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1 let's MASSACHUSETTS INSTITUTE OF TECHNOLOGY INTERPHASE CALCULUS III MIDTERM I INSTRUCTOR: SAMUEL S WATSON s ~ Problem Find the distance between the planes 3x + 2y + z 0 and 3x + 2y + z 6 Since 32 ) is parallel to 32/) the planes are parallel Therefore the distance to 3xt2y +26 is the same for any point on 3x+2y+zO take 00 o) ) pin PJ let Obetheangle it 32 between D and where P Coin ) is on 3*3+26 IPJKOSO ggg/f*q#ay3iddip0lswl9e0 e then 307 C3 2 ) 6 if at 6ft Problem 2 The position of a particle at time t is given by rt) h2t 2 t t2 i Find the positive time t when the velocity of the particle is perpendicular to its acceleration You may express your answer as a radical 2 2T wet%y+) 2 ) F'K7 o ro " Ct ) 0 +4T when t 652 qzy

2 Problem 3 Suppose that ABC is a triangle Define M to be the point which is twothirds of the way from A to the midpoint of BC Define N to be the point which is twothirds of the way from B to the midpoint of AC! a) Express the vector AM in terms of the vectors u AB! and v AC!! b) Express AN in terms of u and v c) What is the distance between M and N? B B s p s E i*fe Etn#* A C A C EAT it * 8 a) AT b) c) MT APT + AT # + at ) } AT BF It I + It }t±u thu#tsutts0 IT st e) Eu +8 ) AT 8 so Mt 0 Problem 4 Find a vector which is perpendicular to the line represented parametrically by h2 parametrically by h t3ti t t4i and the line represented vectors parallel to these lines are /0) & 0 3) so : F tie 0 ) < 0 e 3) /}? Y 33 es is orthogonal to both 33 )

3 Problem 5 Region A consists of all the points in 3D space satisfying the spherical coordinate in equalities Ω apple 0 apple µ apple º and 0 apple apple º 4 Region B consists of all the points in 3D space satisfying the cylindrical coordinate inequality r apple p 2 0 apple µ apple º and 0 apple z apple Without calculating any volumes which region is larger? Sketch both regions carefully and explain how you can be sure one is larger than the other without calculating the volume of either #? AE#n:IEFEIeet:IEn so B 's volume a eager B Problem 6 xy 3 a) Find lim xy)!00) x 2 and b) explain a winning strategy for the limit game to demonstrate that your answer to + y2 a) is correct Note: for b) this means saying how ± should be chosen in terms of to ensure that the function is within of its limit for all x y) within ± of 0 0)) a) y g r4osf mh fast sin O in polar & this Is of the form something going to 0) something bounded) b) since the adversary 's choice of choosing 8 Fe gives Iflxg) r2cososin3o E r2 < < ret E so the limit is # r is constrained by r < 8 for all K g) within or of 007

4 Problem 7 x3 + 3x + y4 a) Find the critical points of f x y) which depicts some level curves of f ) 2y2 and place dots at those locations in the figure below b) To find the maximum value of f x y) for any point x y) satisfying 2 x 2 and to look at the values of f at the critical points found in a)? Explain why or why not 2 y 2 is it sufficient If a) when e { critical points b) No D y3 4g) Boundary El above shown critical y { and } to 0 So six have we must also be points } taken account into Problem 8 Find a vector which is tangent to the hyperboloid x2 vector normal to the hyperboloid at that point y2 + z2 8 at the point x y z) 3 4 5) Hint: first find the 2_y2ez2 ) a dots the normal vector to 445) with plane Zx 2 y 2g 2Z) to give 0 & so 2<3 at is tangent ) so so ) 30 ) is eg to 4 [ or any 30 ) J 345)0 ] with Tto

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