Calculus, Midterm 3, Wednesday May 21, 2014 Instructor: Frank Liou
Exam Time: 8:10 am to 9:10 am Total Score: Points
Name:
Student ID:
TA’s Name:
(1) Please do not turn this page until told to do so.
(2) You can use either Chinese or English to write this exam.
(3) You can use pen or pencils. You have to write everything clearly.
(4) You cannot use your calculator, iPhone, iPad or any other electronic devices during this exam.
(5) Please place your personal belongings under your seat to give free way to the ex- aminers.
(6) No notes, books, or classmates may be used as resources for this exam. It is a violation of the University honor code too, in any way, assist another person in the completion of this exam. Please place your own work covered up as much as possible during the exam that the others will not be tempted or distracted. Thank you for your cooperation.
(7) Read directions to each problem carefully. Show all work for full credit. In most cases, with no supporting work will not receive full credit. The best way to get maximum partial credit is to write neatly and be organized.
(8) If you use any result from the homework, you need to prove it.
(9) Make sure you have 10pages, including the cover page.
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Total ( Points)
Score ( Points)
(1) (20 Points) Letz= exp 1
x2+ 1
cosπy.
(a) (8 Points) Find ∂z
∂x and ∂z
∂y atx= 0 and y= 1/2.
(b) (12 Points) Let x = u
u2+v3+ 1 and y = v
u+v4+ 1.Find ∂z
∂u and ∂z
∂v at the pointu= 0 and v= 1.
(2) (10 Points) Letf(x, y) = sin(x−y) exp −x2−y2
andP(√ 2,√
2).
(a) (7 Points) Find the maximum rate of change of f(x, y) at P.
(b) (3 Points) Find the directional derivative of f(x, y) at P alongi+√ 3j.
(3) (10 Points) Find the tangent plane and the normal line to the level surface x2+ y2+z2 = 6xyz−3 atP(−1,1,−1).You need to verify that the point P lies on the level surface.
(4) (15 Points) Find parametric equation for the line tangent to the curve of the inter- section of the level surfacesx3+ 3y2+y3+ 4xy−z2= 0 and x2+y2+z2= 11 at P(1,1,3).You need to verify that the point lies on both of the surface.
(5) (20 Points) The intersection of the planex+y+2z= 2 and the paraboloidz=x2+y2 is an ellipse. Use the method of Lagrange multiplier to find the points on this ellipse that are nearest to and farthest from the the origin. You will receive no credit if you use other method.
(6) (20 Points) Letf(x, y) =xyexp −x2−y2 . (a) (5 Points) Find all critical points off(x, y).
(b) (5 Points) Find D=fxxfyy−(fxy)2.
(c) (10 Points) Classify all critical points of f(x, y).
(This page is blank).
(7) (Bonus: 20 Points) Evaluate the following the double integrals.
(a) (10 Points) Z Z
R
sinx
x dA, where R is the triangle in the xy-plane bounded by thex-axis, the line y=x and the line x=π.
(b) (10 Points) Z Z
D
ex2+y2dA, whereD is the region{(x, y)∈R2 :x2+y2 ≤1}.
