Math 227
Extra Practice Problems for Test #2
B)
Find the domain and range and describe the level curves
for the function f(x,y).
1) f(x, y) = 9x + 8y
2) f(x, y) = 1
6x2 + 9y2
3) f(x, y) = ln (5x + 3y)
4) f(x, y) = 36 - x2 - y2
5) f(x, y) = C)
5x2
y
Match the surface show below to the graph of its level
curves.
6)
D)
A)
1
7)
C)
D)
A)
8)
B)
A)
2
11) Find an equation for the level surface of the
function f(x, y, z) = x + ey+z that passes
B)
through the point 1, ln(5), ln(8) .
Find the limit.
C)
12)
4x2 + 4y2 + 2
lim
(x, y) → (0, 0) 4x2 - 4y2 + 1
13)
lim
(x, y) → (0, 1)
14)
2 3
lim
- (x, y) → (5, 2) x y
15)
lim
x ln y
(x, y) → (4, 3)
16)
lim
-5xz - 2xy
P → (1, - 1, 0) x2 + y2 - z 2
17)
lim
ln z x2 + y2
P → (5, 5, 1)
y5 sin x
x
D)
At what points is the given function continuous?
18) f(x, y) = ex+y
19) f(x, y) = xy
x + y
20) f(x, y) = x - y
2
2x + x - 6
21) f(x, y, z) = yz cos Solve the problem.
9) Find an equation for the level curve of the
function f(x, y) = 64 - x2 - y2 that passes
through the point 1
x
Find two paths of approach from which one can conclude
that the function has no limit as (x, y) approaches (0, 0).
3y
22) f(x, y) = 3x2 + 3y2
3, 2 .
10) Find an equation for the level curve of the
function f(x, y) = x2 + y2 that passes through
23) f(x, y) = the point 3, 4 .
3
x3 + y6
x3
24) f(x, y) = 2
40) f(x, y) = xy2 + yex + 5
xy
x2 + y2
41) f(x, y) = Find all the first order partial derivatives for the
following function.
25) f(x, y) = 2x - 2y2 - 7
Use the limit definition of the partial derivative to
compute the indicated partial derivative of the function at
the specified point.
∂f
42) Find at the point (4, 8): f(x, y) = 7x2 + 4xy +
∂x
2
26) f(x, y) = (8x5 y2 + 4)
27) f(x, y) = 1
x2 + y2
5y2
43) Find y3
28) f(x, y) = ln x5
29) f(x, y) = 30) f(x, y) = x
x + y
∂f
at the point (-3, 7): f(x, y) = 4 - 10xy +
∂y
2xy2
e-x
x2 + y2
44) Find ∂f
at the point (-9, -5, 10): f(x, y, z) = 4
∂y
x2 y + 6y2 + 3z
x
x + y
Solve the problem.
45) Evaluate 31) f(x, y) = x3 + 6x2 y + 9xy3
y) = x2 - y2 + 10x; x = cost, y = sin t.
32) f(x, y, z) = x2 y + y2 z + xz 2
46) Evaluate 33) f(x, y, z) = ln (xy)z
z) = 47) Evaluate 2
2
2
35) f(x,y,z) = xe(x + y + z )
3
dw
at t = π for the function w(x, y,
2
dt
xy
; x = sin t, y = cost, z = t2 .
z
34) f(x ,y, z) = (sin xy)(cos yz 2 )
36) f(x, y, z) = dw
1
at t = π for the function w(x,
dt
2
dw
at t = 8 for the function w(x, y) =
dt
ey - ln x; x = t2 , y = ln t.
cos y
xz 2
48) Evaluate dw
at t = 2 for the function w(x, y, z)
dt
2
1
= exyz ; x = t, y = t, z = .
t
Find all the second order partial derivatives of the given
function.
37) f(x, y) = x2 + y - ex+y
49) Evaluate ∂w
at (u,v) = (1, 4) for the function
∂u
w(x, y) = xy - y2 ; x = u - v, y = uv.
38) f(x, y) = cos xy2
39) f(x, y) = x ln (y - x)
4
50) Evaluate ∂w
at (u, v) = (1, 5) for the function
∂v
60) Find - 2yz 2 - 7ez = 0.
w(x, y) = xy2 - ln x; x = eu+v, y = uv.
51) Evaluate ∂w
at (u, v) = (1, 5) for the function
∂u
61) Find w(x, y, z) = xz + yz - z 2 ; x = uv, y = uv, z = u.
52) Evaluate yz
∂z
at the point (6, 1, -1) for ln x
∂y
2
exy+z = 0.
∂u
at (x, y, z) = (5, 4, 5) for the
∂x
62) Find function u(p, q, r) = p2 - q2 - r; p = xy, q = y2 ,
r = xz.
π
2
∂x
at the point 1, , 7 for ex cos yz =
28
∂y
0.
Provide an appropriate answer.
∂w
63) Find when r = -4 and s = 1 if w(x, y, z) =
∂r
∂u
53) Evaluate at (x, y, z) = (5, 4, 5) for the
∂z
function u(p, q, r) = p2 q2 - r; p = y - z,
q = x + z, r = x + y.
54) Evaluate ∂y
at the point (4, 6, 4) for -6x2 + 4 ln xz
∂x
xz + y^2, x = 3r + 1, y = r + s, and z = r - s.
64) Find ∂u
at (x, y, z) = (2, 2, 0) for the
∂y
∂w
when u = -3 and v = 4 if w(x, y, z) =
∂u
u
xy2
, x = , y = u + v, and z = u · v.
v
z
1
function u(p, q, r) = epq cos(r); p = ,
x
q = x2 ln y, r = z.
65) Find 11π
∂z
when u = 0 and v = if z(x, y) =
2
∂v
sin x + cos y, x = u·v, and y = u + v.
Use implicit differentiation to find the specified
derivative at the given point.
dy
55) Find at the point (1, 1) for 3x2 + 2y3 + 2xy
dx
Compute the gradient of the function at the given point.
66) f(x, y) = -6x + 3y, (-6, -3)
= 0.
67) f(x, y) = 2x2 - 8y, (4, 8)
dy
3
56) Find at the point (-1, 1) for 6x - + 7x2 y2
dx
y
68) f(x, y) = ln(-8x - 9y), (8, -5)
= 0.
57) Find -3x
69) f(x, y) = tan-1 , (-2, 9)
y
dy
at the point (2, 1) for ln x + xy2 + ln y
dx
= 0.
70) f(x, y, z) = 3x + 10y + 2z, (-10, -7, -5)
58) Find dy
at the point (1, 0) for cos xy + yex = 0.
dx
59) Find dy
at the point (1, -1) for -5xy2 + 4x2 y
dx
71) f(x, y, z) = ln(x2 + 3y2 + 2z 2 ), (3, 3, 3)
Find the derivative of the function at the given point in
the direction of A.
72) f(x, y) = 10x + 6y, (3, -5), A = 4i - 3j
- 2x = 0.
5
73) f(x, y) = 6x2 - 3y, (-6, 7), A = 3i - 4j
85) Find the derivative of the function
x y z
f(x, y, z) = + + at the point (8, -8, 8) in
y z x
74) f(x, y) = ln(-10x - 9y), (4, -5), A = 6i + 8j
the direction in which the function increases
most rapidly.
-9x
, (4, -7), A = 12i - 5j
75) f(x, y) = tan-1 y
86) Find the equation for the tangent plane to the
surface -6x - 7y - 8z = -15 at the point
(1, -1, 2).
76) f(x, y, z) = -3x + 6y - 9z, (-9, 10, -2),
A = 3i - 6j - 2k
87) Find parametric equations for the normal line
to the surface -10x - 7y + 7z = 11 at the point
(1, -1, 2).
5x
77) f(x, y, z) = tan-1 , (-8, 0, 0),
8y - 8z
A = 12i - 3j + 4k
88) Find the equation for the tangent plane to the
surface z = -8x2 + 3y2 at the point (2, 1, -29).
78) f(x, y, z) = ln(x2 - 5y2 - 2z 2 ), (-5, -5, -5),
A = 3i + 4j
89) Find parametric equations for the normal line
to the surface z = 4x2 + 2y2 at the point
Answer the question.
79) Find the direction in which the function is
increasing or decreasing most rapidly at the
point Po.
(2, 1, 18).
90) Find the equation for the tangent plane to the
surface x2 - 8xyz + y2 = 6z 2 at the point
f(x, y) = xy2 - yx2 , Po(-2, 1)
(1, 1, 1).
80) Find the direction in which the function is
increasing or decreasing most rapidly at the
point Po.
91) Find parametric equations for the normal line
to the surface x2 + 7xyz + y2 = 9z 2 at the
point (1, 1, 1).
f(x, y, z) = xy - ln(z), Po(2, -2, 2)
Find all local extreme values of the given function and
identify each as a local maximum, local minimum, or
saddle point.
92) f(x, y) = x2 + 2x + y2 + 8y + 9
Solve the problem.
81) Find the derivative of the function
f(x, y) = x2 + xy + y2 at the point (8, 9) in the
direction in which the function increases most
rapidly.
93) f(x, y) = 2xy - 6x + 6y
82) Find the derivative of the function
f(x, y) = x2 + xy + y2 at the point (-6, -5) in
94) f(x, y) = -2xy(x + y) - 9
the direction in which the function decreases
most rapidly.
95) f(x, y) = x2 + 10xy + y2
83) Find the derivative of the function f(x, y) = exy
at the point (0, 7) in the direction in which the
function increases most rapidly.
96) f(x, y) = x3 + y3 - 300x - 75y - 2
97) f(x, y) = 4 - x4 y4
84) Find the derivative of the function
f(x, y, z) = ln(xy + yz + zx) at the point
(-8, -16, -24) in the direction in which the
function increases most rapidly.
98) f(x, y) = 5x2 y + 3xy2
6
99) f(x, y) = 100x2 + 40xy + 16y2
2
2
100) f(x, y) = (x2 - 81) + (y2 - 9)
2
2
101) f(x, y) = (x2 - 16) - (y2 - 4)
Find the extreme values of the function subject to the
given constraint.
102) f(x, y) = 9x2 + 3y2 , x2 + y2 = 1
103) f(x, y) = xy, x2 + y2 = 200
104) f(x, y) = x2 + y2 , xy2 = 128
105) f(x, y) = y2 - x2 , x2 + y2 = 25
106) f(x, y) = 4x + 6y, x2 + y2 = 13
107) f(x, y) = x2 y, x2 + 2y2 = 6
108) f(x, y) = 3x - y + 1, 3x2 + y2 = 9
109) f(x, y) = x2 + 4y3 , x2 + 2y2 = 2
110) f(x, y) = xy, 9x2 + 4y2 = 36
111) f(x, y) = 12x + 3y, xy = 4, x > 0, y > 0
112) f(x, y, z) = x3 + y3 + z 3 , x2 + y2 + z 2 = 4
113) f(x, y, z) = x + 2y - 2z, x2 + y2 + z 2 = 9
114) f(x, y, z) = x2 + y2 + z 2 , x + 2y + 3z = 6
7
Answer Key
Testname: 227PRACTICEPROBT2
1) Domain: all points in the x-y plane; range: all real numbers; level curves: lines 9x + 8y = c
2) Domain: all points in the x-y plane except (0, 0); range: real numbers > 0; level curves: ellipses 6x2 + 9y2 = c
3) Domain: all points in the x-y plane satisfying 5x + 3y > 0; range: all real numbers; level curves: lines 5x + 3y = c
4) Domain: all points in the x-y plane satisfying x2 + y2 ≤ 36; range: real numbers 0 ≤ z ≤ 6; level curves: circles with
centers at (0, 0) and radii r, 0 < r ≤ 6
Domain: all points in the x-y plane except y = 0; range: all real numbers; level curves: parabolas y = cx2
D
A
B
9) x2 + y2 = 5
10) x2 + y2 = 25
5)
6)
7)
8)
11) x + ey+z = 41
12) 2
13) 1
11
14) - 10
15) ln 81
16) 1
17) ln 5 2
18) All (x, y)
19) All (x, y) such that x ≠ - y
3
20) All (x, y) such that x ≠ and x ≠ -2
2
21) All (x, y, z) such that x ≠ 0
22) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = 0, y = t
23) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = t, y = t3/2
24) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = 0, y = t
∂f
∂f
= 2; = -4y
25)
∂y
∂x
26)
∂f
∂f
= 80x4 y2 (8x5 y2 + 4); = 32x5 y1 (8x5 y2 + 4)
∂y
∂x
27)
∂f
= - ∂x
28)
5 ∂f
3
∂f
= - ; = x ∂y y
∂x
29)
e-x(x2 + y2 + 2x) ∂f
2ye-x
∂f
= - ; = - 2
2
∂x
∂y
(x2 + y2 )
(x2 + y2 )
30)
y
x
∂f
∂f
= ; = - ∂x (x + y)2 ∂y
(x + y)2
31)
∂f
∂f
= 3x2 + 12xy + 9y3 ; = 6x2 + 27xy2
∂x
∂y
32)
∂f
∂f
∂f
= 2xy + z 2 ; = x2 + 2yz; = y2 + 2xz
∂y
∂z
∂x
x
3/2
(x2 + y2 )
; ∂f
= - ∂y
y
3/2
(x2 + y2 )
8
Answer Key
Testname: 227PRACTICEPROBT2
33)
z ∂f
z ∂f
∂f
= ; = ; = ln xy
∂x x ∂y y ∂z
34)
∂f
∂f
∂f
= (y cos xy)(cosyz 2 ); = (x cos xy)(cos yz 2 ) - (z 2 sin xy)(sin yz 2 ); = -2(yz sin xy)(sin yz 2 )
∂y
∂z
∂x
35)
2
2
2 ∂f
2
2
2 ∂f
2
2
2
∂f
= (1 + 2x2 ) e(x + y + z ); = 2xye(x + y + z ); = 2xze(x + y + z )
∂y
∂z
∂x
36)
cosy ∂f
siny ∂f
2 cos y
∂f
= - ; = - ; = - 2
2
2
∂y
∂z
∂x
x z
xz
xz 3
37)
∂2 f
∂2 f
∂2 f
∂2 f
= 2 - ex+y; = - ex+y; = = -ex+y
∂y∂x ∂x∂y
∂x2
∂y2
38)
∂2 f
∂2 f
∂2 f
∂2 f
= -y4 cos xy2 ; = - 2x[2xy2 cos (xy2 ) + sin(xy2 )]; = = - 2y[xy2 cos (xy2 ) + sin(xy2 )];
∂y∂x ∂x∂y
∂y2
∂x2
39)
x - 2y ∂2 f
x
y
∂2 f
∂2 f
∂2 f
= ; = - ; = = 2
2
2
2
∂y∂x
∂x∂y
(y - x) ∂y
(y - x)
(y - x)2
∂x
40)
2
2
∂2 f
∂2 f
∂2 f
∂2 f
= 2yex (1 + 2x2 ); = 2x; = = 2y + 2xex
∂y∂x ∂x∂y
∂x2
∂y2
41)
2y
2x
x - y
∂2 f
∂2 f
∂2 f
∂2 f
= - ; = ; = = 3
3
2
2
∂y∂x
∂x∂y
(x + y) ∂y
(x + y)
(x + y)3
∂x
42)
lim 7(4 + h)2 + 32(4 + h) + 320 - 560
∂f
= h
∂x h → 0
= = 43)
lim 102h + h 2
lim
= 102 + h = 102
h → 0
h
h → 0
lim 4 + 30(7 + h) - 6(7 + h)2 + 500
∂f
= h
∂y h → 0
= 44)
lim 7(16 + 8h + h 2 ) + 32h - 112
h → 0
h
lim -54h - 6h 2
lim
= -54 - 6h = -54
h
h → 0
h → 0
lim 324(-5 + h) + 6(-5 + h)2 + 30 + 1440
∂f
= h
∂y h → 0
= lim 216h + 6h 2
lim
= 216 + 6h = 216
h → 0
h
h → 0
45) -10
4 1
46) - 9 π2
47)
3
4
48) 0
49) -40
50) 35e6 - 1
51) 18
9
Answer Key
Testname: 227PRACTICEPROBT2
52) 155
53) 220
54) 4
55) - 1
8
56)
17
57) - 3
10
58) 0
15
59)
14
60) - 47
32
61)
1 - 6e7
1 - 2e7
62)
7
2
63)
∂w
= -32
∂r
64)
∂w 1
= 8
∂u
65)
∂z
= 1
∂v
66) -6i + 3j
67) 16i - 8j
9
8
68)
i + j
19
19
69) - 2
3
i - j
39
13
70) 3i + 10j + 2k
1
2
1
71) i + j + k
3
9
9
72)
22
5
73) - 204
5
74) - 66
25
75)
576
17485
76) - 27
7
77) - 7
65
10
Answer Key
Testname: 227PRACTICEPROBT2
78) - 17
150
79)
5
-8
i + j
89
89
80)
1
(-4i + 4j - k)
33
81) 1301
82) - 545
83) 7
5
2
84)
88
85)
1
4
2
86) -6x - 7y - 8z = -15
87) x = -10t + 1, y = -7t - 1, z = 7t + 2
88) -32x + 6y - z = -29
89) x = 16t + 2, y = 4t + 1, z = -t + 18
90) 6x + 6y - 4z = 8
91) x = 9t + 1, y = 9t + 1, z = -11t + 1
92) f(-1, -4) = -8, local minimum
93) f - 3, 3 = 18, saddle point
94) f(0, 0) = -9, saddle point
95) f(0, 0) = 0, saddle point
96) f(10, 5) = -2252. local minimum; f(10, -5) = -1752, saddle point; f(-10, 5) = 1748, saddle point; f(-10, -5) = 2248,
local maximum
97) f(0, 0) = 4, local maximum
98) f(0, 0) = 0, saddle point
99) f(0, 0) = 0, local minimum
100) f(0, 0) = 6642, local maximum; f(0, 3) = 6561, saddle point; f(0, -3) = 6561, saddle point; f(9, 0) = 6642, saddle
point; f(9, 3) = 0, local minimum; f(9, -3) = 0, local minimum; f(-9, 0) = 81, saddle point; f(-9, 3) = 0, local
minimum; f(-9, -3) = 0, local minimum
101) f(0, 0) = 240, saddle point; f(0, 2) = 256, local maximum; f(0, -2) = 256, local maximum; f(4, 0) = -16, local
minimum; f(4, 2) = 0, saddle point; f(4, -2) = 0, saddle point; f(-4, 0) = -16, local minimum; f(-4, 2) = 0,
saddle point; f(-4, -2) = 0, saddle point
102) Maximum: 3 at (0, ±1); minimum: 9 at (±1, 0)
103) Maximum: 100 at (10, 10) and (-10, -10); minimum: -100 at (10, -10) and (-10, 10)
104) Maximum: none; minimum: 48 at (4, ±4 2)
105) Maximum: 25 at (0, ±5); minimum: -25 at (±5, 0)
106) Maximum: 26 at (2, 3); minimum: -26 at (-2, -3)
107) Maximum: 4 at (±2, 1); minimum: -4 at (±2, -1)
3
3 3
3
108) Maximum: 7 at , - ; minimum: -5 at - , 2
2 2
2
109) Maximum: 4 at (0, 1); minimum: -4 at (0, -1)
3
3
110) Maximum: 3 at 2, 2 and - 2, - 2 ; minimum: -3 at 2
2
2, - 3
2
2 and - 2, 111) Maximum: none; minimum: 24 at (1, 4)
112) Maximum: 8 at (2, 0, 0), (0, 2, 0), (0, 0, 2); minimum: -8 at (-2, 0, 0), (0, -2, 0), (0, 0, -2)
113) Maximum: 9 at (1, 2, -2); minimum: -9 at (-1, -2, 2)
11
3
2
2
Answer Key
Testname: 227PRACTICEPROBT2
114) Maximum: none; minimum: 18
3 6 9
at , , 7
7 7 7
12