math 200 - drexel universitydp399/math200/math200_201835_ex… · math 200 exam 2 - page 2 of 9...
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Math 200Exam 2
May 29th, 2019
Name:
Section:
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? 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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