XM521: Multiple Choice Exercises

Transcription

1 XM521: Multiple Choice Exercises Lesson 1 (1) Determine which of the following choices best describes the xz-plane. (a) All points of the form (0,y,0) (b) All points of the form (x,0,z) (c) The intersection of the x-axis and the z-axis (d) All points of the form (c,y,c) where c is a given constant (2) Consider a box whose edges are all 3 inches in length. What is the distance from one corner of the box to the corner farthest away? (a) 27 inches (b) 2 3 inches (c) 9 inches (d) 3 3 inches (3) Find the distance between points ( 2,0,3) and (4, 1,2). (a) 38 (b) 6 (c) 6 (d) 30 (4) The distance between a point P and a point A = (3,2, 1) is half the distance between A and B = (6,0,5). What is the maximum possible distance between P and B? (a) 3.5 (b) 10.5 (c) 7 (d) 11 Lesson 2 (1) Find the center and radius of the sphere represented by the following equation: (a) Center (1, 1, 2) and radius 2 (b) Center ( 2,6,1) and radius 3 x 2 +y 2 +z 2 +2x 6y +6 = 0. 1

2 (c) Center (1, 2,0) and radius 1 (d) Center ( 1,3,0) and radius 2 (2) Determine the graph of the following equation: x 2 +y 2 +z 2 4x 8y +2z +21 = 0. (a) Point (b) Line (c) Sphere (d) (This symbol denotes the empty set, the set with no elements.) (3) Which of the following equations has no graph in 3-space? (a) z = tan(x) (b) y 2 +x 2 = 6 (c) sin(x)+y 2 = 2 (d) x 3 z 2 = 5 (4) Consider the sphere represented by the equation: x 2 +y 2 +z 2 2x+4y 8z +20 = 0. What is the shortest distance between the point (1,0,4) and this sphere? (a) 0 (b) 1 (c) 2 (d) 3 Lesson 3 (1) Determine which of the following is not a vector quantity: (a) The ocean current at a particular point in the ocean (b) The force of wind acting on a leaf (c) The speed of a plane taking off (d) The acceleration of a plane taking off (2) If v 0, which of the following must be parallel to v? (a) v+w for any vector w (b) kv for a real nonzero scalar k 2

3 (c) Any vector with the same initial point as v (d) None of the above (3) Which of the following is not equivalent to the statement v is the zero vector? (a) v+v = v (b) v has length zero (c) The initial point of v coincides with the terminal point of v. (d) v v 0 (4) If u is parallel to the x-axes and v is parallel to the z-axis, which of the following is necessarily parallel to u+v? Lesson 4 (a) the y-axis (b) the line represented by x = z (c) u v (d) None of the above (1) Find the components of the vector v with initial point ( 2, 5, 1) and terminal point (3,2,4). (a) 5, 3,3 (b) 1,7,5 (c) 5,3, 3 (d) 1,3,3 (2) Let v = 3, 1,2 and w = 2,4,0. Find the value of 2v w. (a) 4,2,4 (b) 7,9,2 (c) 8, 6,4 (d) 6, 2,4 (3) Which of the following could be the first assertion of a correct proof that vector addition is commutative? (a) u+v = u 1 +v 1,u 2 +v 2,u 3 +v 3. (b) Let u = u 1,u 2,u 3 and v = v 1,v 2,v 3 be vectors in 3-space. (c) u 1 +v 1,u 2 +v 2,u 3 +v 3 = v 1 +u 1,v 2 +u 2,v 3 +u 3. 3

4 (d) u+v = v+u. (4) Let u = 1,0,2 and v = 3,2,5. Find x such that 3v+x = 2u x. (a) x = 11 2,2,3 Lesson 5 (b) x = 11 2, 3, 11 2 (c) x = 2,2,3 (d) x = 11, 6, 11 (1) What is 2i+j 4k? (a) 19 (b) 5 (c) 21 (d) 21 (2) Which of the following is a unit vector with the same direction as v = 0i j+2k? (a) 0, 1 3, 2 3 (b) 1 2, 1 2,1 (c) 0, 1 2 5, 5 1 (d) 0, 3, 3 2 (3) Which of the following vectors has length 3 and the same direction as the unit vector 2 3, 5 3? (a) 6, 3 5 (b) (c) (d) 2, , , 5 (4) Let u 1,u 2 = 3, u 1,u 3 = 2, u 2,u 3 = 7. Find u 1,u 2,u 3. (a) 31 (b) 7 (c) 62 4

5 (d) 9 Lesson 6 (1) Let F 1 and F 2 be vectors forming a right angle with F 1 = 3 and F 2 = 1. Find F 1 +F 2. (a) 2+ 3 (b) 2 (c) 4 2 (d) 2 6 (2) Let F 1 = N and F 2 = 2 N. Let π/3 be the angle between F 1 and F 2. What is the angle between F 1 and F 1 +F 2? (a) π/6 (b) π/4 (c) π/3 (d) π/2 (3) Supposethree forces F 1, F 2, andf 3 act on a particle in 3-space, wheref 1 has magnitude 5 and points in the direction of 3/5,4/5,0, F 2 has magnitude 12 and points in the direction of 3/3, 3/3, 3/3, and F 3 has magnitude 5 and points in the direction of 0,1,0. If these are the only forces acting on the particle, what vector points in the same direction in which the particle moves? Lesson 7 (a) 1 2, 1 2, 1 (b) 0,0, (c) , , 3 3 (d) 5,11, 2 (1) Let u = 2i 3j+k and v = i+j 2k. Find u v. (a) 8 (b) 1 (c) 3 (d) 1 (2) Let u = 1,0 and v = (a) π/3 1, 3. What is the angle between u and v? 5

6 (b) π/2 (i.e. they are orthogonal) (c) π/6 (d) π/4 (3) The following is a bad proof of the theorem that u v = v u for vectors u and v in 2-space. Let u = u 1,u 2 and v = v 1,v 2. So which implies u v = v u u 1,u 2 v 1,v 2 = v 1,v 2 u 1,u 2, u 1 v 1 +u 2 v 2 = v 1 u 1 +v 2 u 2. By commutativity of ordinary multiplication, So the above equality is true. u 1 v 1 = v 1 u 1 and u 2 v 2 = v 2 u 2. Which tip in the Guidelines for Writing Proofs does the above argument violate? (a) Avoid leaving large gaps in your proofs. (b) Never assume what you are trying to prove. (c) Variables that appear in a proof either must be given in the statement of the theorem or must be introduced in the proof before they are used. (d) Never write nonsense or statements that are obviously false. (4) Let u = 2,6, v = 1,0 and w = 3, 3. Find (u w)v. (a) 6 (b) 4 3 (c) 8 10 (d) 12 Lesson 8 (1) Let v = 2,0, 2 and suppose that v has its initial point at the origin. Find the angle that v makes with the x-axis. (a) π/6 (b) π/2 6

7 (c) π/4 (d) π/4 (2) Which of the following is a direction cosine of v = 2i+3j k? (a) 1 14 (b) 3 2 (c) 2 14 (d) (3) Let v = 2, 1,1. Let α be the angle between v and the x-axis, β the angle between v andthey-axis, andγ theanglebetweenvandthez-axis. Computecos 2 α+cos 2 β+cos 2 γ. (a) 1 2 (b) 3 (c) 1 (d) 0 (4) Consider a diagonal D of a box with dimensions 47 by 7 by 10. Which of the following is an angle that D makes with one of the box s edges? (a) π/3 (b) π/4 (c) π/6 (d) π/8 (5) Let a be any number such that 1 < a < 0. Which describes the set of endpoints of vectors whose direction cosine in direction i equals a? (a) a line (b) a double-cone opening in the x directions (c) a (single) cone opening in the negative x direction (d) the set may be empty, depending on the value of a (6) Let a and b be any numbers such that 0 < a,b < 1. Which describes the set of endpoints of vectors whose direction cosine in direction i equals a and in direction j equals b? (a) a line (b) a half-line (c) two half-lines 7

8 Lesson 9 (d) the set may be empty, depending on the values of a and b (1) Suppose that v is a vector in 2-space, v = 2, e 1 and e 2 are orthogonal unit vectors, and the angle between v and e 1 is π/6. Which of the following might be an orthogonal decomposition of v? (a) e 1 + 3e 2 (b) 3e 1 +e 2 (c) 2(e 1 +e 2 ) (d) e 1 + 3e 2 (2) Find the orthogonal projection of v = 2i j+k on b = i+j (a) 2i+2j (b) i+ 1 2 j (c) 1 2 (i+j) (d) k (3) A box is moved along the floor using a rope that applies a force of 7 lb at an angle of π/6. How much work (in foot-pounds) is done if the box is moved 10 feet? (a) 70 (b) 35 (c) 50 2 (d) 35 3 Lesson 10 (1) Let u = 4i j+3k and v = ai 2j+bk. If u v = 5i+2j 6k, what is a+b? (a) 3 (b) 2 (c) 1 (d) 0 (2) Which of the following is not true? (a) i k = j (b) i (j+k) = (i k)+(i j) (c) (i+j) (j+k) = i+j+k 8

9 (d) j k = i (3) If u, v and w are vectors in 3-space, which of the following must be equal to u (v w)? Lesson 11 (a) (u v) w (b) (w v) u (c) w (v u) (d) v (u w) (1) Let u and v be nonzero vectors in 3-space. Which of the following is nonzero? (a) v (u v) (b) u v where u and v are orthogonal (c) u u (d) ( v) (v u) (2) Let u = 0,2,0 and v = (a) 20 (b) 90 (c) 45 (d) 60 2,1, 1. What is the angle between u and v? (3) Let u = 2, 1,0 and v = 1,3, 1 be adjacent sides of a parallelogram P. Find the area of P. (a) 3 6 (b) 30 (c) 7 (d) 4 2 Lesson 12 (1) Let u = 1,0, 2, v = 2,3,1, and w = 1,1,3. Calculate the scalar triple product u (v w). (a) 0 (b) 3 (c) 8 9

10 (d) 2 (2) Let u = 2,1,1, v = 0, 2,1, and w = 1, 3,1 be adjacent edges of a parallelepiped P. Find the volume of P. (a) 6 (b) 2 (c) 5 (d) 0 (3) If u, v, and w are adjacent edges of a parallelepiped P, which of the following does not give the volume of P? Lesson 13 (a) None of the below (b) w (v u) (c) v (u w) (d) u (w v) (1) Consider the line passing through ( 2,1) and parallel to the vector 3,2. Which of the following is a point on the line? (a) ( 1,2) (b) ( 5, 2) (c) (0,3) (d) (4,5) (2) Which of the following lines is not parallel to x,y,z = 2 t, 6+3t,2+t? (a) x,y,z = 4+t,1 3t, 3+t (b) x,y,z = 1,7,2 +t 1,3,1 (c) x,y,z = 2+t,1 3t, 3 t (d) x,y,z = 1,0, 3 +t 2,6,2 (3) Where does the line x,y = 2 t,3+t intersect the line y = 2x? (a) ( 5 2,5) (b) ( 1, 2) (c) ( 5 3, 10 3 ) (d) (4,1) 10

11 Lesson 14 (1) Which of the following equations represents a line in 3-space that passes through the points (6, 1,2) and (0,3,5)? (a) x,y,z = 6, 1,2 +t 0,3,5 (b) x,y,z = 6t,3 4t,5 3t (c) x,y,z = 6, 3, 1 +t 1,1,1 (d) x,y,z = 6 3t, 1+2t,2+3t (2) Which of the following pairs of lines are skew? (a) x 1,y 1,z 1 = 3t,2 t,4+t and x 2,y 2,z 2 = 6 3t, 3+t,5 t (b) x 1,y 1,z 1 = 2 t, 1+2t,3 2t and x 2,y 2,z 2 = 1, 2 3t, t (c) x 1,y 1,z 1 = 1+t, 1 2t,3 t and x 2,y 2,z 2 = 2+2t, 4t,1 2t (d) x 1,y 1,z 1 = 2+4t,1 2t, 1+t and x 2,y 2,z 2 = 3 2t, 2+4t,1 3t (3) The following is a bad proof of the claim that lines L 1 : x,y,z = 2+t,3 t,2t and L 2 : x,y,z = 2t,4 2t, 1+4t are parallel in 3-space. L 1 : x,y,z = 2+t,3 t,2t 2,3,0 +t 1, 1,2 1, 1,2 L 2 : x,y,z = 2t,4 2t, 1+4t 0,4, 1 +t 2, 2,4 2, 2,4 2 1, 1,2 Which of the following tips in the Guidelines for Writing Proofs does it violate? (a) Never write statements that are obviously false. (b) Never assume what you are trying to prove. (c) Write complete sentences. (d) Minimize ambiguity by making sure that quantifier phrases precede the expressions to which they apply. Lesson 15 (1) Which of the following is an equation of a plane passing through the point ( 2,1,5) and perpendicular to n = 1,3,1? 11

12 (a) 2x+y +5z = 6 (b) 3x+4y +6z = 0 (c) x+3y +z = 10 (d) x+2y 4z = 7 (2) Find an equation of the plane through the points (3,2, 2), (1,0,2), and (2, 1,1). (a) (x 1) y +(z 2) = 0 (b) 3(x 3)+(y 2)+2(z +2) = 0 (c) 6(x 1) 2y +2(z 1) = 0 (d) 4(x 2)+2(y +1) 3(z 1) = 0 (3) What can you say about the line x,y,z = 2+2t,4 t, 1+t and the plane 4x 2y +2z = 6? (a) The line is perpendicular to the plane. (b) Every point of the line is also on the plane. (c) The line and the plane are parallel, but do not share any points. (d) None of the above. (4) Let l be the line defined by the parametric equations x(t) = 2t+1, y(t) = 3t+2, z(t) = 4t+3. Which describes all the planes containing l? (a) the plane 3(x 1)+2(y 2) 3(z 3) = 0 (b) all planes ax+by +cz +d = 0 satisfying 2a+3b+4c = 0 (c) all planes ax+by +cz +d = 0 satisfying 2a+3b+4c = 0 and a+2b+3c+d = 0 (d) all planes 2x+3y +4z +d = 0 (5) Which of the following is true? (a) Given any line l in 3-space and a point P not on l, there can be more than one plane containing both l and P. (b) Given any three distinct points in 3 space that do not lie on a common line, there is a plane that contains the triangle they form. (c) If two lines in 3-space do not intersect, no plane contains them both. (d) If a line and plane are parallel, then any vector perpendicular to the line is perpendicular to the plane. Lesson 16 12

13 (1) Find the acute angle of intersection between the following planes: 2 2x+2y +2z = 6 and 2x y +z = 8. (a) π/8 (b) π/6 (c) π/4 (d) π/3 (2) Find the distance between the point (2,0,2) and the plane 4x+4y 2z = 6. (a) 1 3 (b) 1 2 (c) 1 (d) 0 (3) Which of the following planes passes through the points (4, 1,2) and ( 2,3,1) and is parallel to the plane x+y 3z = 4? Lesson 17 (a) x+y 3z +3 = 0 (b) x+y 3z +2 = 0 (c) x+y 3z +5 = 0 (d) None of the above (1) Find an equation for the trace of the surface x 2 +4y 2 z 2 = 4 in the plane x = 2. (a) 4x 2 +4y 2 z 2 = 4 (b) x 2 +4y 2 z 2 = 0 (c) 4y 2 = z 2 (d) 4y 2 z 2 = 8 (2) Which of the following describes the trace of the surface 3x 2 y+z 2 = 9 in the plane z = 1? (a) A single point (b) An ellipse (c) A hyperbola (d) A parabola 13

14 (3) Consider the line in the yz-plane defined by y = z and the surface generated by revolving this line around the y-axis. Find the equation of the trace of this surface in the plane y = y 0. Lesson 18 (a) x 2 +z 2 = y 2 0 (b) (y y 0 ) 2 = z 2 (c) y 2 z 2 = x 2 (d) x 2 +(y y 0 ) 2 +z 2 = 0 (1) Which of the following is not true about the surface given by 2x 2 +3y 2 6z 2 = 0? (a) It represents an elliptic cone. (b) Its trace in the plane z = 1 represents a hyperbola. (c) Its trace in the xy-plane represents a single point. (d) The surface is symmetric over the xy-plane. (2) Determine the surface represented by the equation x 2 +4y 2 4x 8y 4z +8 = 0. (a) An elliptic cone (b) A hyperboloid of one sheet (c) An elliptic paraboloid (d) A single point (3) Considerthe surfacerepresented by theequation z2 4 x2 y2 3 is a trace of this surface in some plane? = 1. Which of the following Lesson 19 (a) A point (b) An ellipse (c) A hyperbola (d) All of the above (1) Find the rectangular coordinates of the point with cylindrical coordinates (r, θ, z) = (2, π 4, 1). (a) ( 1, 2,1) (b) ( 3, 1,1) (c) (2,1, 1) 14

15 (d) ( 2, 2, 1) (2) Which of the following rectangular-coordinate equations describes the same set of points as the cylindrical-coordinate equation z = r 2 sin(2θ)? (a) z = x 2 +y 2 (b) z = (x 2 +y 2 ) 2 sin(2tan(y/x)) (c) z = x 2 y 2 (d) z = 2xy (3) Which of the following describes the graph of the cylindrical-coordinate equation r = secθtanθ? Lesson 20 (a) A plane (b) A sphere of radius 1 (c) A parabolic cylinder (d) A paraboloid (1) Convert the point (ρ,θ,φ) = (3, π 6, π 2 ) from spherical coordinates to rectangular coordinates. (a) ( 3 3 2, 3 2,0) (b) (0, 3 2, ) (c) ( 3 2, 3 3 2, π 2 (d) 3 2, 3 3 2,0 (2) LetS bethesurfacedefinedbyρ = 2secφ. Whichofthefollowingrectangular-coordinate equations describes S? (a) x 2 z 2 = y (b) xy = 2 (c) z = 2 (d) x = 1 (3) What best describes the region in 3-space satisfying the inequalities: ρ 1, π 2 φ π, and 0 θ π/2. (a) All points lying outside the unit sphere and below the xy-plane (b) Half of a cylinder (c) All points above the xy-plane 15

16 (d) one-eighth of a sphere Lesson 21 (1) Give the domain for the following vector-valued function: (a) (b) {t : 1 2 < t 1} (c) {t : t 2} (d) {t : t 1} F(t) = 1 t,ln(2t 1), 4t (2) Which of the following best describes the graph of the vector-valued function: F(t) = 2cost,2sint,3, 0 t 2π? (a) A circular helix increasing in z as t increases (b) A straight line lying in the plane z = 3 (c) A circle of radius 2 lying in the plane z = 3 and centered at (0,0,3) (d) An ellipse on the plane z = 3 whose length in the x-direction is twice that in the y-direction (3) The following is a bad proof of the claim that r(t) = tsint,0, tsint lies on the plane x = z: x(t) = tsint; y(t) = 0; z(t) = tsint r(t) = x(t),y(t),z(t) x = tsint = ( tsint) = z Which of the following tips from the Guidelines for Writing Proofs does this bad proof violate? (a) In addition to mathematical notation and symbols, use words when appropriate to help the reader understand what you re doing. Lesson 22 (b) Variables must be introduced before they are used. (c) Never assume what you are trying to prove. (d) Quantifier phrases must precede the expressions to which they apply. (1) Let r(t) = 3cosπt,4e 2t,6. Find r ( 1). (a) 3,8e 2,6 16

17 (b) 0,4,0 (c) 0, 8 e 2,0 (d) 3,8,6 (2) Which of the following rules for differentiation and integration is not true? (a) kr(t)dt = k r(t)dt. (b) dr/dt = 0 if r(t) is constant (c) d dt (r 1(t) r 2 (t)) = dr 1 /dt dr 2 /dt. (d) d dt f(t)r(t) = f(t) d dtr(t), where f is a real-valued function. (3) Let r (t) = 4t,sint, 2e 2t and r(0) = 1,0,2. Find r(t). (a) 2t 2 +1, cost+1, e 2t +3 Lesson 23 (b) 2t+1,cost 1,2e t (c) t 2 +1,sint,e t +1 (d) 2t 2,sint cost,e t 2 (1) Which of the following represents the line tangent to r(t) = where t = 1? (a) 1 e,2, 3 2 (b) 1 e,2,3 +t 1 e,2, 3 2 (c) 1 e,2, 3 2 +t e t,2, 3 2 t (d) t 1 e,2,3 e t,2t,3 t at the point (2) Let r 1 (t) = 2t,e t,sint and r 2 (t) = cost, t,t 2. Find the derivative of r 1 (t) r 2 (t). (a) 4cost 2sint+3 t +4 (b) 2sintcost+3te t 5 (c) 4sint(1+t)+te t 1 (d) cost(2+t 2 ) e t (1+t) (3) Which of the following pairs of vector-valued functions have graphs that intersect at the point (2,0,5)? (a) r 1 (t) = 2tant,3+t 2, 5t and r 2 (t) = 5t 2,cost+e t,2t (b) r 1 (t) = t,0,7 t and r 2 (t) = sint,e t 1,5t 17

18 (c) r 1 (t) = t+2,sint,5e t and r 2 (t) = 2t,t+1,5 t Lesson 24 (d) r 1 (t) = 5t,2 t,2sint and r 2 (t) = 4 t,3 2t,t 2 (1) Which of the following is a smooth function of the parameter t? (a) lnt,2t,t+3 (b) t 2 +3t,cost,5t (c) e t +e t,3t 2,cost (d) sin(t 2 ), 2 t +1,4t (2) Let r(t) = 3e t,2t 2 and consider the change of parameter t = g(τ) = 2τ +1. Find the derivative with respect to τ of r(g(τ)). (a) 6e 2τ+1,16τ +8 (b) 3e τ,4τ (c) 2e τ +1,4τ 2 +1 (d) 3e 2τ+1,4τ 2 +2τ (3) Find a change of parameter t = g(τ) for the circular helix r(t) = cost,sint,t,0 t 2π such that the helix is traced downwards and clockwise (when facing down) as τ increases. Lesson 25 (a) t = 1 τ, 0 τ 1 (b) t = πτ, 0 τ 1 (c) t = 4π τ, 0 τ 1 (d) t = 2π(1 τ), 0 τ 1 (1) Considerthegraphin2-spaceofthesmoothvector-valuedfunctionr(t) = 3sint, 3cost,2t. Find its arc length from t = 0 to t = 1. (a) 3π +2 (b) 5 (c) 13 (d) 1 π 2 (2) Find an arc length parametrization of the line x = 5t 6, y = t+4 that has the same orientation and has reference point at t = 0. 18

19 (a) x = 5 26 s 6; y = 1 26 s+4 (b) x = 5 13 s 6; y = 5 13 s+4 (c) x = s 6; y = 5s+4 (d) x = 5s+6; y = s 4 (3) Consider the arc length parametrization x = 3 10 sint, y = 3 10 cost, z = t 10. (You should verify that this is an arc length parametrization even though we ve used t to denote the parameter.) Starting at t = 0, what are the final coordinates after moving along the graph for a distance of π units? Lesson 26 (a) ( 3 10,0, π 10 ) (b) ( 3 10, 3 10, π 10 ) (c) (3,0,π) (d) (0, 3 10, π 10 ) (1) Given r(t) = t 2 +2t,3t, find the unit tangent vector T(t) at t = 0. (a) 2 3 2, 1 2 (b) 2 13, 3 13 (c) 3 13, 3 13 (d) 2 3, 3 (2) Let r(t) = 3t,sint+2,cost+3. Find the binormal vector B(t). (a) 3sintcost,4tsint, 4tcost (b) 0, sint, cost (c) 1 2, 3 2 cost, 3 2 sint (d) 3 2, 2cost+sint,3sint (3) Which of the following is an equation for the normal plane for r(t) = 2t,e t,3t 2 at t = 0? (a) e y +6z = 0 (b) 2x+y = 5 19

20 (c) 2+e y +3z 2 = 0 (d) 2x+y 1 = 0 (4) Let q(s), 10 < s < 10 be an arc-length parameterization of a curve C. Which of the following is false? (a) If q(s 1 ) = q(s 2 ), then T(s 1 ) and T(s 2 ) may point in different directions. (b)ifp(u)isanyotherarclengthparametrization ofc, theneitheru = s+coru = s+c for some constant c. (c) The reparametrization g(v) = q(logv), e 10 < v < e 10 is smooth, but h(w) = q(w 3 ), 10 1/3 < w < 10 1/3 is not. Lesson 27 (d) The tangent vector is given by q (s) = B(s) N(s). (1) Which of the following curves in a plane does not have constant curvature? (a) A circle (b) A parabola (c) A straight line (d) None of the above (2) Calculate the curvature κ(t) of r(t) = 3sint, 3cost,4t. (a) 3 25 (b) 7t2 15 (c) 1 5+t 2 (d) 10 3 (3) If r (t) = 1 5 4cos(et ),5sin(e t ),3cos(e t ), what is the curvature of r? (a) e 5t (b) 5 (c) e t (d) 1 5 (4) LetC beacurvethat doesnot cross itself. Let r(t), a t b, beasmoothparameterization of C and κ(t) the curvature. Let q(s), c s d, be an arc-length parameterization of C and κ(s) the nonzero curvature. Which of the following might be false? (a) The curvatures are given by κ(t) = r (t) and κ(s) = q (s). (b) If q(s 0 ) = r(t 0 ), then κ(s 0 ) = κ(t 0 ). 20

21 (c) If q(s 0 ) = r(t 0 ), then T(t 0 ) = ±q (s 0 ), depending on the orientation of q(s). Lesson 28 (d) If q(s 0 ) = r(t 0 ), then N(t 0 ) = q (s 0 )/ q (s 0 ), assuming N(t 0 ) exists. (1) Consider a curve defined in 2-space by r(t) = sint,2cost. Find the radius of curvature at the point where t = π. (a) 4π 2 (b) 7 3 (c) π 3 (d) 1 2 (2) Let C be a curve in 2-space. Let φ(s) = 4s+6 be the angle measured counterclockwise from the positive x direction to the unit tangent vector, with s being an arc length parameter. Which of the following can be said about the curve C? (a) C has constant, nonzero curvature (b) C has zero curvature (c) C only has constant curvature when s > 0 (d) C has increasing curvature (3) For the curve r(t) = 2t 2,3t, what is lim t κ(t)? (a) 1 (b) 1 (c) 0 (d) The limit does not exist (4) Let r(t) = x(t),y(t),z(t) be a smooth parametrization of a curve C. Which of the following arguments is fallacious? (a) If r(t) is a smooth parametrization of a curve C such that r (t) = r (t) for all t, then r(t) = r(t)+ a,b,c for some constant vector a,b,c. Thus, C is a translation of C. (b)ifthebinormalb(t) = a,b,c forsomenon-zeroconstant vector a,b,c forallt, then since T(t) B(t) = 0, we have r (t) a,b,c = 0. It follows that ax(t)+by(t)+cz(t) = d, for some constant d. Thus C lies in a plane. (c) If κ(t) = k for some constant k for all t, then C bends at a constant rate. Thus, if k = 0, then C is (part of) a line, and if k > 0, then C is (part of) a circle. 21

22 (d) If T (t) = a,b,c for some constant vector a,b,c for all t, then there is some constant vector a 0,b 0,c 0 such that for all t, T(t) = a,b,c t+ a 0,b 0,c 0. Since T(t) T (t) = 0 for all t, it follows that a,b,c = 0. Thus T(t) is constant and C is (part of) a line. Lesson 29 (1) Consider the position function r(t) = e 2t,4t 2,cos(t 1) of a particle in 3-space. What is the particle s acceleration at time t = 1? (a) 0,8,0 (b) 4e 2,8, 1 (c) e 2,4,1 (d) e 2,12,0 (2) Consider a particle moving along a smooth curve C in 3-space. At a given point, which of the following vectors must be orthogonal to the acceleration vector? (a) the normal vector (b) the velocity vector (c) the position vector (d) the binormal vector (3) Let a particle move along a smooth curve C is 3-space. Suppose that, at some point, the particle has velocity v = 2, 1, 2 and curvature 4. If a is the acceleration at this point, find v a. (a) 4 (b) 0 (c) 108 (d) 5 Lesson 30 (1) Let r(t) = 3t 2, 2t 3,t Find the scalar tangential component of acceleration at time t = 1. (a) 54 7 (b) 72 5 (c) 8 22

23 (d) 17 8 (2) At time t = 0, a particle is launched so that it shoots out at a height of 10 feet with initial velocity 1, 0. If g represents the acceleration due to gravity, find the position function r(t) of this particle. (a) 10t,gt 2 (b) 10t, (g +1) (c) 10, (gt+t) (d) t, 1 2 gt2 +10 (3) An object is launched from the ground with initial speed 800ft/sec, making an angle of 30 with the ground. How high does the object rise? Lesson 31 (a) 2000 ft. (b) 2500 ft. (c) 5000 ft. (d) 7500 ft. (1) For the function z(x,y) = y/x, which of the following describes the level curve z = 2? (a) A hyperbola (b) A parabola (c) A single point (d) A straight line with one point removed (2) If w(x,y,z) = z 3 +x 2 y 2 and σ is the level surface w = 1, which of the following is the trace of σ in the plane z = 1? (a) A circle of radius 1 (b) A single point (c) A pair of intersecting lines (d) None of the above (3) On a scale of 1 to 10, with 1 being of no importance and 10 being of great importance, what is the importance of the concept of limit and its rigorous epsilon-delta definition (which we will be studying in upcoming lessons)? (a) 1 (b) 17 23

24 (c) 2π (d) 10 Lesson 32 (1) If D R 2 and D is a closed set, which of the following must be true about D? (a) If D is unbounded, then D = R 2. (b) If C is an open subset of R 2, then the union C D is also closed. (c) If p is any point in R 2, then the set D {p} is also closed. (d) (a) and (c) (2) If U R 2 and U is open, which of the following must be true about U? (a) If U is nonempty and p U, then there exists a closed set D such that p D U. (b) If W is another open set in R 2, then U W is open. (c) If W is another open set in R 2, then C = {p R 2 : p U and p W} is open. (d) (a) and (b) (3) If D R 2 has no accumulation points, which of the following must be true? Lesson 33 (a) D must be finite. (b) D must be finite or unbounded. (c) D cannot be closed. (d) D must be open. (1) If f(x) = 3x and ǫ > 0, which of the following is the largest value of δ that would make the implication 0 < x 2 < δ implies f(x) 6 < ǫ true? (a) δ = 3ǫ (b) δ = 2ǫ (c) δ = ǫ/3 (d) δ = ǫ/6 (2) If f(x,y) = 3x+4y and ǫ > 0 is arbitrary, which of the following values of δ would make the implication 0 < x 2 +y 2 < δ implies f(x,y) < ǫ true? 24

25 (a) δ = ǫ/3 (b) δ = ǫ/4 (c) δ = ǫ/7 (d) b and c (3) Assuming (0,0) is an accumulation point of the domain of f(x,y), which of the following statements implies that the lim (x,y) (0,0) f(x,y) 2? (a) The limit of f(x,y) along the curve x(t) = 1+t 2,y(t) = t is equal to 4. (b) There exists ǫ > 0 such that for any δ > 0 there exists (x,y) in the domain of f satisfying 0 < x 2 +y 2 < δ and f(x,y) 2 ǫ. Lesson 34 (c) f(0,0) = 1. (d) all of the above (1) Which of the following statements is true about the function f(x,y) = x2 y 2 x 2 +y 2? (a) lim (x,y) (0,0) f(x,y) = 0. (b) lim (x,y) (0,0) f(x,y) = +. (c) lim (x,y) (0,0) f(x,y) = 1. (d) none of the above (2) If f(x,y) = x2 y x 4 +y 2 for all (x,y) (0,0) and f(0,0) = k for some real number k, which of the following is true about f? (a) f is not continuous (0,0). (b) f might be continuous at (0,0) depending on the value of k. (c) There are exactly two values of k that make f continuous at (0,0). (d) b and c (3) If f(x,y) = (2+ x4 y 4 x 2 +y 2 )( 1+e 1/(x2 +y 2) ) for all (x,y) (0,0), and f(0,0) = k, which of the following is true about f? (a) f is not continuous (0,0). (b) f might be continuous at (0,0) depending on the value of k. (c) There are exactly two values of k that make f continuous at (0,0). (d) b and c 25

26 Lesson 35 (1) If f(x,y) = x 2 ye x+y, c = f x (1,1), and d = f y (1,1), then which of the following is equal to (c,d)? (a) (3e 2,2e) (b) (2e 2,3e 2 ) (c) (e,e) (d) (3e 2,2e 2 ) (2) If f(x,y) = exy sin(x+y) ln(x 2 +1), what is f x (π,0)? (a) 0 (b) 2lnπ (c) 1 ln(1/(1+π 2 )) (d) none of the above (3) If f(x,y) = sin(xy)+x 3 y y 2, g(x,y) = f x (x,y), and h(x,y) = g y (x,y), then which of the following describes h(x, y)? (a) xysin(xy)+3x 2 (b) xsin(xy)+3x 2 (c) sin(xy) Lesson 36 (d) cos(xy) xysin(xy)+3x 2 (1) If ω and c are constants, then the function u(x,t) = (sincωt)(sinωx) satisfies which of the following equations? (a) u xx +u tt = 0 (b) u tt = c 2 u xx (c) u xx = c 2 u tt (d) u t = c 2 u xx (2) Which of the following functions satisfies Laplace s equation u xx +u yy = 0? (a) u(x,y) = x 3 3xy 2 (b) u(x,y) = e x+y sin(x y) (c) u(x,y) = ln(x 2 +y 2 ) (d) all of the above 26

27 (3) If f(x,y,z) has continuous partial derivatives of all orders, what is the largest possible number of distinct third-order partial derivatives f could have? (a) 6 (b) 9 (c) 10 (d) 16 Lesson 37 (1) If f x (1,2) and f y (1,2) exist, which of the following must be true? (a) f is differentiable at (1,2). (b) f is continuous at (1,2). (c) The function g(y) = f(1,y) is continuous at y = 2. (d) none of the above (2) Which of the following statements is true about the function defined by f(x,y) = { x, for x 0; 0, for x < 0. (a) f is differentiable at any point not on the y-axis. (b) f is continuous everywhere. (c) f y exists everywhere, but at some points f x does not exist. (d) all of the above (3) Which of the following statements is not necessarily true? (a) If f is differentiable at (x 0,y 0 ), then f x and f y are continuous at (x 0,y 0 ). (b) If f is not continuous (x 0,y 0 ), then it is not differentiable there. (c) If f(0,y) is not continuous at y = 1, then f y (0,1) does not exist. (d) If f x and f y are differentiable in an open disk centered at (x 0,y 0 ), then f is differentiable in that disk. Lesson 38 (1) If f is a differentiable function of x and y, which are differentiable functions of t, and if (x(0),y(0)) = (0,0), f x (0,0) = 2, x (0) = 3, f y (0,0) = 4, and df dt t=0 = 10, what is y(h) lim h 0 h? (a) 1 27

28 (b) 2 (c) 3 (d) 4 (2) If f is a differentiable function of x and y, which are differentiable functions of u, which is a differentiable function of t, which of the following gives d dt f(x(u(t)),y(u(t)))? (a) x du u dt + y du u dt (b) f u u x x t + f u (c) f dx t dt + f dy t dt (d) f dx x du u y y t du dt + f dy du y du dt. (3) Which of the following best describes the strategy employed to prove the chain rule for f(x(t),y(t))? (a) Using properties of limits, compute directly lim h 0 f(x(t+h),y(t+h)) f(x(t),y(t)) h. (b) Apply the chain rule for functions of one variable twice, once for x and once for y. (c) Apply the chain rule for functions of one variable to f(x,y) and the product rule to (x(t),y(t)). (d) Use the definition of differentiability to express f t lim t 0 of that expression. in a convenient form, then take Lesson 39 (1) Whichofthefollowingstatementsaboutthechainruleforfunctionsoftheformf(x(u,v),y(u,v)) is always true? (a) f u = f x u u + f u (b) f u f v = f x ( x y u u x v (c) f x + f y = f u u x f v v y (d) None of the above )+ f y ( y u y v ) (2) Which of the following statements best describes the proof of the chain rule for functions of the form f(x(u,v),y(u,v))? (a) The proof is similar to the proof of the chain rule for functions of one variable, but the details are beyond the scope of this course. (b) The proof follows from basic facts about implicit differentiation for functions of one variable. (c) The proof follows from the tree diagram expressing the relationships between f, x, y, u, and v. 28

29 (d) The proof follows from the chain rule for functions of the form f(x(t),y(t)) and the fact that u f(x(u,v),y(u,v)) (u,v)=(u 0,v 0 ) = d du f(x(u,v 0),y(u,v 0 )) u=u0. (3) If z = f(u) and u = g(x,y), where g and f are differentiable, what is z dx? (a) df (b) f (c) u Lesson 40 g du x x x u x + u y y x. (d) None of the above (1) If L is the normal line to the surface x 2 + y 2 +z 2 = 25 at the point ( 3,0,4), then L passes through which of the following points? (a) (4,0, 3) (b) (1,1,1) (c) (0,1,0) (d) ( 9,0,12) (2) At which point on the surface z = 8 3x 2 2y 2 is the tangent plane perpendicular to the line x = 2 3t, y = 7+8t, z = 5 t? (a) (0,0,8) (b) (1/2, 2, 3/4) (c) ( 3,1, 21) (d) (2,7,5) (3) At which point does the normal line to the surface z = 2 xy pass through the origin? (a) (0,0,2) Lesson 41 (b) (1,1,1) (c) ( 1, 1,1) (d) All of the above (1) What is the directional derivative of f(x,y) = y 2 lnx at (1,4) in the direction of v = 3i 3j? (a) 48 (b)

30 (c) 7 2/2 (d) 48 (2) If f is differentiable at (x 0,y 0 ), u and w are nonzero vectors with u+w 0, and c is a positive real number, which of the following equations is not always true? (a) D cu f(x 0,y 0 ) = D u f(x 0,y 0 ) (b) D u f(x 0,y 0 ) = D u f(x 0,y 0 ) (c) u+w D u+w f(x 0,y 0 ) = u D u f(x 0,y 0 )+ w D w f(x 0,y 0 ) (d) None of the above (3) If f x (x 0,y 0 ) = 1 and f y (x 0,y 0 ) = 2, and u is a unit vector such that D u f(x 0,y 0 ) = 2, what is u? Lesson 42 (a) u = j or u = ( 4/5)i+(3/5)j (b) u = i or u = j (c) u = ± 2 2 (i+j) (d) u = (1/2)i ( 3/2)j (1) If f(x,y) = xcosy +ysinx, which of the following vectors is perpendicular to the level curve f(x,y) = 0 at (0,0)? (a) 3i (b) 2i+j (c) i j (d) 4j (2) On a certain mountain, the elevation z above a point (x,y) in the xy-plane at sea level is z = x 2 4y 2 ft. The positive x-axis points east, and the positive y-axis points north. If a climber is at the point ( 20,5,1100) and wants to travel a level path, in which of the following directions should the climber begin walking? (a) arctan(1/ 3) degrees west of north (b) 45 degrees west of south (c) arctan(2) degrees north of east (d) 30 degrees south of west (3) Suppose f is differentiable, f(0,0) = 0, and for all (x,y) satisfying x 2 + y 2 1 we have f(x,y) 3. What must be true about f(x,y) for (x,y) lying on the circle x 2 +y 2 = 1? 30

31 (a) f(x,y) is constant on the circle x 2 +y 2 = 1. Lesson 43 (b) f(x,y) = 0 for some (x,y) on the circle x 2 +y 2 = 1. (c) f(x,y) 3 for all (x,y) on the circle x 2 +y 2 = 1. (d) b and c (1) If r(t) = x(t)i + y(t)j + z(t)k and w = f(x,y,z) are both differentiable, and r(t 0 ) = x 0 i+y 0 j+z 0 k, then which of the following expressions is equal to w (t 0 )? (a) f (x 0,y 0,z 0 )x (t 0 )+f (x 0,y 0,z 0 )y (t 0 )+f (x 0,y 0,z 0 )z (t 0 ) (b) f(x (t 0 ),y (t 0 ),z (t 0 )) (c) r (t 0 ) f x (x 0,y 0,z 0 ),f y (x 0,y 0,z 0 )),f z (x 0,y 0,z 0 ) (d) f x (x 0,y 0,z 0 ),f y (x 0,y 0,z 0 ),f z (x 0,y 0,z 0 ) r (t 0 ) (2) If f is a differentiable function of one variable, w = f(t), and t = x 2 +y 2 +z 2, which of the following equations is true? (a) w x +w y +w z = 2f (t)(x+y +z)/t (b) (w x ) 2 +(w y ) 2 +(w z ) 2 = (f (t)) 2 (c) (w x ) 2 +(w y ) 2 +(w z ) 2 = t 2 f (t) (d) a and b (3) Suppose w = (x x2 n) k where n > 2. What nonzero values of k make the following statement true? n 2 w x 2 = 0 for all (x 1,...,x n ). i (a) k = 1 Lesson 44 (b) k = n/2 (c) k = 1+ n 2 (d) k = 1 n 2 i=1 (1) What is the directional derivative of f(x,y,z) = x 3 y 2 z 5 2xz + yz + 3x at the point ( 1, 2,1) in the direction of the negative z-axis? (a) 20 (b) 10 (c) 0 31

32 (d) 5 (2) If f(x 1,x 2,x 3,x 4 ) = x 2 1 x2 2 +x2 3 x2 4, which of the following is a unit vector pointing in the direction in which f decreases most rapidly at the point (1,1,1,1)? (a) 0, 2 2,0, 2 2 (b) 0, 2 2, 2 2,0 (c) 1 2, 1 2, 1 2, 1 2 (d) 1 2, 1 2, 1 2, 1 2 (3) If P is a tangent plane to the surface x 2/3 + y 2/3 + z 2/3 = 1, what is the sum of the squares of the x-, y-, and z- intercepts of P? (a) 1 (b) 0 (c) 1 (d) 2/3 Lesson 45 (1) Suppose the function f(x,y) is defined on the closed domain D. Which of the following statements are true? (a) If f has a relative maximum on D, then f has an absolute maximum on D. (b) If f has an absolute minimum on D, then f has a relative minimum on D. (c) If D is also bounded, then f has both an absolute maximum and absolute minimum on D. (d) b and c (2) Suppose f(x, y) is differentiable on the domain D. Which of the following statements are true? (a) If (x 0,y 0 ) is a relative minimum, then f(x 0,y 0 ) = 0. (b) If (x 0,y 0 ) is a relative maximum and f(x 0,y 0 ) 0, then (x 0,y 0 ) is a boundary point of D. (c) If f has a relative minimum on D, then it has at least one relative maximum on D. (d) None of the above (3) Suppose f is a continuous function. On which of the following domains D must f have an absolute maximum and an absolute minimum? (a) D = {(x,y) : x 1, y 1} 32

33 (b) D = {(x,y) : 1 < x < 1, 1 < y < 2} Lesson 46 (c) D = {(x,y) : 0 x 1, x 2 < y < x} (d) D = {(x,y) : x + y 1} (1) Suppose f is differentiable on R 2 and f xx (1,2) = 7, f yy = 3, and f xy = 4. What can we say about the point (1,2)? (a) (1,2) is a relative max. (b) (1,2) is a relative min. (c) (1,2) is a saddle point. (d) None of the above (2) If f(x,y) = xy+ 2 x + 4 y minimum of f? is defined on the set {(x,y) : x > 0, y > 0}, what is the absolute (a) 0 (b) 1 (c) 6 (d) f doesn t have an absolute minimum on this domain. (3) If f(x,y) = 4x 2 3y 2 +2xy is defined on the set {(x,y) : 0 x 1, 0 y 1}, what is the absolute maximum of f? (a) 13/3 (b) 4 (c) 11/3 (d) 10/3 Lesson 47 (1) Suppose f(x,y) is differentiable, g(x,y) = e xy +x 2 +y e 1, and on the closed curve g(x,y) = 0, f has a constrained relative minimum at (1, 1). Which of the following vectors might be f(1, 1)? (a) 2e 1,e+1 (b) e 1,e 1 (c) 1, 1 (d) e 2,e 1 (2) What is the minimum value of f(x,y) = x 2 y 2 subject to the constraint 4x 2 +y 2 = 8? 33

34 (a) 1/2 (b) 0 (c) 1 (d) 8 (3) Suppose f(x,y) and g(x,y) are differentiable, the curve C is defined by g(x,y) = 0, g(x,y) 0 on C, and f = 3 g at the point P on C. If f xx (P) > 0 and f xx (P)f yy (P) fxy 2 (P) > 0, what must be true about P? Lesson 48 (a) P is a constrained relative maximum of f. (b) P is a constrained relative minimum of f. (c) P is a constrained absolute minimum of f. (d) None of the above (1) Whatisthemaximumvalue off(x,y,z) = xyz alongtheconstraint surfacex 2 +y 2 +z 2 = 1? (a) 2/8 (b) 7/27 (c) 3/9 (d) None of the above (2) Suppose (a,b,c) is a point on the surface xy z 2 = 1 closest to the origin. On which of the following curves does the point (a,b) lie? (a) y = x (b) x = 0 (c) x = y (d) x = y +1 (3) Which of the following might be an extremum of the function f(x,y,z) = x 4 +y 4 +z 4 on the constraint x+y +z = 1? (a) (1/4, 1/4, 1/4) (b) (1/3, 1/3, 1/3) (c) (1/2, 1/2, 1/2) (d) (0,0,0) 34