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Unformatted text preview: March 2001 PURDUE UNIVERSITY Study Guide for the Advanced Placement Exam in Multivariate Calculus
Students who pass this exam will receive 4 credit hours for MA 261 This study guide describes brieﬂy the topics to be mastered prior to attempting to obtain credit by examination for MA 261. The material covered is the calculus of several variables, and it can be studied from many textbooks, almost all of them entitled CALCULUS or CALCULUS WITH ANALYTIC GEOMETRY. The textbook currently used at Purdue is CALCULUS – Early Transcendentals, 4th edition, Stewart, Brooks/Cole. IMPORTANT: 1. Study all the material thoroughly. 2. Solve a large number of exercises. 3. When you feel prepared for the examination, solve the practice problems. 4. Come to the examination rested and conﬁdent. The subject matter of the calculus of several variables extends the student’s ability to analyze functions of one variable to functions of two or more variables. Graphs of functions of two variables or of equations involving three variables may be thought of as surfaces in three dimensions. Tangents, normals, and tangent planes to these surfaces are part of the subject matter of the calculus of several variables. The concept of volume is deﬁned for threedimensional solids. These new concepts require the introduction of partial derivatives and multiple integrals. Most of the problems to be solved require the repeated application of ideas and techniques from the calculus of one variable. Accordingly, a good grasp of the notions of diﬀerentiation and integration for functions of one variable is a necessary prerequisite for the study of the calculus of several variables. Several of the concepts from plane analytic geometry are also generalized in the course, leading to a brief study of three dimensional analytic geometry, including such topics as planes and the quadric surfaces (whose cross sections are conics), and three dimensional coordinate systems, including rectangular, cylindrical and spherical coordinates, and the relationships among these. The topics to be studied prior to attempting the attached practice problems are listed below 1. Analytic Geometry of Three Dimensions Angle between two vectors, scalar product, cross product, planes, lines, surfaces, curves in 3 dimensional space. 2. Partial Diﬀerentiation Functions of several variables, partial derivatives, diﬀerential of a function of several variables, partial derivatives of higher order, chain rule, extreme value problems, directional derivatives, gradient, implicit functions. 3. Multiple Integrals Double integrals, iterated integrals in rectangular and polar coordinates, applications, surface integrals, triple integrals in rectangular, cylindrical and spherical coordinates. 4. Line and surface integrals, independence of path, Green’s theorem, divergence theorem. 1 A set of 52 practice problems is attached. Naturally, this set does not cover all points of the course, but if you have no diﬃculty with it, you will probably do well in the examination. The correct answers are given on the last page. The examination consists of 25 multiple choice questions and, like the set of practice problems, is almost entirely manipulative. This does not mean that you do not need a thorough understanding of the concepts, but rather that you are not asked to quote any deﬁnitions or theorems or oﬀer proofs of any theorems. You are expected to perform the manipulations required with a high degree of understanding and accuracy. Two hours are allowed for this examination. Calculators are not needed for the examination. If you wish you may bring a nongraphing, nonprogrammable calculator to the examination. Books or notes are not allowed and formulas will not be provided. You should know all the formulas that are needed for solving the practice problems. SPECIAL NOTE: A word of advice concerning the taking of the actual examination for credit. No one does well on an examination when he or she is excessively fatigued. Therefore you are urged to provide yourself an adequate rest period before taking the actual examination. If your trip to the campus necessitates travel into the late hours of the night or an extremely early departure from your home, you should allow for a one night rest in the Lafayette area before taking the examination. Many students who are unsuccessful with the examination tell us that failing to take the above precautions contributed strongly to their inability to complete their examination successfully. Most such students ﬁnd that their ﬁrst year was somewhat less rewarding than it might have been because of the time spent retracing materials studied in high school. Please consult your advanced credit schedule for the actual time and place of the examination. It is usually given both morning and afternoon. 2 MA 261 PRACTICE PROBLEMS 1. If the line has symmetric equations
x −1 2 = ﬁnd a vector equation for the line that contains the point (2, 1, −3) and is parallel to . A. = (1 + 2t) − 3t + (−2 + 7t) r i j k C. = (2 + 2t) + (1 − 3t) + (−3 + 7t) r i j k E. = (2 + t) + + (7 − 3t) r ij k B. = (2 + t) − 3 + (7 − 2t) r i j k D. = (2 + 2t) + (−3 + t) + (7 − 3t) r i j k y −3 = z +2 7, 2. Find parametric equations of the line containing the points (1, −1, 0) and (−2, 3, 5). A. C. x = 1 − 2t, y = −1 + 3t, z = 5t x = 1 − 3t, y = −1 + 4t, z = 5t D. x = −2t, y = 3t, z = 5t B. x = t, y = −t, z = 0 E. x = −1 + t, y = 2 − t, z = 5 3. Find an equation of the plane that contains the point (1, −1, −1) and has normal vector 1 2 i + 2j + 3k . A. D. x−y−z+
9 2 =0 B. x + 4y + 6z + 9 = 0 E.
1 2 C. x −1
1 2 = y +1 2 = z +1 3 x−y−z =0 x + 2y + 3z = 1 4. Find an equation of the plane that contains the points (1, 0, −1), (−5, 3, 2), and (2, −1, 4). A. 6x − 11y + z = 5 D. = 18 − 33 + 3 r i j k B. 6x + 11y + z = 5 E. x − 6y − 11z = 12 C. 11x − 6y + z = 0 k j 5. Find parametric equations of the line tangent to the curve (t) = t + t2 + t3 at the point (2, 4, 8) r i A. x = 2 + t, y = 4 + 4t, z = 8 + 12t D. B. x = 1 + 2t, y = 4 + 4t, z = 12 + 8t E. x = 2 + t, y = 4 + 2t, z = 8 + 3t C. x = 2t, y = 4t, z = 8t x = t, y = 4t, z = 12t 6. The position function of an object is k (t) = cos t + 3 sin t − t2 r i j Find the velocity, acceleration, and speed of the object when t = π . Velocity A. B. C. D. E. Acceleration Speed √ 1 + π4 k −3 −2π j k −−π 2 i √ −3 +2π ij k −−2 ik 10 + 4π 2 √ 3 −2π k j −−2k i 9 + 4π 2 √ −2 −3 −2π j k ik 9 + 4π 2 √ − 2 i k −3 − 2π j k 5 3 7. A smooth parametrization of the semicircle which passes through the points (1, 0, 5), (0, 1, 5) and (−1, 0, 5) is A. (t) = sin t + cos t + 5 0 ≤ t ≤ π r i j k, C. (t) = cos t + sin t + 5 π ≤ t ≤ 3π r i j k,
2 π 2≤ E. (t) = sin t + cos t + 5 r j k, t≤ 2 3π 2 B. (t) = cos t + sin t + 5 0 ≤ t ≤ π r i j k, D. (t) = cos t + sin t + 5 0 ≤ t ≤ r i j k, π 2 3 3 8. The length of the curve (t) = 2 (1 + t) 2 + 2 (1 − t) 2 + t , −1 ≤ t ≤ 1 is r i3 j k 3 √ √ √ √ 3 B. 2 C. 1 3 D. 2 3 A. 2 E. √ 2 9. The level curves of the function f (x, y ) = A. circles B. lines C. parabolas 1 − x2 − 2y 2 are D. hyperbolas E. ellipses 10. The level surface of the function f (x, y, z ) = z − x2 − y 2 that passes through the point (1, 2, −3) intersects the (x, z )plane (y = 0) along the curve A. z = x2 + 8 B. z = x2 − 8 C. z = x2 + 5 D. z = −x2 − 8 E. does not intersect the (x, z )plane 11. Match the graphs of the equations with their names: (1) x2 + y 2 + z 2 = 4 (2) x + z = 4 (3) x + y = z (4) x2 + y 2 = z (5) x + 2y + 3z = 1 A. 1b, 2c, 3d, 4a, 5e D. 1b, 2d, 3a, 4c, 5e
2 2 2 2 2 2 2 2 (a) paraboloid (b) sphere (c) cylinder (d) double cone (e) ellipsoid B. 1b, 2c, 3a, 4d, 5e E. 1d, 2a, 3b, 4e, 5c C. 1e, 2c, 3d, 4a, 5b 12. Suppose that w = u2 /v where u = g1 (t) and v = g2 (t) are diﬀerentiable functions of t. If g1 (1) = 3, g2 (1) = 2, g1 (1) = 5 and g2 (1) = −4, ﬁnd dw when t = 1. dt A. 6 B. 33/2 C. −24
∂w ∂r . D. 33 E. 24 13. If w = euv and u = r + s, v = rs, ﬁnd A. e(r+s)rs (2rs + r2 ) D. e(r+s)rs (1 + s) B. e(r+s)rs (2rs + s2 ) E. e(r+s)rs (r + s2 ). C. e(r+s)rs (2rs + r2 ) 4 14. If f (x, y ) = cos(xy ), A. −xy cos(xy ) ∂2 f ∂ x∂ y = B. −xy cos(xy ) − sin(xy ) E. − cos(xy )
∂z ∂x . C. − sin(xy ) D. xy cos(xy ) + sin(xy ) 15. Assuming that the equation xy 2 + 3z = cos(z 2 ) deﬁnes z implicitly as a function of x and y , ﬁnd A.
y2 3−sin(z 2 ) B. −y 2 3+sin(z 2 ) C. y2 3+2z sin(z 2 ) D. −y 2 3+2z sin(z 2 ) E. −y 2 3−2z sin(z 2 ) 16. If f (x, y ) = xy 2 , then ∇f (2, 3) = A. 12 + 9 i j B. 18 + 18 i j C. 9 + 12 i j D. 21 E. √ 2. 17. Find the directional derivative of f (x, y ) = 5 − 4x2 − 3y at (x, y ) towards the origin A. −8x − 3
y √ B. −8x2 −32 x +y
2 C. − √ 8x−3 64x2 +9 D. 8x2 + 3y 8x E. √ 2+3y2 . x +y 2 u 18. For the function f (x, y ) = x2 y , ﬁnd a unit vector for which the directional derivative D f (2, 3) is zero. u A. + 3 i j B.
i+3 √j 10 C. − 3 i j D. i−3 √j 10 E. 3− i √ j. 10 19. Find a vector pointing in the direction in which f (x, y, z ) = 3xy − 9xz 2 + y increases most rapidly at the point (1, 1, 0). A. 3 + 4 i j B. + ij C. 4 − 3 i j D. 2 + ik E. − + . ij 20. Find a vector that is normal to the graph of the equation 2 cos(π xy ) = 1 at the point ( 1 , 2). 6 √ ij C. 12 + ij D. j E. 12 − . ij A. 6 + ij B. − 3 − 21. Find an equation of the tangent plane to the surface x2 + 2y 2 + 3z 2 = 6 at the point (1, 1, −1). D. 2x + 4y − 6z = 0 A. −x + 2y + 3z = 2 E. x + 2y − 3z = 6. B. 2x + 4y − 6z = 6 C. x − 2y + 3z = −4 22. Find an equation of the plane tangent to the graph of f (x, y ) = π + sin(π x2 + 2y ) when (x, y ) = (2, π ). A. 4π x + 2y − z = 9π B. 4x + 2π y − z = 10π C. 4π x + 2π y + z = 10π D. 4x + 2π y − z = 9π E. 4π x + 2y + z = 9π . 5 23. The diﬀerential df of the function f (x, y, z ) = xey A. df = xey B. df = xe C. df = e E. df = e D. df = e
2 2 −z 2 is −z 2 dx + xey dx dy dz 2 −z 2
2 dy + xey 2 −z 2
2 dz
−z 2 y 2 −z 2 y 2 −z 2 y 2 −z 2 dx − 2xyey dx + 2xye −z 2 dy + 2xzey dy − 2xze dz dz y 2 −z 2 y 2 −z 2 y 2 −z 2 (1 + 2xy − 2xz ) 24. The function f (x, y ) = 2x3 − 6xy − 3y 2 has A. a relative minimum and a saddle point C. a relative minimum and a relative maximum E. two relative minima. 25. Consider the problem of ﬁnding the minimum value of the function f (x, y ) = 4x2 + y 2 on the curve xy = 1. In using the method of Lagrange multipliers, the value of λ (even though it is not needed) will be √ 1 2 D. √2 E. 4. A. 2 B. −2 C. 26. Evaluate the iterated integral A. − 8 9 B. 2 3x
1 0 1 x B. a relative maximum and a saddle point D. two saddle points dydx. C. ln 3 D. 0 E. ln 2. 27. Consider the double integral, R f (x, y )dA, where R is the portion of the disk x2 + y 2 ≤ 1, in the upper halfplane, y ≥ 0. Express the integral as an iterated integral. A. C. E.
−1 − 1 √1−x2 f (x, y )dydx −1 0 1 √1−x2 f (x, y )dydx. 00 1 √1−x2
√ 1−x2 f (x, y )dydx B. D. 0 √1−x2
−1 0 1 √1−x2 √ 0 − 1−x2 f (x, y )dydx f (x, y )dydx 28. Find a and b for the correct interchange of order of integration: 4b 2 2x 0 x2 f (x, y )dydx = 0 a f (x, y )dxdy . A. a = y 2 , b = 2y √ D. a = y, b = y 2 B. a = y , b = 2 √ y E. cannot be done without explicit knowledge of f (x, y ). C. a = y , b = y 2 29. Evaluate the double integral R ydA, where R is the region of the (x, y )plane inside the triangle with vertices (0, 0), (2, 0) and (2, 1). A. 2 B.
8 3 C. 2 3 D. 1 E. 1 3. 30. The volume of the solid region in the ﬁrst octant bounded above by the parabolic sheet z = 1 − x2 , below by the xy plane, and on the sides by the planes y = 0 and y = x is given by the double integral A. D. 1x
00 10 0 x (1 (1 − x2 )dydx − x2 )dydx E. B. 1 1−x
0 x
2 dydx. 1 1−x2
0 0 x dydx C. 1 x −1 −x (1 − x2 )dydx 6 31. The area of one leaf of the threeleaved rose bounded by the graph of r = 5 sin 3θ is A.
5π 6 B. 25π 12 C. 25π 6 D. 5π 3 E. 25π 3. 32. Find the area of the portion of the plane x + 3y + 2z = 6 that lies in the ﬁrst octant. √ √ √ √ B. 6 7 C. 6 14 D. 3 14 A. 3 11 √ E. 6 11. 33. A solid region in the ﬁrst octant is bounded by the surfaces z = y 2 , y = x, y = 0, z = 0 and x = 4. The volume of the region is A. 64 B.
64 3 C. 32 3 D. 32 E. 16 3. 34. An object occupies the region bounded above by the sphere x2 + y 2 + z 2 = 32 and below by the upper nappe of the cone z 2 = x2 + y 2 . The mass density at any point of the object is equal to its distance from the xy plane. Set up a triple integral in rectangular coordinates for the total mass m of the object. 4 √16−x2 √32−x2 −y2 4 √16−x2 √32−x2 −y2 z dz dy dx B. −4 −√16−x2 √ z dz dy dx A. −4 −√16−x2 √ 2 √4−x2 √32−x2 −y2 √ √ z dz dy dx −2 − 4−x2 − x2 +y 2 √ 4 √16−x2 32−x2 −y2 xy dz dy dx. E. −4 −√16−x2 √ 2 2 C.
x +y − x 2 +y 2 x 2 +y 2 D. 4 √16−x2 √32−x2 −y2 √2 2 z dz dy dx 00
x +y 35. Do problem 34 in spherical coordinates. A. C. E. ρ3 cos ϕ sin ϕ dρ dϕ dθ 0 2π π √32 3 2 4 ρ sin ϕ dρ dϕ dθ 0 0 0 √ 2π π 32 4 ρ cos ϕ dρ dϕ dθ. 0 0 0
0 0 2π π 4 √32 B. D. 2π
0 2π
0 0 π 4 0 π 2 0 √ 0 √32
32 ρ cos ϕ sin ϕ dρ dϕ dθ ρ3 cos ϕ sin ϕ dρ dϕ dθ 36. The double integral A. D. π1
0 0 r9 sin2 θ dr dθ r8 sin θ dr dθ 1 √1−x2
0 0 y 2 (x2 + y 2 )3 dydx when converted to polar coordinates becomes B. E.
π 2 37. Which of the triple integrals converts 2 √4−x2 √ 2 √ dz dy dx −2 − 4−x2 x 2 +y 2 π 2 0 1
0 0 1
0 r9 sin2 θ dr dθ. π 2 0 1
0 r8 sin2 θ dr dθ C. π1
0 0 r8 sin θ dr dθ D. from rectangular to cylindrical coordinates? 2π 2 2 π22 B. 0 0 r r dz dr dθ A. 0 0 r r dz dr dθ π22
0 0 r C. r dz dr dθ E. 38. If D the solid region above xy plane that is between z = is the x2 + y 2 + z 2 dV = z = 1 − x2 − y 2 , then D A.
14π 3 2π 2 0 2 2
−2 r r dz dr dθ. 2π 2 2
0 −2 r r dz dr dθ 4 − x2 − y 2 and D. 8π E. 15π . B. 16π 3 C. 15π 2 7 39. Determine which of the vector ﬁelds below are conservative, i. e. F = grad f for some function f . i j 1. F (x, y ) = (xy 2 + x) + (x2 y − y 2 ) . y x 2. F (x, y ) = i + j .
y x j k i 3. F (x, y, z ) = yez + (xez + ey ) + (xy + 1)ez . A. 1 and 2 B. 1 and 3 C. 2 and 3 D. 1 only E. all three 40. Let F be any vector ﬁeld whose components have continuous partial derivatives up to second order, let f be any real valued function with continuous partial derivatives up to second order, and let ∇ = ∂ + ∂ + ∂ . Find the incorrect statement. i ∂x j ∂y k ∂z A. curl(grad f ) = 0 D. curl F = ∇ × F B. div(curl F ) = 0 E. div F = ∇ · F C. grad(div F ) = 0 41. A wire lies on the xy plane along the curve y = x2 , 0 ≤ x ≤ 2. The mass density (per unit length) at any point (x, y ) of the wire is equal to x. The mass of the wire is √ √ √ B. (17 17 − 1)/8 C. 17 17 − 1 A. (17 17 − 1)/12 √ √ E. ( 17 − 1)/12 D. ( 17 − 1)/3 r i j 42. Evaluate C F · d where F (x, y ) = y + x2 and C is composed of the line segments from (0, 0) to (1, 0) and from (1, 0) to (1, 2). A. 0 B.
2 3 C. 5 6 D. 2 E. 3 43. Evaluate the line integral x dx + y dy + xy dz C where C is parametrized by (t) = cos t + sin t + cos t for − π ≤ t ≤ 0. r i j k 2 A. 1 B. −1 C.
1 3 1 D. − 3 E. 0 44. Are the following statements true or false? 1. The line integral C (x3 + 2xy )dx + (x2 − y 2 )dy is independent of path in the xy plane. 2. C (x3 + 2xy )dx + (x2 − y 2 )dy = 0 for every closed oriented curve C in the xy plane. 3. There is a function f (x, y ) deﬁned in the xy plane, such that i j grad f (x, y ) = (x3 + 2xy ) + (x2 − y 2 ) . B. 1 and 2 are false, 3 is true E. all three are true A. all three are false D. 1 is true, 2 and 3 are false C. 1 and 2 are true, 3 is false √ 45. Evaluate C y 2 dx + 6xy dy where C is the boundary curve of the region bounded by y = x, y = 0 and x = 4, in the counterclockwise direction. A. 0 B. 4 C. 8 D. 16 E. 32 8 46. If C goes along the xaxis from (0, 0) to (1, 0), then along y = along the y axis, then C xy dy = 1 √1−x2 y dy dx A. − 0 0 1 √1−x2 x dy dx D. 0 0 B. E. 0 1 √1−x2
0 0 √ 1 − x2 to (0, 1), and then back to (0, 0) C. − 1 √1−x2
0 0 y dy dx x dy dx r i j 47. Evaluate C F · d, if F (x, y ) = (xy 2 − 1) + (x2 y − x) and C is the circle of radius 1 centered at (1, 2) and oriented counterclockwise. A. 2 B. π C. 0 D. −π E. −2 48. Green’s theorem yields the following formula for the area of a simple region R in terms of a line integral over the boundary C of R, oriented counterclockwise. Area of R = R dA = B. C y dx C. C x dx D. 1 C y dx − x dy E. − x dy A. − C y dx 2 49. Evaluate the surface integral √ A. 8 6 B.
8 3 √ 6 Σ x dS where Σ is the part of the plane 2x + y + z = 4 in the ﬁrst octant. C.
8 3 √ 14 D. √ 14 3 E. √ 10 3 50. If Σ is the part of the paraboloid z = x2 + y 2 with z ≤ 4, is the unit normal vector on Σ directed n · dS = (x, y, z ) = x + y + z , then Fn upward, and F i j k Σ A. 0 B. 8π C. 4π D. −4π E. −8π 51. If F (x, y, z ) = cos z + sin z + xy , Σ is the complete boundary of the rectangular solid region bounded i j k n by the planes x = 0, x = 1, y = 0, y = 1, z = 0 and z = π , and is the outward unit normal on Σ, then 2 n F · dS = Σ A. 0 B.
1 2 C. 1 D. π 2 E. 2 n 52. If F (x, y, z ) = x + y + z , Σ is the unit sphere x2 + y 2 + z 2 = 1 and is the outward unit normal on i j k n Σ, then Σ F · dS = A. −4π B.
2π 3 C. 0 D. 4π 3 E. 4π 9 ANSWERS 1–C, 2–A, 3–B, 4–B, 5–A, 6–D, 7–B, 8–D, 9–E, 10–B, 11–A, 12–E, 13–B, 14–B, 15–D, 16–C, 17–E 18–D, 19–A, 20–C, 21–E, 22–A, 23–D, 24–B, 25–E, 26–B, 27–C, 28–B, 29–E, 30–A, 31–B, 32–D, 33–B, 34–B, 35–A, 36–E, 37–B, 38–C, 39–B, 40–C, 41–A, 42–D, 43–D, 44–E, 45–D, 46–B, 47–D, 48–A, 49–B, 50–E, 51–A, 52–E 10 ...
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This note was uploaded on 03/20/2011 for the course MA 261 taught by Professor Stefanov during the Spring '08 term at Purdue University.
 Spring '08
 Stefanov
 Calculus

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