded by the surface x + y + z = 1 and the coordinate planes. 16 Vector Calculus 16.1 Vector Fields Use MATLAB to plot vector fields. 1. [1073/6] Plot the vector field F(x,y) = yi xj x 2 + y [1073/10] Plot the vector field F(x, y, z) = i + k. 3. [1073/14] Plot the vector field F(x, y) = [cos(x + y), x]. 4. [1074/16] Plot the vector field F(x, y, z) = [1, 2, z]. 5. [1074/18] Plot the vector field F(x, y, z) = [x, y, z]. 6. [1074/22] Find the gradient vector field of f (s,t) = 2s + 3t. 7. [1074/24] Find the gradient vector field of f (x,y,z) = x 2 ye y/z. 8. [1074/26] Find and plot the gradient vector field of f (x,y) = 1 ( 2 x 2 y 2). 9. [1074/34] At time t = 1, a particle is located at (1,3). If it moves in a velocity field F(x,y) = [ xy 2, y 2 10 ], find its approximate location at time t = [1074/36] Consider vector field F(x, y) = i + x j. (a) Plot the vector field along with some flow lines. What shape do these flow lines appear to have? (b) If parametric equations of the flow lines are x = x(t), y = y(t), what differential equations do these functions satisfy? Deduce that dy/dx = x. (c) If a particle starts at the origin in the velocity field given by F, find an equation of the path it follows. 10

11 16.2 Line Integrals 1. [1084/8] Evaluate the line integral C x2 dx + y 2 dy, where C consists of the arc of the circle x 2 + y 2 = 4 from (2,0) to (0,2) followed by the line segment from (0,2) to (4,3). 2. [1084/12] Evaluate the line integral C x2 + y 2 + z 2 ds, where C : x = t, y = cos2t, z = sin2t, 0 t 2π. 3. [1085/20] Evaluate the line integral C F dr, where F = [ x + y 2, xz, y + z ] and C is r = [ t 2,t 3, 2t ], 0 t [1085/22] Evaluate the line integral C F dr, where F = [x, y, xy] and C is r = [cost, sint, t], 0 t π. 5. [1085/26] Evaluate the line integral C f ds correct to four decimal places, where f = z ln(x + y) and C is parameterized as r = [ 1 + 3t, 2 +t 2, t 4], 1 t 1. [ ] x 6. [1085/28] Let F = x 2 + y, y and C be the 2 x 2 + y 2 parabola y = 1 + x 2 from ( 1,2) to (1,2). Parameterize C as r, then compute C F dr. 7. [1085/30] Evaluate the line integral C F dr, where F = [x, z, y] and C is r = [ 2t, 3t, t 2], 1 t [1085/34] A thin wire has the shape of the first-quadrant part of the circle with center the origin and radius a. If the density function is δ = kxy, find the wire s mass and center of mass. 9. [1086/36] Find the mass and center of mass of a wire in the shape of the helix x = t, y = cost, z = sint, 0 t 2π, if the density at any point is the square of the distance from the point to the origin. 10. [1086/40] Find the work done by the force field F(x,y) = x 2 i + ye x j on a particle that moves along the parabola x = y from (1,0) to (2,1) The Fundamental Theorem for Line Integrals 1. [1094/3] Determine whether or not F = [ xy + y 2, x 2 + 2xy ] is conservative. If it is, find a potential function f ; that is, a function f such that F = f. 2. [1094/6] Determine whether or not F = [ye x, e x + e y ] is conservative. If it is, find a potential function f. 3. [1094/8] Determine whether F = [ 2xy + y 2, x 2 2xy 3] is conservative. If it is, find a potential function f. 4. [1094/10] Determine whether F = [lny + y/x, lnx + x/y] is conservative. If it is, find a potential function f. 5. [1094/12] Let F = [ 3 + 2xy 2, 2x 2 y ] and let C be the arc of the hyperbola y = 1/x from (1,1) to ( 4, 1 4). (a) Find a potential function f for F. (b) Use the Fundamental Theorem for Line Integrals (FTLI) to evaluate C F dr. 6. [1094/14] Let F = [ (1 + xy)e xy, x 2 e xy] and let C be given by r = [cost, 2sint], 0 t π/2. (a) Find a potential function f for F. (b) Use the FTLI to evaluate C F dr. 7. [1095/16] Let F = [ y 2 z + 2xz 2, 2xyz, xy 2 + 2x 2 z ] and C be given by r = [ t, t + 1, t 2], 0 t 1. (a) Find a potential function f for F. (b) Use the FTLI to evaluate C F dr. 8. [1095/18] Let F = [siny, xcosy + cosz, ysinz] and C be given by r = [sint, t, 2t], 0 t π/2. (a) Find a potential function f for F. (b) Use the FTLI to evaluate C F dr. 9. [1095/20] Use the FTLI to evaluate C F dr where F = [siny, xcosy siny] and C is any path from (2,0) to (1,π). 10. [1095/24] Find the work done by the force field F = [2x + y, x] in moving an object from P(1, 1) to Q(4,3) Green s Theorem 1. [1101/2] Evaluate the line integral C ydx xdy where C is the circle with center the origin and radius 4. Do this (a) directly and (b) using Green s Theorem. 2. [1102/4] Evaluate the line integral C x2 y 2 dx + xydy where C consists of the arc of the parabola y = x 2 from (0,0) to (1,1) and the line segments from (1,1) to (0,1) and from (0,1) to (0,0). Do this (a) directly and (b) using Green s Theorem. 3. [1102/6] Evaluate C ( x 2 + y 2) dx + ( x 2 y 2) dy by using Green s Theorem. Here C is the triangle with vertices (0,0), (2,1), and (0,1), traversed counterclockwise. 4. [1102/8] Evaluate C y4 dx + 2xy 3 dy using Green s Theorem. Here C is ellipse x 2 + 2y 2 = 2, traversed counterclockwise. 5. [1102/10] Evaluate ( C 1 y 3 ) ( dx + x 3 + e y2) dy using Green s Theorem. Here C is the boundary of the region between the circles x 2 + y 2 = 4 and x 2 + y 2 = [1102/12] Use Green s Theorem to evaluate C F dr, where F = [ e x + y 2, e y + x 2] and C consists of the arc of the curve y = cosx from ( π/2,0) to (π/2,0) and the line segment from (π/2,0) to ( π/2,0), positively oriented. 7. [1102/14] Use Green s Theorem to evaluate [ C F dr, where x ] F = 2 + 1,tan 1 x. Here C is the triangle from (0,0) to (1,1) to (0,1) and back to (0,0), traversed counterclockwise. 11

12 8. [1102/16] Verify Green s Theorem by evaluating the line integral and double integral involved. Here C is the ellipse 4x 2 + y 2 = 4, P = 2x x 3 y 5, and Q = x 3 y 8. Use a CAS. 9. [1102/18] A particle starts at the origin, moves along the x-axis to (5,0), then along the quarter-circle x 2 + y 2 = 25, x 0, y 0 to the point (0,5), and then down the y-axis back to the origin. Use Green s Theorem to find the work done on this particle by the force field F = [ sinx, siny + xy x3]. 10. [1102/20] If a circle C with radius 1 rolls along the outside of the circle x 2 + y 2 = 16, a fixed point P on C traces out a curve, x = 5cost cos5t, y = 5sint sin5t, called an epicycloid. Graph the epicycloid and use Green s Theorem with F = [0,x] to find the area it encloses Curl and Divergence 1. [1109/2] Find (a) the curl and (b) the divergence of the vector field F = [ 0, x 3 yz 2, y 4 z 3]. 2. [1109/4] Let F = [sin yz, sin xz, sin xy]. Find its curl and divergence. 3. [1109/6] Find the curl and divergence of the vector field F = ln(2y + 3z)i + ln(x + 3z)j + ln(x + 2y)k. 4. [1109/8] Find the curl and divergence of F = [arctan(xy),arctan(yz),arctan(xz)]. 5. [1109/10] The vector field F is shown below in the xy-plane and looks the same in all other horizontal planes. (In other words, F is independent of z and its z-component is 0.) 16.6 Parametric Surfaces and Their Areas 1. [1120/6] Identify the surface r = [3 cost, s, sint], s [1120/12] Graph the surface x = cos u, y = sin u sin v, z = cosv, 0 u 2π, 0 v 2π. 3. [1120/18] Graph the surface x = sin u, y = cos u sin v, z = sinv. 4. [1120/24] Find a parametric representation of the part of the cylinder x 2 + z 2 = 9 that lies above the xy-plane and between the planes y = 4 and y = 4. Then graph this surface. 5. [1121/30] Find parametric equations for the surface obtained by rotating the curve x = 1/y, y 1, about the y-axis. Then graph the surface. 6. [1121/36] Let r = [sin u, cos u sin v, sin v]. Find its tangent plane at the point P corresponding to u = π/6, v = π/6. 7. [1121/42] Find the area of the part of the cone z = x 2 + y 2 that lies between the plane y = x and the cylinder y = x [1121/48] Find the area of the helicoid (or spiral ramp) r = [ucosv, usinv, v], 0 u 1, 0 v π. 9. [1121/54] Correct to 4 decimal places, find the area of the part of the surface z = ( 1 + x 2) / ( 1 + y 2) that lies above the rotated square x + y 1. Illustrate. 10. [1122/60] Let x = acoshucosv, y = bcoshusinv, z = csinhu, 0 u π, 0 v 2π. (a) Show these equations yield a hyperboloid of one sheet. (b) Graph the hyperboloid using a = 1, b = 2, c = 3. (c) Compute the surface area of the hyperboloid in part (b) to 4 decimal places Surface Integrals (a) Is div F positive, negative, or zero? Explain. (b) Determine whether curl F = 0. If not, in which direction does curl F point? 6. [1109/14] Determine if F = [ xyz 4, x 2 z 4, 4x 2 yz 3] is conservative. If so, find a potential function f for F. 7. [1109/16] Determine if F = i + sinzj + ycoszk is conservative. If so, find a potential function f for F. 8. [1109/18] Determine if F = e x [sinyz, zcosyz, ycosyz] is conservative. If so, find a potential function f for F. 9. [1109/20] Is there a vector field G on R 3 such that curl G = [x,y,z]? Explain. 10. [1109/22] Show that any vector field that has the form F(x,y,z) = f (y,z) i + g(x,z) j + h(x,y) k is incompressible. 1. [1133/8] Evaluate the surface integral S x2 + y 2 ds, where S is the surface with parametric equations x = 2uv, y = u 2 v 2, z = u 2 + v 2, u 2 + v [1133/12] Evaluate the surface integral S yds, where S is the surface z = 2 ( 3 x 3/2 + y 3/2), 0 x 1, 0 y [1133/16] Evaluate the surface integral S y2 ds, where S is the part of the sphere x 2 + y 2 + z 2 = 1 that lies above the cone z = x 2 + y [1133/20] Evaluate the surface integral S x2 + y 2 + z 2 ds, where S is the part of the cylinder x 2 + y 2 = 9 between the planes z = 0 and z = 2, together with its top and bottom disks. 5. [1133/24] Evaluate the surface integral S F ds (the flux of F across S) where F = xi yj + z 3 k and S is the part of the cone z = x 2 + y 2 between the planes z = 1 and z = 3 with downward orientation. 12

13 6. [1133/28] Compute the flux of F = [yz, xz, xy] across S, the surface z = xsiny, 0 x 2, 0 y π, pointing upward. 7. [1133/32] Compute the flux of F = [y, z y, x] across S, the surface of the tetrahedron with vertices (0,0,0), (1,0,0), (0,1,0), and (0,0,1) with outward orientation. 8. [1133/34] Find S xyzds, where S is the surface z = x2 y 2, 0 x 1, 0 y [1134/40] Find the mass of a thin funnel in the shape of a cone z = x 2 + y 2, 1 z 4, having density δ = 10 z. 10. [1134/42] Let S be the part of the sphere x 2 + y 2 + z 2 = 25 that lies above the plane z = 4. If S has constant density k, find (a) its center of mass and (b) its moment of inertia I z Stokes Theorem 1. [1139/2] Use Stokes Theorem to evaluate S curl F ds where F = x 2 sinzi + y 2 j + xyk and S is the part of the paraboloid z = 1 x 2 y 2 that lies above the xy-plane, oriented upward. 2. [1139/4] Use Stokes Theorem to evaluate S curl F ds where F = [ tan 1 ( x 2 yz 2), x 2 y, x 2 z 2] and S is the cone x = y 2 + z 2, 0 x 2, oriented in the direction of the positive x-axis. 3. [1139/6] Use Stokes Theorem to evaluate S curl F ds where F = [ e xy, e xz, x 2 z ] and S is the half of the ellipsoid 4x 2 + y 2 + 4z 2 = 4 that lies to the right of the xz-plane, oriented in the direction of the positive y-axis. 4. [1139/8] Use Stokes Theorem to evaluate C F dr, where C the boundary of the part of the plane 3x + 2y + z = 1 in the first octant, oriented counterclockwise as viewed from above, and F = [1, x + yz, xy z]. 5. [1139/10] Use Stokes Theorem to evaluate C F dr, where F = [2y, xz, x + y] and C is the curve of intersection of the plane z = y + 2 and the cylinder x 2 + y 2 = [1139/12] Let F = [ x 2 y, 1 3 x3, xy ] and C be the curve of intersection of the hyperbolid paraboloid z = y 2 x 2 and the cylinder x 2 + y 2 = 1, oriented counterclockwise as viewed from above. (a) Use Stokes Theorem to evaluate C F dr. (b) Graph the hyperboloid and cylinder on the same plot. (c) Find parametric equations for C and then graph the curve of intersection. 7. [1139/14] Verify that Stokes Theorem is true for F = 2yzi + yj + 3xk with S the part of the paraboloid z = 5 x 2 y 2 above the plane z = 1, oriented upward. 9. [1139/17] A particle moves along the line segments from the origin to the points (1,0,0), (1,2,1), (0,2,1) and back to the origin under the influence of the force field F = [ z 2,2xy,4y 2]. Find the work done. 10. [1140/18] Evaluate C (y + sinx) dx + ( z 2 + cosy ) dy + x 3 dz where C is the curve r = [sint, cost, sin2t], 0 t 2π. Note that C lies on the surface z = 2xy The Divergence Theorem 1. [1145/2] Verify that the Divergence Theorem is true for the vector field F = [ y 2 z 3, 2yz, 4z 2] where E is the solid that is enclosed by the paraboloid z = x 2 + y 2 and the plane z = [1145/4] Verify that the Divergence Theorem is true for the vector field F = x 2 i yj + zk where E is the solid cylinder y 2 + z 2 9, 0 x [1145/6] Use the Divergence Theorem to calculate the surface integral S F ds (i.e., the flux across S) for the vector field F = [ x 2 yz, xy 2 z, xyz 2] where S is the surface of the box enclosed by the six planes x = 0, x = a, y = 0, y = b, z = 0, and z = c, where a,b,c are positive numbers. 4. [1145/8] Use the Divergence Theorem to calculate the flux of F across S for the vector field F = [ x 3 + y 3,y 3 + z 3,x 3 + z 3] where S is the sphere with center the origin and radius [1145/10] Use the Divergence Theorem to calculate the flux of F across S for the vector field F = [z,y,xz] where S is the surface of the tetrahedron enclosed by the coordinate planes and plane a x + y b + c z = 1 where a,b,c are positive numbers. 6. [1145/12] Use the Divergence Theorem to calculate S F ds for the vector field F = [ xy + 2xz, x 2 + y 2, xy z 2] where S is the surface of the solid bounded by the cylinder x 2 + y 2 = 4 and the planes z = y 2 and z = [1146/14] Use the Divergence Theorem to calculate S F ds for F = r 2 r where r = [x,y,z] and S is the sphere of radius a with center at the origin. 8. [1146/16] Use the Divergence Theorem to calculate S F ds for the cube cut from the first octant by planes x = π 2, y = π 2, z = π 2. Here F = sinxcos2 yi + sin 3 ycos 4 zj + sin 5 zcos 6 xk. 9. [1146/18] Let F = [ ztan 1 ( y 2), z 3 ln ( x ), z ]. Find the flux of F across the part of the paraboloid x 2 + y 2 + z = 2 that lies above the plane z = 1 and is oriented upward. 10. [1146/24] Use the Divergence Theorem to evaluate the surface integral S 2x + 2y + z2 ds where S is the sphere x 2 + y 2 + z 2 = [1139/16] Let C be a simple closed smooth curve that lies in the plane x + y + z = 1. Show that C zdx 2xdy + 3ydz depends only on the area of the region enclosed by C and not on the shape of C or its location in the plane. 13