MATH 223 FINAL EXAM STUDY GUIDE ( )

Transcription

1 MATH 3 FINAL EXAM STUDY GUIDE ( ) The following questions can be used as a review for Math 3 These questions are not actual samples of questions that will appear on the final eam, but the will provide additional practice for the material that will be covered on the final eam When solving these problems keep the following in mind: Full credit for correct answers will onl be awarded if all work is shown Eact values must be given unless an approimation is required redit will not be given for an approimation when an eact value can be found b techniques covered in the course The answers, along with comments, are posted as a separate file on 1 A sonic boom carpet is a region on the ground where the sonic boom is heard directl from the airplane and not as a reflection The width of the carpet, W, can be epressed as a function of the air temperature on the ground directl below the airplane, t, and the vertical temperature gradient at the t airplane s altitude, d Suppose Wtd (, ) = k for some positive constant k d (a) If d is fied, is the width of the carpet an increasing or decreasing function of t (b) If t is fied, is the width of the carpet an increasing or decreasing function of d Describe the following sets of points in words, write an equation, and sketch a graph: (a) The set of points whose distance from the line L is 5 The line L is the intersection of the plane = 3and the -plane (b) The set of points whose distance from the z-plane is three (c) The set of points whose distance from the z-ais and the -plane are equal 3 B setting one variable constant, find a plane that intersects (a) parabola (b) sinusoidal curve (c) line(s) cos + z = 3 in a: 4 onsider the function f(, ) = (a) Plot the level curves of the function for z =, 1, 0,1, (b) Imagine the surface whose height above an point (, ) is given b f(, ) Suppose ou are standing on the surface at the point where = 1, = (i) What is our height? (ii) If ou start to move on the surface parallel to the -ais in the direction of increasing, does our height increase or decrease? (iii) Does our height increase or decrease if ou start to move on the surface parallel to the - ais in the direction of increasing? (iv) Are ou ascending or descending when ou go from (1,) in the direction making an angle of 3π/4 with the -ais? (v) Suppose ou start to move in the direction ou found in part (iv) at a rate of 3mph, at what rate is our height changing with respect to time

2 5a Describe the level curves(contour lines) for each of the following functions: 1 (i) f (, = (ii) g (, = ( + ) (iii) h(, = cos 5b Describe the level surfaces for each of the following functions: (i) f( z,, ) z = (ii) 1 z gz (,, ) = e (iii) f (,, z) = ln( + z ) 6 The figure at the right shows the level curves of the temperature T in degrees elsius as a function of t hours and depth h in centimeters beneath the surface of the ground from midnight ( t = 0) one da to midnight ( t = 4) the net (a) Sketch a graph of the temperature as a function of time at 0 centimeters (b) Sketch a graph of the temperature as a function of the depth at noon from S J Williamson, Fundamentals of Air Pollution, (Reading: Addison-Wesle, 1973) 7 Given the table of some values of a linear function, complete the table and find a formula for the function onsider the planes: I 3 5 z = II 5= + 3 III 5+ 3 = IV 3+ 5 = V z = VI + 1= 0 List all of the planes which: (a) Are parallel to the z-ais (b) Are parallel to 3= 5+ z+ 7 (c) ontain the point (1, 1, 6) i + 3k 3i k (d) Are normal to ( ) ( )

3 9 A portion of the graph of a linear function is shown (a) Find an equation for the linear function (b) Find a vector perpendicular to the plane (c) Find the area of the shaded triangular region 10 Match each of the following functions (a) (f), given b a formula, to the corresponding tables, graphs, and/or contour diagrams (i) (i) There ma be more than one representation or no representations for a formula (a) f(, ) = (b) f(, ) = (c) f(, ) = 1 1 (d) f(, ) = (e) f (, ) 6 3 = (f) f(, ) = + (i) Table 1 (ii) Table (iii) Table 3 (iv) (v) (vi) (vii) (viii) (i)

4 11 Let v = 3i + j k and w= 4i 3 j + k Find each of the following: (a) A vector of length 5 parallel to w (b) A vector perpendicular to v but not perpendicular to w (c) v w (d) The angle between v and w (e) The component of v in the direction of w (f) A vector perpendicular(orthogonal) to both v and w 1 onsider the vectors u = i j + 3k and v = ai + aj k (a) For what value(s) of a are u and v perpendicular? (b) For what value(s) of a are u and v parallel? (c) Find an equation of the plane normal to u and containing the point (1,, 3) (d) Find a parameterization for the line parallel to u and containing the point (1,, 3) 13a Find a vector parallel to the line of intersection of the two planes + 3z = 7 and 3 = z (b)find parametric equations for the line in part(a) 14 Find the following: ln + 3 arctan + ( ) (a) ( 3 ) ( ) (b) H H + T f if f( HT, ) = ( ) 3 5 H (c) + 15 Find an equation for the tangent plane to: (a) f (, = e at ( =, ) (1, ) (b) ( 1) 4( ) ( z 3) = at (3,3, 6) 16 A ball is thrown from ground level with initial speed v (m/sec) and at an angle of α with the v sin( α) horizontal It hits the ground at a distance sv (, α ) = where g = 9 8m/sec g (a) Find the differential ds (b) What does the sign of s α (0, π 3) tell ou? (c) Use the linearization of s about (0, π 3) to estimate the change in α that is needed to get approimatel the same distance if the initial speed changes to 19 m/sec 17 The depth of a lake at the point ( is, ) given b h (, ) = + 3feet A boat is at (-1,) (a) If the boat sails in the direction of the point (3,3), is the water getting deeper or shallower? (b) In which direction should the boat sail for the depth to remain constant? Give our answer as a vector (c) If the boat moves on the curve rt () = ( t 1) i+ ( t+ ) jfor t in minutes, at what rate is the depth changing when t =?

5 18 alculate the following: z (a) grad 1+ (b) curl ( + + z ) i ( + z) j + ( z) k cos sec 3 z div i + j + e k ( ) ( ) (c) ( ) ( ) ( ) 3 (d) The maimum rate of change of f( z,, ) = tan + zat ( π 4,3,1) (e) The directional derivative of f (,, z) = e + z at the point (1, 1, 0) in the direction of i 3 j + 4k z z (f) The potential function for G = i + j + e cos( e ) k 19 Which of the following vector fields are conservative? If the are conservative find the corresponding potential function (a) F(,, z) = e i + e j + (cos z) k (b) F(,, z) = i + j + ( z ) k (c) F(, ) = ( + i ) + ( j 3 (d) F(, = ( i + ( + 8 ) j 0 The contour plot for f( is, ) shown at the right Determine if each quantit is positive, negative, or zero (a) (1,1) f (b) f ( 1,1) f (d) f (, ) f (f) f u ( 1, 1) where 1 3 u = i _ j (c) (, ) (e) (1,1) 1 Let w (, ) = 3cos( π (a) Find (b) Find w u dw dt = (1, 1 ) t 1 if and w v (1, 1 ) if = e t and = ln t = u + v and v = u A steel bar with circular cross section of radius 6 cm and length 50 cm is being heated The radius and the length increases b 005 cm for each 1 degree centigrade rise in temperature What is the rate of change in the volume of the steel bar?

6 3a Find and classif all of the critical points for 3 3 f(, ) = + + (b) Find the quadratic Talor polnomial of f (, near the critical point, where the point is a local minimum of f (, 4 Let f (, = K + 4 (a) Verif that the point (0, 0) is a critical point (b) Determine the values of K, if an, for which (0, 0) can be classified as the following (i) a saddle point (ii) a local minimum (iii) a local maimum 5 Find an equation for each surface: (a) + = 8 in clindrical coordinates (b) = in clindrical coordinates (c) z = + in spherical coordinates (d) z = 10 in spherical coordinates 6 Determine (without calculation) whether the integrals are positive, negative, or zero Let D be the region inside the unit circle centered at the origin, T be the top half of the region, B be the bottom half of the region, L be the left half of the region, and R be the right half of the region (a) e da (b) cos T da (c) ( ) B + da (d) e da L R 7 Evaluate each of the integrals: 3 6 (a) cos(3 )sin( + 5) dd (b) 0 0 π π ρ sinφdρdφdθ (c) dd (d) z ( ) 3 e dzdd (e) da where R is shown below (f) R π 5 sinθ rdrdθ Hint: sketch the region first π 4 0 1

7 8 Set up integrals needed to find the following: (a) The volume between the sphere ρ = and the cone z = r (artesian, clindrical, and spherical) (b) The volume between z = 0 and z = + + (artesian and clindrical) (c) The volume of the solid in the first octant bounded from above b + z = 16 and = 1 (artesian in the order dddz and lindrical in the order rddrdθ ) (d) The volume of the tetrahedron under the portion of the plane shown at the right, bounded b the planes = 0, = 0, and 1 z = 0 (artesian) 6 9 A pile of dirt is approimatel in the shape of an inverted cone of height 4m with base radius of 4m The densit of the dirt is proportional to the distance from the ape of the pile of dirt Set up an integral for the mass of the dirt pile 30 Give parametric equations for the following curves: (a) A circle of radius 3 on the plane = 1centered at (,1, 0) oriented clockwise when viewed from the origin (b) A line perpendicular to z = 3+ 7 and through the point (1,, 3) (c) The curve = ( + ) 3 oriented from (,64) to (0,8) (d) The intersection of the surfaces z = + and z = 6 (e) The ellipse with major diameter a along the -ais and minor diameter b along the ais, centered at the origin 31 A child is sliding down a helical slide Her position at time t seconds after the start is given in feet r= 3cost i + 3sin t j+ (10 tk ) The ground is the -plane b ( ) ( ) (a) When is the child 6 feet from the ground? (b) How fast is the child traveling at seconds? (c) At time t = π seconds, the child leaves the slide tangent to the slide at that point What is the equation of the tangent line? 3 The surface of a hill is represented b z = 1 3, where and are measured horizontall A projectile is launched from the point (1,1, 7) and travels in a line perpendicular to the surface at that point (a) Find parametric equations for the path (b) Does the projectile pass through the point (1,1,8)?

8 33 Match the vector field to its sketch (a) i + j (b) i j (c) i + j (d) i (e) i + j (f) i + j (i) (ii) (iii) (iv) (v) (vi) 34 Given the plot of the vector field, F, list the following quantities in increasing order (i) F dr 1 (ii) F dr (iii) F dr 3 1 3

9 35 Evaluate F dr : (a) F = ( + z) i + zj + k is the line from (, 4, 4) to (1, 5, ) (b) F = i + z sin( z) j + sin( z) k is the curve from A(0,0,1) to B (3,1, ) as shown below (e) ( ) ( ) (c) F = i j + zk is the circle of radius 3 centered on the z-ais in the plane z = 4 oriented clockwise when viewed from above 3 (d) F= 4 i+ ( + j is the curve = sin( ) from (0,0) to ( π,0) 3 3 F = + sin( ) i + ln( + 1) j is the circle of radius 5 centered at (0,0) in the plane oriented counterclockwise (f) F = i + j is the parameterized path r ( t) = ((3 t) + sin( π t / 3)) i + ((3 t) + log( t + 1)) j, 0 t 3 36 alculate the flu of F through the surface, S, given below: (a) F= 3i+ 4 j+ ( z k ) S is a square of side on the plane z = oriented upward (b) F = 5i + zj k S is = + z for 0 8, oriented in the negative -direction 3 5 (c) F = i ( z j + ( + 7 z) k S is the closed clinder centered on the -ais with radius 3, length 5, oriented outward (d) F = i + j + zk S is the part of the surface z = 5 ( + ) above the disk of radius 5 centered at the origin, oriented upward 37 (a) Evaluate S grad z dr 3 ( ) where is the square of side centered at (1,1) in the -plane, oriented counterclockwise (b) Evaluate curl( i ( + z) j + zk ) da where S is the cube of side 4 centered at (,1, 3), oriented outward (c) Evaluate d + zd dz where is the circle of radius a in the z plane centered at the origin, c oriented clockwise when viewed from the positive -ais r 38 onsider the flu of the vector field H = for p 0 out of the sphere of radius 5 centered at p r the origin For what value of p is the flu a maimum? What is that maimum value?

10 39 Let S be the ellipsoid z = 64 and let 1 S be the sphere + + z = 1, both oriented r outward Let F = 3, r 0 r (a) Find div F (b) alculate the flu out of the sphere (c) Using the answers to part (a) and (b), find the flu out of the ellipsoid 40 The vector fields below have the form F = Fi 1 + Fj Assume F1 and F depend onl on and For each vector field, circle the best answers (a) (b) (c) P P P (i) F dr is positive negative zero (ii) divf( P) is positive negative zero (iii) curlf at P has positive k component negative k component zero k component (iv) F could be a gradient field could not be a gradient field Let F= (75 ) i j zk and let S 1, S 5, and S6 be spheres of radius 1, 5, and 6 respectivel, centered at the origin (a) Where is divf = 0? (b) Without computing the flu, order the flu out of the spheres from smallest to largest 4 In the region between the circles : + = 4 and : = in the -plane, the vector field F has curlf = 3k If and 1 are both oriented counterclockwise when viewed from above, find the value of F dr F dr 1 43 A particle travels along the heli given b = cost, = sin t, z = t, for 0 t 4π and is subject to a force F = i + j + 5k Find the total work done b the force for 0 t 4π

11 44 Determine if each of the following quantities is a vector (V), a scalar (S), or is not defined (ND) Assume that u and v are 3-D vectors, r = i + j + zk, S is a smooth surface, is a smooth curve, G is a differentiable 3-D vector field, and f is a differentiable scalar function of,, and z (a) ( curlg) r (b) div( G r ) (c) f u ( abc,, ) (d) ( divg) r (e) u v r (f) curl( fg ) (g) ( curlg) dr (h) ( divg) da (i) gradg S 45 True or False? (a) If all of the contours of a function gare (, ) parallel lines, then the function must be linear (b) If curlf is parallel to the -ais for all,, and z and if is a circle in the -plane, then the circulation of F around must be zero (c) If f is a differentiable function, then f u ( ab, ) f( ab, ) (d) If F is a divergence free vector field defined everwhere and S is a closed surface oriented inward, then F da = 0 S (e) If G is a curl free vector field defined everwhere and is a simple closed path, then G dr = 0 46 Use the portion of the contour diagram of f( shown, ) below to estimate the following: (a) gradf (15,78) (b) f u (15,76) in the direction i + j (c) A critical point of f(, ) (d) f dr where is the path from (15,76) to (4,76) (e) f (, ) da where R is the rectangle 9 15, R d d d 3 3 z d 47 alculate ( i + j + e k ) dr when viewed from above + = 16, z = where is 8, oriented counterclockwise

12 48 Suppose S is the surface obtained b taking the union of the upper hemisphere of radius centered at (0, 0, 4) and an open clinder of radius centered around the z-ais, with 0 z 4 If the d d d d orientation of S is awa from the origin, evaluate the integral ( curl( i j + k )) da S 49 Suppose W is a solid consisting of cube and three solid clinders of height 6 and radius (See picture below) The cube is centered at the origin and has side 5 One clinder is centered on the z-ais and the other two clinders are centered on the -ais Let S be the whole surface of W ecept for the circular disk of the clinder centered on the z-ais with outward orientation Find F d da d, given that F i z = j 3 zk S 50 onsider the line L 1 parameterized b r1 ( t) = ( t) i + (3 + t) j + (4 5t) k and the line L parameterized b r ( t) = (5 8t) i + ( 3 + t) j + (6 + t) k Decide whether the following statements are True or False (a) The lines L 1 and L are perpendicular to each other (b) The line L1is normal to the plane 8 z = 0 (c) The line L is parallel to the plane = 5z (d) The line L is parallel to the vector 16 i + j (e) The plane + 4 z 9 = 0 contains the line L