MA 261 PRACTICE PROBLEMS1. If the line ` has symme tric equationsx¡12=y¡3=z+27,¯nd a ve ctor equation for the line `0that contains the poin t (2; 1; ¡3) and is parallel to `.A. ~r =(1+2t)~i ¡ 3t~j +(¡2+7t)~k B. ~r =(2+t)~i ¡ 3~j +(7¡ 2t)~kC. ~r =(2+2t)~i +(1¡ 3t)~j +(¡3+7t)~k D. ~r =(2+2t)~i +(¡3+t)~j +(7¡ 3t)~kE. ~r =(2+t)~i +~j +(7¡ 3t)~k2. Find parametric equations of the line containing the points (1; ¡1; 0) and (¡2; 3; 5).A. x =1¡ 3t; y = ¡1+4t; z =5t B. x = t; y = ¡t; z =0C. x =1¡ 2t; y = ¡1+3t; z =5t D. x = ¡2t; y =3t; z =5tE. x = ¡1+t; y =2¡ t; z =53. Find an equation of the plane that contains the point (1; ¡1; ¡1) and has normal vector12~i +2~j +3~k.A. x ¡ y ¡ z +92=0 B. x +4y +6z +9=0 C.x¡112=y+12=z+13D. x ¡ y ¡ z =0 E.12x +2y +3z =14. 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 B. 6x +11y + z =5 C. 11x ¡ 6y + z =0D. ~r =18~i ¡ 33~j +3~k E. x ¡ 6y ¡11z =125. Find parametric equations of the line tangent to the curve ~r(t)=t~i + t2~j + t3~k at the point (2; 4; 8)A. x =2+t; y =4+4t; z =8+12t B. x =1+2t; y =4+4t; z =12+8tC. x =2t; y =4t; z =8t D. x = t; y =4t; z =12t E. x =2+t; y =4+2t; z =8+3t6. The position function of an object is~r(t)=cost~i +3sint~j ¡ t2~kFind the velocity, acceleration, and speed of the object when t = ¼.Velocity Acceleration SpeedA. ¡~i ¡ ¼2~k ¡3~j ¡ 2¼~kp1+¼4B.~i ¡ 3~j +2¼~k ¡~i ¡ 2~kp10 + 4¼2C. 3~j ¡ 2¼~k ¡~i ¡ 2~kp9+4¼2D. ¡3~j ¡ 2¼~k~i ¡ 2~kp9+4¼2E.~i ¡ 2~k ¡3~j ¡ 2¼~kp517. A smooth parametrization of the semicircle which passes through the point s (1; 0; 5), (0; 1; 5) and(¡1; 0; 5) isA. ~r(t)=sint~i +cost~j +5~k; 0 · t · ¼ B. ~r(t)=cost~i +sint~j +5~k; 0 · t · ¼C. ~r(t)=cost~i +sint~j +5~k;¼2· t ·3¼2D. ~r(t)=cost~i +sint~j +5~k; 0 · t ·¼2E. ~r(t)=sint +cost~j +5~k;¼2· t ·3¼28. The length of the curve ~r(t)=23(1 + t)32~i +23(1 ¡ t)32~j + t~k, ¡1 · t · 1isA.p3B.p2C.12p3D.2p3E.p29. The level curves of the function f(x; y)=p1 ¡ x2¡ 2y2areA. circles B. lines C. parabolas D. hyperbolas E. ellipses10. The level surface of the function f(x; y; z)=z¡x2¡y2that passes through the point (1; 2; ¡3) intersectsthe (x; z)-plane (y =0)alongthecurveA. z = x2+8 B. z = x2¡ 8C.z = x2+5 D. z = ¡x2¡ 8E. does not in tersect the (x; z)-plane11. Matc h the graphs of the equations with their names:(1) x2+ y2+ z2= 4 (a) paraboloid(2) x2+ z2= 4 (b) sphere(3) x2+ y2= z2(c) cylinder(4) x2+ y2= z (d) double cone(5) x2+2y2+3z2= 1 (e) ellipsoidA. 1b, 2c, 3d, 4a, 5e B. 1b, 2c, 3a, 4d, 5e C. 1e, 2c, 3d, 4a, 5bD. 1b, 2d, 3a, 4c, 5e E. 1d, 2a, 3b, 4e, 5c12. Suppose that w = u2=v where u = g1(t)andv = g2(t) are di®eren tiable functions of t.Ifg1(1) = 3,g2(1) = 2, g01(1) = 5 and g02(1) = ¡4, ¯nddwdtwhen t =1.A. 6 B. 33=2C.¡24 D. 33 E. 2413. If w = euvand u = r + s, v = rs, ¯nd@[email protected]. e(r+s)rs(2rs + r2)B.e(r+s)rs(2rs + s2)C.e(r+s)rs(2rs + r2)D. e(r+s)rs(1 + s)E.e(r+s)rs(r + s2).214. If f(x; y)=cos(xy),@2f@x@y=A. ¡xy cos(xy)B.¡xy cos(xy) ¡ sin(xy)C.¡sin(xy)D. xy cos(xy)+sin(xy)E.¡cos(xy)15. Assuming that the equation xy2+3z =cos(z2)de¯nesz implicitly as a function of x and y,¯nd@[email protected]¡sin(z2)B.¡y23+sin(z2)C.y23+2z sin(z2)D.¡y23+2z sin(z2)E.¡y23¡2z sin(z2)16. If f(x; y)=xy2,thenrf(2; 3) =A. 12~i +9~j B. 18~i +18~j C. 9~i +12~j D. 21 E.p2.17. Find the directional derivative of f(x; y)=5¡ 4x2¡ 3y at (x; y) towards the originA. ¡8x ¡ 3B.¡8x2¡3ypx2+y2C.¡8x¡3p64x2+9D. 8x2+3y E.8x2+3ypx2+y2.18. For the function f (x; y)=x2y, ¯nd a unit vector ~u for which the directional derivative D~uf(2; 3) iszero.A.~i +3~j B.i+3~jp10C.~i ¡ 3~j D.i¡3~jp10E.3~i¡~jp10.19. Find a vector pointing in the direction in which f(x; y; z)=3xy ¡ 9xz2+ y increases most rapidly atthe point (1; 1; 0).A. 3~i +4~j B.~i +~j C. 4~i ¡ 3~j D. 2~i +~k E. ¡~i +~j.20. Find a vector that is normal to the graph of the equation 2 cos(¼xy )=1atthepoint(16; 2).A. 6~i +~j B. ¡p3~i ¡~j C. 12~i +~j D.~j E. 12~i ¡~j.21. Find an equation of the tangent plane to the surface x2+2y2+3z2= 6 at the point (1; 1; ¡1).A. ¡x +2y +3z =2 B. 2x +4y ¡6z =6 C. x ¡ 2y +3z = ¡4D. 2x +4y ¡6z =0 E. x +2y ¡ 3z =6.22. Find an equation of the plane tangent to the graph of f (x; y)=¼ +sin(¼x2+2y)when(x; y)=(2;¼).A. 4¼x +2y ¡z =9¼ B. 4x +2¼y ¡ z =10¼ C. 4¼x +2¼y + z =10¼D. 4x +2¼y ¡ z =9¼ E. 4¼x +2y + z =9¼.323. The di®erential df of the function f(x; y; z)=xey2¡z2isA. df = xey2¡z2dx + xey2¡z2dy + xey2¡z2dzB. df = xey2¡z2dx dy dzC. df = ey2¡z2dx ¡ 2xyey2¡z2dy +2xzey2¡z2dzD. df = ey2¡z2dx +2xyey2¡z2dy ¡ 2xzey2¡z2dzE. df = ey2¡z2(1 + 2xy ¡ 2xz)24. The function f(x; y)=2x3¡ 6xy ¡ 3y2hasA. a relative minimum and a saddle point B. a relative maximum and a saddle pointC. a relative minimum and a relative maximum D. two saddle pointsE. two relative minima.25. Consider the problem of ¯nding the minim u m value of the function f (x; y)=4x2+ y2on the curvexy = 1. In using the method of Lagrange multipliers, the value of ¸ (ev en though it is not needed) willbeA. 2 B. ¡2C.p2D.1p2E. 4.26. Evaluate the iterated integralR31Rx01xdydx.A. ¡89B.2C.ln3D.0E.ln2.27. Consider the double integral,RRRf(x; y)dA,whereR is the portion of the disk x2+y2· 1, in the upperhalf-plane, y ¸ 0. Express the integral as an iterated integral.A.R1¡1Rp1¡x2¡p1¡x2f(x; y)dydx B.R0¡1Rp1¡x20f(x; y)dydxC.R1¡1Rp1¡x20f(x; y)dydx D.R10Rp1¡x2¡p1¡x2f(x; y)dydxE.R10Rp1¡x20f(x; y)dydx.28. Find a and b for the correct interchange of order of integration:R20R2xx2f(x; y)dydx =R40Rbaf(x; y)dxdy.A. a = y2;b=2y B. a =y2;b=py C. a =y2;b= yD. a =py;b =y2E. cannot be done without explicit knowledge of f(x; y).29. Evaluate the double integralRRRydA,whereR is the region of the (x; y)-plane inside the triangle withvertices (0; 0), (2; 0) and (2; 1).A. 2 B.83C.23D. 1 E.13.30. The volume of the solid region in the ¯rst octant bounded above by the parabolic sheet z =1¡ x2,below by the xy plane, and on the sides by the planes y =0andy = x is given by the double in tegralA.R10Rx0(1 ¡ x2)dydx B.R10R1¡x20xdydx C.R1¡1Rx¡x(1 ¡ x2)dydxD.R10R0x(1 ¡ x2)dydx E.R10R1¡x2xdydx.431. The area of one leaf of the three-leaved rose bounded by the graph of r =5sin3µ isA.5¼6B.25¼12C.25¼6D.5¼3E.25¼3.32. Find the area of the portion of the plane x +3y +2z = 6 that lies in the ¯rst octant.A. 3p11 B. 6p7C.6p14 D. 3p14 E. 6p11.33. A solid region in the ¯rst octant is bounded