MATH 127 - Final Exam

Name (in print):

Circle your laboratory section:

Satbir Malhi Satbir Malhi Joseph Brennan Connor SmithTR 8 AM TR 9 AM WF 9 AM WF 10 AM

Arturo Gil Iurii Posukhovskyi Arturo Gil Iurii PosukhovskyWF 11 AM TR 12 PM WF 12 PM TR 1 PM

Hamid Mofidi Yufei Yu Hamid Mofidi Yufei YuTR 2 PM MW 3 PM TR 3 PM MW 4 PM

Exam Instructions:

Date and Time: Wednesday, December 13th, 2017 at 4:30-7:00pm.

Calculator: Scientific Calculator (no programmable graphing calculators).

You may not use your phone as a calculator

• Write all of the work on this exam - nothing else will be graded. You must show yourwork to earn credit. Your work must be legible and any work which you do not wantgraded must be scratched or clearly crossed out.

• This exam is closed book and no notes will be permitted. Cell phones, computers andelectronic devices are not permitted. Each student must be prepared to produce, uponrequest, a card with a photograph for identification. Once you have completed theexam, find the graduate teaching assistant who teaches your laboratory section andturn the exam in to them.

• Exams will be scanned after grading to ensure grade corrections are accurate.

For Instructor’s Use Only:Number: MC 6 7 8 9 10 11 12 13 Bonus Total

Grade:Max Grade: 12 10 8 10 8 16 16 8 21 10 100

1

Multiple Choice(Circle the correct answer)

Two points each

1. Circle ALL the limits that do not exist. (4 points)

(a) lim(x,y)!(0,0)

2x2 + 3y2

xy

(b) lim(x,y)!(0,0)

x3

x2 + y2

(c) lim(x,y)!(0,0)

x2

x2 + y2

(d) lim(x,y)!(0,0)

xy

3x2 + 2y2

2. Find an equation of the tangent line to the curve of intersection of the surface f(x, y) =9� x2 � y2 with the plane y = 1 at the point (1,1).

(a) y = 1

(b) z � 7 = �2(x� 1)

(c) z � 7 = 7(x� 1)

(d) z � 7 = 6(x� 1)

2

3. Find the direction for which the directional derivative of f(x, y) = x2 � 3xy + 2y2 at(1,-1) is a maximum

(a)

*5p74

74,�7

p74

74

+

(b)

⌧5

12,� 7

12

�

(c)

*�p2

2,�

p2

2

+

(d)

*�p2

10,�7

p2

10

+

4. Let f(x, y, z) = ex cos y + z, x = s2t5, y =s

tand z = s2. Find

@f

@twhen s = 2⇡ and

t = 1

(a) 0

(b) 40⇡2e4⇡2

(c) 20⇡2e4⇡2

(d) e4⇡2

5. Find the local maxima, local minima, and saddle points, if any, for z = x3 + 6xy + y3

(a) (0,0) is a saddle point; (-2,-2) is a local maximum

(b) (0,0) is a saddle point; (-1,-1) is a local maximum

(c) (0,0) is a local minimum; (-2,-2) is a saddle point

(d) (0,0) is a local minimum; (-1,-1) is a saddle point

3

6. Suppose f is di↵erentiable function such that

f(1, 3) = 1, fx(1, 3) = 2, fy(1, 3) = 4

fxx(1, 3) = 2, fxy(1, 3) = �1, and fyy(1, 3) = 4

(a) (2 points) Find rf(1, 3)

(b) (4 points) At the point (1,3) what is the rate of change of f in the direction ofh1, 1i.

(c) (4 points) Find the linear approximation of f(1.2, 2.9).

4

7. (8 points) Use the Change of Variables formula and the map G(u, v) = (u� v, u + v)

to compute

ZZ

D

x + y dA where D is the region with boundaries y = x, y = 3x, and

x+ y = 4.

5

8. (5 points each) Rewrite each of the double integrals below in the form specified. (Ifnecessary, break into several integrals.)

(a)

Z 2

0

Z x2

0

f(x, y) dy dx; change the order of integration to dx dy

(b)

Z ⇡2

0

Z 2

1

f(r, ✓) dr d✓; change to rectangular form.

6

9. (8 points) Use multiple integrals to find the volume of the tetrahedra with vertices(1,0,1), (0,0,0), (0,0,1), (0,1,1).

7

10. (8 points each) Compute the following line integrals.

(a)

Z

C

(x2 + y2) dx� 2xy dy where C is the boundary of the triangle formed by the

lines x = 0, y = 1 and y � x = 0 oriented clockwise.

(b)

Z

C

x5 ds where C is the arc of y =1

4x4 from

✓1,

1

4

◆to (2,4).

8

11. (8 points each) Use Green’s Theorem, Stokes’ Theorem or the Divergence Theorem tocompute the following:

(a) (8 points) Find the flux of F(x, y, z) = hxy2, yz2, zx2i through the solid surface Swith outward pointing normals where S is region that lies between the cylindersx2 + y2 = 4 and x2 + y2 = 9 and between the planes z = �1 and z = 1.

9

(b) Evaluate

Z

C

F · dr where F(x, y, z) = hxy, 3z, 2yi and C is the intersection of the

plane x + z = 4 and the cylinder x2 + y2 = 9 oriented counterclockwise whenviewed from above.

10

12. (8 points) Find the maximum and minimum values of f(x, y, z) = xyz on the spherex2 + y2 + z2 = 12

11

13. Consider the vector field F(x, y) = h3x2z, z2, x3 + 2yzi

(a) (3 points) Show that F is conservative.

(b) (4 points) Find the potential function for f(x, y) for F such that F = rf

(c) (6 points) If F is a force field, find the work done by F along the curve C

parametrized by r(t) =

⌧ln t

ln 2, t

32 , t cos(⇡t)

�for 1 t 4

12

Bonus:(10 points)

Let F = hF1, F2, F3i be any infinitely di↵erentiable vector field. Show that div(curl(F)) = 0

13