Math 200Exam 2

May 29th, 2019

Name:

Section:

? For free response questions, you must show all work. Answers without proper justifica-tion will not receive full credit. Partial credit will be awarded for significant progress towardsthe correct answer. Cross o↵ any work that you do not want graded.

? This is a closed-book exam. You may not use any books or notes on this exam.

? You have 50 minutes to complete this exam. When time is called, stop writingimmediately and turn in your exam at the front of the room - be sure to place your exam inthe appropriate folder.

? You may not use any electronic devices including (but not limited to) calculators,cell phones, or tablets. Using such a device will be considered a violation of the university’sacademic integrity policy and, at the very least, will result in a grade of 0 for this exam.

Page: 2 3 4 5 6 7 8 Total

Points: 20 20 15 15 10 10 10 100

Score:

Solutions

Math 200 Exam 2 - Page 2 of 9 5/29/2019

1. (20 points) Find the absolute extrema for the function f(x, y) = 2xy � x � y over the regionbounded between y = x2 and y = x. State the extreme values and the points where they occur.

Region CriticalPointsti f 2y I

b 4 2 1b

cool 7 crit.pt EI

Boundary 1 y x'i 0,1 Boundary 2 y x o i

b x _2x x x x b x 2x x x x2 3 x 2

2 2 2xb x _6 2 l 2xb x 4 2

cp cp x I E I12 plugintof only f y x togetyual

testis in R

f ya4 Yeft's1 f Fg lo 17ftf 1,1 O

f o.o O

Absmax 0 at 1,1 and o.oAbsmin lot 7 F at Fo E Fg

Math 200 Exam 2 - Page 3 of 9 5/29/2019

2. (20 points) Identify all critical points of the function f(x, y) = x2y � 2y2 � x2. Then, classifyeach critical point as a relative maximum, relative minimum, or saddle point.

f 2xy Zx fy x 4y2xy 2x o 2 4 1 02 4y o X oor

sub.in

ofyxo O24y O yoyx2 4 o x I2

CriticalPoints o.o 2,1 C2,1

f 2y 2 fyy 4 f y 2x fy

x y f fy f D Type

to 4 4 16 Saddle

Math 200 Exam 2 - Page 4 of 9 5/29/2019

3. (15 points) Consider the region R = {(x, y) : 1 x2 + y2 9 and 0 y x}, shown below.

Use polor coordinates to evaluate the double integral

ZZ

R

1px2 + y2

dA

ConvertegR I E x y E 9 IE r e 9 r 1 to r 3

y O O O

y x 0 17 y x It I tan0 1Integrand f

ftp dA oF4 rI.rdrdO

Idrdot

Math 200 Exam 2 - Page 5 of 9 5/29/2019

4. Consider the surface

S : z = x2 � 2y2 + 3

(a) (10 points) Find an equation for the tangent plane to S at the point P (1, 1).

(b) (5 points) Find a set of parametric equations for the normal line to S at P .

2 I l I 2 3 2

z xZ Zy 13 XZZyl z 13 0

LetF x yZ 2y22 3

TF 2x 4y 1

TF 1,1 2 2 4 1

Tangentplane 2 x D 4 y i z 2 o

2x 4y z 4

iii

Math 200 Exam 2 - Page 6 of 9 5/29/2019

5. (5 points) Which of the following is the local linear approximation for f(x, y) = 2yex at thepoint P (0, 2)?

(a) L(x, y) = 2y

(b) L(x, y) = 4x+ 2y + 8

(c) L(x, y) = 2x+ 2y

(d) L(x, y) = 4x+ 2y

(e) L(x, y) = 4x� 2y

6. (5 points) Suppose f is a di↵erentiable function of x and y, and define g(u, v) = f(2u+v, u+v2).

Use the table of values shown below to calculate@g

@v

����(u,v)=(1,2)

.

(x, y) f g fx fy

(1, 2) 2 4 -1 3(4, 5) 3 -2 5 2(5, 4) 2 5 -2 4

(a) -8

(b) -4

(c) 11

(d) 12

(e) 13

f 2ye fy 2e

1 0,21 4 116,2 2

f6,4 4

x y 4 x o 12 y 2 4

4 2 1

fx Zun

Y u uz4 4

y 1,4 5

C II I zu Elen 4

2 y5C 1214

Math 200 Exam 2 - Page 7 of 9 5/29/2019

7. (5 points) Consider the function f(x, y) = 2x ln(y)+x. Compute the directional derivative forf at the point P (�1, 1) in the direction of the vector ~v = h3,�4i.

(a) 1/5

(b) 1

(c) 11/5

(d) 11

(e) 17

8. (5 points) Consider the function f(x, y) = x2 sin y + 3x. Which of the following vectors pointsin the direction in which f increases the fastest at the point P (�1, 0)?

(a) h�3,�1i

(b) h�3, 1i

(c) h1, 3i

(d) h3, 1i

(e) h4, 1i

Tf any I

ofCi i L l 2

Ia ET l3j

Defti D f Li 2 3 4

I5

Tf say 3 xkosy

Cy0ft o 3 i

Math 200 Exam 2 - Page 8 of 9 5/29/2019

9. (5 points) Which of the following double integrals is equal to

Z 4

0

Z px

x/2f(x, y) dydx?

(a)

Z px

x/2

Z 4

0f(x, y) dxdy

(b)

Z 4

0

Z py

y/2f(x, y) dxdy

(c)

Z 2

0

Z 2y

y2f(x, y) dxdy

(d)

Z 2

0

Z y2

2yf(x, y) dxdy

(e)

Z 4

0

Z 2y

y2f(x, y) dxdy

10. (5 points) Which of the following double integrals is equal to

Z ln 2

0

Z 2

eyf(x, y) dxdy?

(a)

Z 2

ey

Z ln 2

0f(x, y) dydx

(b)

Z ln 2

0

Z 2

exf(x, y) dydx

(c)

Z ln 2

0

Z 2

lnxf(x, y) dydx

(d)

Z 2

0

Z lnx

0f(x, y) dydx

(e)

Z 2

1

Z lnx

0f(x, y) dydx

h

D

Math 200 Exam 2 - Page 9 of 9 5/29/2019

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