MA 261 EXAM 1 Name1. Find the equation of the plane containing (0, 1, 2) and whose normal is perpendicularto both ¯a =¯i +¯j,¯b =¯j −¯k.A. x + y + z =3B. −x + y + z =3C. x − y − z =3D. x + y + z = −3E. None of the above2. The distance between the plane2x + y +2z =4and the point (1, 7, 2) isA. 1B. 2C. 3D. 4E. None of the above1MA 261 TEST 1 Name3. A unit tangent vector to the graph of y =2x3at (1, 2) is given byA.¯i +6¯j√37B.¯i +4¯j√17C.¯i −¯j√2D.2¯i +3¯j√13E.¯i +2¯j√54. A particle is moving with acceleration 4¯j +6t¯k. If the position at time t =1is¯r(1) =¯i +3¯j +¯k and the velocity at time t =0is¯v(0) =¯i +¯j, then the position attime t =2isA. 4¯i +10¯j +10¯kB.¯i +4¯j +10¯kC.¯i +83¯j +4¯kD. 2¯i +10¯j +8¯kE. 2¯i +8¯j +8¯k2MA 261 TEST 1 Name5. Which of the following surfaces represents the graph of z =x24+ y2in the 1st octant.3MA 261 TEST 1 Name6. If f (x, y)=3x2+ yxx2+ y2,(x, y) 6=(0, 0), let ` be the limit of f (x, y)as(x, y) → (0, 0)along the y-axis, and let m be the limit of f (x, y)as(x, y) → (0, 0) along the liney = x.ThenA. ` =3,m=2B. ` =0,m=2C. ` =0,m=32D. ` =3,m=3E. ` =12,m=127. Find a value of a for which the function z = 4 cos(x + ay) satisfies∂2z∂y2=9∂2z∂x2.A. a =2B. a =0C. a =12D. a =1E. a =34MA 261 TEST 1 Name8. Find the maximal directional derivative off(x, y, z)=ex+ ey+ e2zat (1, 1, −1).A. e√3 − 2eB.√2e2+4e−4C.1e√2 − 4e−3D.√2e2+ e−4E.√e2+2e−49. Find symmetric equations of the line containing (1, 2, 3) and perpendicular to theplane 2x +3y − z =8.5MA 261 TEST 1 Name10. Find the length of the curve¯r(t)=t22¯i +7¯j +t33¯k,0≤ t ≤ 2.11.(a) Complete the following definition of fyat (0, 0):fy(0, 0) = limh→0(b) If f (x, y)=(x+y33x2+4y2, (x, y) 6=(0, 0)0, (x, y)=(0, 0), compute fy(0, 0) by evaluating theabove limit.fy(0, 0) =6MA 261 TEST 1 Name12. A right circular cylinder has a radius and altitude that vary with time. At a certaininstant the altitude is increasing at 0.5 ft/sec and the radius is decreasing at 0.2 ft/sec.How fast is the volume changing if at this time the radius is 20 feet and the altitudeis 60
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