Calculus III: Practice Final - Columbia University

1 Calculus III: Practice FinalName:Circle one:Section 6 Section 7. Read the problems carefully. Show your work unless asked otherwise. Partial credit will be given for incomplete work. The exam contains 10 problems. The last page is the formula sheet, which you may detach. Good luck!Question:12345678910 TotalPoints:10101010101010101010100 Score:1 Calc III (Spring 13) Practice FinalPage 2 of 121. (10 points) Circle True or False. No justifation is needed.(a) The curve traced by cos2(t),sin2(t) is a (b) The plane 3x+ 2y z= 0 is perpendicular to the linex= 3t,y= 2t,z= (c) The functionf(x,y,z) ={sin(x+y+z)x+y+zifx+y+z6= 01ifx+y+z= 0is continuous at (0,0,0).TrueFalse(d) If the acceleration is constant, then the trajectory must be a straight (e) The complex numbere2+3ihas magnitude III (Spring 13) Practice FinalPage 3 of 122. In the following, computeV W,V W, and the cosine of the angle betweenVandW.}

2 (a) (5 points)V= 2, 1,1 , W= 1,3, 2 .(b) (5 points)V=i+ 3j, W= 3j III (Spring 13) Practice FinalPage 4 of 123. Use the contour plot off(x,y) to answer the questions. No justification is needed.(a) (3 points) Mark any three critical points off. Label themA,B, andC. Identifywhether they are local minima, local maxima or saddle points.(b) (2 points) Draw a vector at (1,1) indicating the direction of fat (1,1).(c) (3 points) Determine the sign of1. f x(3,4):2. f y(2,3) (5,3) whereuis the South East direction:(d) (2 points) Give a (admittedly rough) numerical estimate of f x(1,1).Calc III (Spring 13) Practice FinalPage 5 of 124. (a) (5 points) Write parametric equations for the tangent line at 1,0,1 to the curvetraced by t2,lnt,t3 .(b) (5 points) Write an equation of the normal plane to the curve at the same III (Spring 13) Practice FinalPage 6 of 125.

3 (a) (5 points) Letf(x,y) =xy/(x2+y2). Find lim(x,y) (0,0)f(x,y) or show that thelimit does not exist.(b) (5 points) Letf(x,y) = xy0et2dt. Find f xand f III (Spring 13) Practice FinalPage 7 of 126. A ball of unit mass is thrown with the initial velocity ofi+j. It experiences the forceof gravity of magnitude 10 units in the jdirection and a force due to the wind ofmagnitude 1 unit in theidirection. Suppose the ball is initially at (0,4).(a) (7 points) Find the position of the ball at timet.(b) (3 points) Where is the ball when it hits the ground?Calc III (Spring 13) Practice FinalPage 8 of 127. Supposeu=exywherex=st+s+tandy=st s t.(a) (2 points) Find the value ofuwhens= 2 andt= 2.(b) (8 points) Find an approximate numerical value ofuwhens= andt= III (Spring 13) Practice FinalPage 9 of 128. (10 points) Find all the critical points of the functionf(x,y) =x3+y3 3xy.

4 Determineif they are local maxima, local minima or saddle III (Spring 13) Practice FinalPage 10 of 129. (10 points) A race track is in the shape ofan ellipse with minor radius 1 km and ma-jor radius 2 km as shown. A car is goingalong this track at a constant speed of 100km/h. Find the tangent and normal com-ponent of its accelaration when it as track21 PCalc III (Spring 13) Practice FinalPage 11 of 1210. (10 points) You want to design a cylindrical cup that can hold 100 ml coffee. Tominimize the material to be used, you decide to minimize the surface area. What is theradius and height of the optimal cup? (Ignore the thickness of the walls.)LIST OF USEFUL (1)ddxxn=nxn 1(2)ddxsinx= cosx(3)ddxcosx= sinx(4)ddxtanx= sec2x(5)ddxcotx= csc2x(6)ddxsecx= secxtanx(7)ddxcscx= cscxcotx(8)ddxex=ex(9)ddxln|x|=1x(10)ddx arcsinx=1 1 x2(11)ddxarccosx= 1 1 x2(12)ddxarctanx=11+ (1) sin2x+ cos2x= 1(2) tan2x+ 1 = sec2x(3) 1 + cot2x= csc2x(4) sin(x y) = sinxcosy cosxsiny(5) cos(x y) = cosxcosy sinxsiny(6) sin2x=1 cos 2x2(7) cos2x=1+cos curvesFor a parametric space curve given byr(t)(1) Curvature =|r (t) r (t)||r (t)|3.

5 (2) Tangent component of accelerationaT=|r (t)| =r (t) r (t)|r (t)|.(3) Normal component of accelerationaN= |r (t)|2=|r (t) r (t)||r (t)|.1