2018 final exam pdf PDF

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The final exam of math 1920 multivariable for engineers in 2018...

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Math 1920

Name

Final exam

13 December 2018

9:00–11:30 am

Place an X in the box by your discussion section number

          

201 202 203 204 205 206 207 208 209 210 211

Joseph Fluegemann Joseph Fluegemann Romin Abdolahzadi Gokul Nair Manki Kim Andres Fernandez Andrew Horning Thomas Reeves Romin Abdolahzadi Gokul Nair Manki Kim

MW 7:30–8:20 pm MW 8:35–9:25 pm TR 8:00–8:50 am TR 8:00–8:50 am TR 8:00–8:50 am TR 8:00–8:50 am TR 8:00–8:50 am TR 8:00–8:50 am TR 9:05–9:55 am TR 9:05–9:55 am TR 9:05–9:55 am

          

212 213 214 215 216 217 218 219 220 221 222

Thomas Reeves Andres Fernandez Andrew Horning Prairie Wentworth-Nice Prairie Wentworth-Nice Chaitanya Tappu Ilya Amburg Chaitanya Tappu Ilya Amburg Itamar Oliveira Itamar Oliveira

Instructions—please read now • Write your name and check the box with your discussion section number right now. • There are 9 problems and this booklet has 13 sheets. You may use the back of each sheet as scratch paper. • You have 150 minutes to complete the exam. You may leave early, but if you finish within the last 15 minutes, please remain in your seat. • Show your work and simplify your answers. To receive full credit, your answers must be neatly written and logically organized. • You are allowed a two-sided letter size sheet of notes. No books or electronic devices or any other resources are allowed. • Academic integrity is expected of all Cornell University students at all times, whether in the presence or absence of members of the faculty. You may not give, use, or receive unauthorized aid in this examination. Please sign below to indicate that you have read and agree to these instructions.

Signature

TR 9:05–9:55 am TR 9:05–9:55 am TR 9:05–9:55 am MW 7:30–8:20 pm MW 8:35–9:25 pm TR 12:20–1:10 pm TR 12:20–1:10 pm TR 1:25-2:15 pm TR 1:25-2:15 pm TR 1:25–2:15 pm TR 2:30–3:20 pm

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Math 1920

13 December 2018

Final exam

2

1 (10 points). Suppose a function  y z g (u, v ) satisfies  g(0, 1)  3 and ∇g(0, 1)  h2, 1i. Define a function f (x, y, z) by f (x, y, z )  g xe , y cos(x + z ) . Let P  (0, 1, 0). (a) Find ∇ fP .

(b) Find the equation of the tangent plane at P to the surface f (x, y, z)  3.

Math 1920

13 December 2018

2 (8 points). Evaluate the double integral

∫

1 0

∫

Final exam 1 10

√3

x 2 e y dy dx. x

3

Math 1920

13 December 2018

Final exam

4

3 (12 points). Let W be the solid region given in spherical coordinates by 1 ≤ ρ ≤ R, 12 π ≤ φ ≤ π, 0 ≤ θ ≤ 2π, where R > 1 is a constant. (a) Sketch the solid W . (b) Find the centroid of W . (Recall that the centroid of a solid is the center of mass with respect to a constant mass density.)

Math 1920

13 December 2018

Final exam

5

4 (12 points). Determine the local minima, local maxima and saddle points of the function f (x, y)  (x 2 + y 2 )2 − 2c 2 (x 2 − y 2 ), where c > 0 is a constant. (The answers depend on c .)

Math 1920

13 December 2018

Final exam

6

5 (12 points). Let D be the portion of the unit sphere x 2 + y 2 + z 2  1 contained in the octant x ≥ 0, y ≥ 0, z ≥ 0. Let a, b, and c be positive constants. What is the maximum value of the function f (x, y, z )  x 2a y 2b z 2c over the region D? At what point of D is this maximum attained? (The answers depend on a, b, c .)

Math 1920

13 December 2018

Final exam

7

6 (12 points). Let G(u, v )  (uv −1 , u 2 v ). Let D0 be the square in the uv-plane given by 1 ≤ u ≤ 2, 1 ≤ v ≤ 2. (a) Sketch the region D  G(D0 ), i.e. the region in the x y-plane given by x  uv −1 , y  u 2 v , 1 ≤ u ≤ 2, 1 ≤ v ≤ 2. (b) Use G to evaluate the integral

∬

D

x y dx dy.

Math 1920

13 December 2018

Final exam

8

7 (16 points). Let a, b , c , d be constants and let F be the vector field F(x, y) 

1 hax + b y, cx + dyi. x2 + y2

The domain D of F is { (x, y) ∈ R2 | (x, y) , (0, 0) } , the plane with the origin removed. (a) Is D simply connected? Explain your answer. (b) Calculate curlz (F) 

∂F2 ∂F1 . − ∂y ∂x

(c) Let C be the circle of radius 1 about the origin oriented counterclockwise. Calculate

∫

C

For what values of a, b , c , d is the integral equal to 0? (d) For what values of a, b , c , d is F conservative? For those values, find a potential of F. (You may continue your solution on the next sheet.)

F · dr.

Math 1920

13 December 2018

Use this space to continue your solution of Problem 7.

Final exam

9

Math 1920

13 December 2018

Final exam

10

8 (20 points). Let S be the surface in R3 defined by x 2 + y 2 − z 2  1 and 0 ≤ z ≤ h, where h is a positive constant. Let S be oriented by the outward pointing normal vector. (a) Sketch the surface S . In your sketch indicate the orientation of the boundary of S . (b) Parametrize the surface S . Specify the domain D of the parametrization. (c) Calculate the outward pointing normal vector field N to the surface. (d) Verify Stokes’ Theorem for the surface S and the vector field F(x, y, z)  h2yz, 0, x yi . (You may continue your solution on the next sheet.)

Math 1920

13 December 2018

Use this space to continue your solution of Problem 8.

Final exam

11

Math 1920

13 December 2018

9 (18 points). Let F be the vector field defined by F 

Final exam

12

1 hx, y, zi. (x 2 + y 2 + z 2 )3/2

(a) Compute the upward flux of F through the surface S0 defined by x 2 + y 2 ≤ R2 , z  h, where R > 0 and h > 0 are constant. (Continued on next page.)

Math 1920 9 (continued).

13 December 2018

Final exam

13

(b) Compute ∇ · F.

(c) Compute the upward flux of F through the surface S defined by z  e 1−x

2 −y 2

, x 2 + y 2 ≤ 1....