MV Calc, Ch 14A practice test Good Luck to: ______________________________________ I ♥ MV Calc
Directions: You may not use a calculator. To receive any partial credit, show all pertinent work that supports
your conclusions.
1.a) If
lim f ( x ) = L , where f ( x ) is a function of one variable, then the definition of lim f ( x ) = L is as
x →a
x → x0
follows: Definition 1: Given any number
∈> 0 , we can find a δ > 0 such that if x − x0 < δ , then
2
7
f ( x ) − L <∈ . Use the definition above to prove that lim x + 1 = .
x →2 3
3
Definition 2
b) Recall the definition of a limit on a point of
a surface. Let
f ( x, y ) be a function of two
variables, and assume that f is defined at
all points of some open disk centered at
( x0 , y0 ) , except possibly at ( x0 , y0 ) .
We define
lim
( x , y )→( x0 , y0 )
f ( x, y ) = L if given any number ∈> 0 , we can find a number δ > 0 such that f ( x, y )
satisfies f ( x, y ) − L <∈ whenever the distance between
0<
2
( x − x0 ) + ( y − y0 )
2
( x, y ) and ( x0 , y0 ) satisfies
< δ . Use the diagram shown above to explain how the diagram conveys the
above definition 2.
2. a) Show that the value of
,
xyz
approaches 0 as ( x, y, z ) → ( 0, 0,0 ) along any line x = at , y = bt
x + y4 + z4
2
z = ct .
b) Show that the limit
x = t2,
lim
( x , y , z )→( 0,0,0)
xyz
does not exist by letting ( x, y, z ) → ( 0, 0,0 ) along the curve
x + y4 + z4
2
y = t, z = t .
Conc c
(in ppm)
14
100
100
96
96
0
2
6
15
Time t in months
16
18
100
100
99
98
95
93
93
82
20
99
97
90
70
22
97
95
86
58
24
95
92
80
36
3. An experiment to measure the toxicity of formaldehyde yielded the data in the table shown above. The
values show the percent, P = f(t, c), of rats surviving an exposure to formaldehyde at a concentration of c (in
parts per million, ppm) after t months. Estimate ft(18, 6) and fc(18, 6). Interpret your answers in terms of
formaldehyde toxicity.
4. In each case, give a possible contour diagram for the function
a)
f x > 0 and f y > 0
GO ON TO THE NEXT PAGE
b)
f x > 0 and f y < 0
c)
f ( x, y ) if
f x < 0 and f y > 0
d)
f x < 0 and f y < 0
5. A contour diagram for
z = f ( x, y ) is shown above.
∂z
positive or negative? Explain your response.
∂x
∂z
b) Is
positive or negative? Explain your response.
∂y
a) Is
c) Estimate
f ( 2,1) ,
∂z
∂z
, and
.
∂x ( 2,1)
∂y ( 2,1)
6. The Laplace Equation is known as
Equation.
∂2 z
∂x
2
+
∂2 z
∂y
2
= 0 . Verify that z = e x sin y + e y cos x satisfies Laplace’s
I ♥ MV Calc
MV Calc, Ch 14A Solutions to practice test
Directions: You may not use a calculator. To receive any partial credit, show all pertinent work that supports
your conclusions.
1.a) If
lim f ( x ) = L , where f ( x ) is a function of one variable, then the definition of lim f ( x ) = L is as
x →a
x → x0
follows: Definition 1: Given any number
∈> 0 , we can find a δ > 0 such that if x − x0 < δ , then
Definition 2
2
7
f ( x ) − L <∈ . Use the definition above to prove that lim x + 1 = .
x →2 3
3
3
3
2
Solution: Let ∈> 0 . Choose δ = ∈ . Suppose that x − 2 < δ = ∈ ⇒ x − 2 <∈
2
2
3
2
4 2
7
⇒ x − = x + 1 − <∈ .
3
3 3
3
b) Recall the definition of a limit on a point of
a surface. Let
f ( x, y ) be a function of two
variables, and assume that f is defined at
all points of some open disk centered at
( x0 , y0 ) , except possibly at ( x0 , y0 ) .
We define
lim
( x , y )→( x0 , y0 )
f ( x, y ) = L if given any number ∈> 0 , we can find a number δ > 0 such that f ( x, y )
satisfies f ( x, y ) − L <∈ whenever the distance between
0<
2
( x − x0 ) + ( y − y0 )
2
( x, y ) and ( x0 , y0 ) satisfies
< δ . Use the diagram shown above to explain how the diagram conveys the
above definition 2.
Solution: From your text see page 939.
2. a) Show that the value of
xyz
approaches 0 as ( x, y, z ) → ( 0, 0,0 ) along any line x = at , y = bt
x + y4 + z4
2
, z = ct .
lim
Solution:
t →0
abct 3
atbtct
abct
lim
= lim 2
=
=0
2
2
4
4
4
4
2 2
4 4
4 4
t →0 a t + b t + c t
t → 0 a + b 4 t 2 + c 4t 2
a t +b t +c t
b) Show that the limit
x = t2,
lim
( x , y , z )→( 0,0,0)
xyz
does not exist by letting ( x, y, z ) → ( 0, 0,0 ) along the curve
x + y4 + z4
2
y = t, z = t .
t2 ⋅t ⋅t
t2 ⋅t ⋅t
1
=
lim
=
t →0 t 4 + t 4 + t 4
t →0 t 4 + t 4 + t 4
3 . From parts a and b we can see that along two different
xyz
lim
2
x
,
y
,
z
→
0,0,0
(
) (
) x + y4 + z4
parametric paths two different limits were obtained. Therefore,
does not exist.
Solution:
lim
0
2
6
15
Conc c
(in ppm)
14
100
100
96
96
Time t in months
16
18
100
100
99
98
95
93
93
82
20
99
97
90
70
22
97
95
86
58
24
95
92
80
36
3. An experiment to measure the toxicity of formaldehyde yielded the data in the table shown above. The
values show the percent, P = f(t, c), of rats surviving an exposure to formaldehyde at a concentration of c (in
parts per million, ppm) after t months. Estimate ft(18, 6) and fc(18, 6). Interpret your answers in terms of
formaldehyde toxicity.
Solution: We have ft (18, 6 ) ≈
∆P 90 − 93
=
= −1.5 percent per month. Eighteen months after rats are
∆t 20 − 18
exposed to a formaldehyde concentration of 6 ppm, the percent of rats surviving is decreasing at a rate of
about 1.5 per month. In other words, during the eighteenth month, an additional 1.5% of the rats die (how
morbid
).
We have f c (18, 6 ) ≈
∆P 82 − 93
=
= −1.22 percent /ppm. If the original concentration increases by 1 ppm,
∆C 15 − 6
the percent surviving after 18 months decreases by about 1.22.
4. In each case, give a possible contour diagram for the function
a) f x > 0 and f y > 0
b) f x > 0 and f y < 0
f ( x, y ) if
c) f x < 0 and f y > 0
d) f x < 0 and f y < 0
Solution:
a) Since f x > 0 , the values on the contours increase
as you move to the right. Since
f y > 0 , the values
on the contours increase as you move upward . See
figure below.
b) Since f x > 0 , the values on the contours increase
as you move to the right. Since
f y < 0 , the values
on the contours decrease as you move upward . See
figure below.
c) Since f x < 0 , the values on the contours decrease
as you move to the right. Since
f y > 0 , the values
on the contours increase as you move upward . See
figure below.
5. A contour diagram for
d) Since f x < 0 , the values on the contours decrease
as you move to the right. Since
on the contours decrease as you move upward . See
figure below.
z = f ( x, y ) is shown above.
∂z
positive or negative? Explain your response.
∂x
∂z
Solution:
> 0 because as x increases the level curves are increasing.
∂x
∂z
b) Is
positive or negative? Explain your response.
∂y
∂z
Solution:
< 0 because as y increases the level curves are decreasing.
∂y
a) Is
c) Estimate
f ( 2,1) ,
f ( 2,1) ≈ 10 ,
∂z
∂z
, and
.
∂x ( 2,1)
∂y ( 2,1)
∂z
4
∂z
≈ = 2,
≈ −4
∂x ( 2,1) 2
∂y ( 2,1)
f y < 0 , the values
6. The Laplace Equation is known as
∂2 z
∂x
2
+
∂2 z
∂y
2
= 0 . Verify that z = e x sin y + e y cos x satisfies Laplace’s
Equation.
Solution:
∂z
∂  ∂z  ∂ x
= e x sin y − e y sin x ,
e sin y − e y sin x = e x sin y − e y cos x
 =
∂x
∂x  ∂x  ∂x
(
)
∂z
∂  ∂z  ∂ x
= e x cos y + e y cos x ,
e cos y + e y cos x = −e x sin y + e y cos x
 =
∂y
∂y  ∂y  ∂y
(
∂2 z
∂x
2
+
∂2 z
∂y
2
= e x sin y − e y cos x − e x sin y + e y cos x = 0
)