2022W1_MATH_100C_ALL_2022W1.OWDAYLFG3E04.Webwork-Assignment-6-5
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School
University of British Columbia *
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Course
math 100
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
4
Uploaded by HighnessBravery13778
Lea Grandin
2022W1
MATH
100C
ALL
2022W1
Assignment Webwork-Assignment-6 due 10/27/2022 at 11:59pm PDT
Problem 1.
(1 point)
Use the figures below to evaluate the indicated derivative, or state
that it does not exist. If the derivative does not exist, enter
dne
in
the answer blank. The graph to the left (in black) gives
f
(
x
)
, while
the graph to the right gives
g
(
x
)
(which is constant for values of
x
greater than 20).
f
(
x
)
g
(
x
)
d
dx
f
(
g
(
x
))
|
x
=
10
=
(If the derivative does not exist, enter
dne
.)
Answer(s) submitted:
•
2/3
(correct)
Correct Answers:
•
0.666667
Problem 2.
(1 point)
A table of values for
f
,
g
,
f
0
,
and
g
0
is given below:
x
f
(
x
)
g
(
x
)
f
0
(
x
)
g
0
(
x
)
1
3
2
4
6
2
1
8
5
7
3
7
2
7
9
(a) If
F
(
x
) =
f
(
f
(
x
))
, find
F
0
(
2
)
.
(b) If
G
(
x
) =
g
(
g
(
x
))
, find
G
0
(
3
)
.
(a)
F
0
(
2
) =
(b)
G
0
(
3
) =
Answer(s) submitted:
•
20
•
63
(correct)
Correct Answers:
•
20
•
63
Problem 3.
(1 point)
Consider the function
r
(
x
) =
f
(
g
(
h
(
x
)))
. Given that
h
0
(
1
) =
3
,
h
(
1
) =
4
,
g
0
(
4
) =
3
g
(
4
) =
6
and
f
0
(
6
) =
2
,
Find the value of
r
0
(
1
)
or enter NA if there is not enough informa-
tion to compute this derivative.
r
0
(
1
)
=
Answer(s) submitted:
•
18
(correct)
Correct Answers:
•
18
Problem 4.
(1 point)
If 16
+
4
f
(
x
)+
3
x
2
(
f
(
x
))
3
=
0 and
f
(
-
2
) =
-
1, find
f
0
(
-
2
)
.
f
0
(
-
2
) =
Answer(s) submitted:
•
-3/10
(correct)
Correct Answers:
•
-3/10
Problem 5.
(1 point)
Find
y
0
if
y
=
e
-
1
/
x
2
y
0
=
Hint:
Recall the definition of the derivative of the natural expo-
nential function and use the chain rule.
Answer(s) submitted:
•
2/(eˆ(1/xˆ2) *xˆ3)
(correct)
Correct Answers:
•
(2/x**3)*e**(-1/x**2)
1
Problem 6.
(1 point)
(a) Let
f
(
x
) =
√
8
+
5
x
4
. Find
f
0
(
x
)
.
f
0
(
x
) =
(b) Let
f
(
x
) =
e
√
8
+
5
x
4
. Find
f
0
(
x
)
.
f
0
(
x
) =
Answer(s) submitted:
•
10xˆ3(8+5xˆ4)ˆ-0.5
•
10xˆ3(8+5xˆ4)ˆ-0.5 * eˆ((8+5xˆ4)ˆ0.5)
(correct)
Correct Answers:
•
1/[2*sqrt(8+5*xˆ4)]*5*4*xˆ3
•
eˆ[sqrt(8+5*xˆ4)]*1/[2*sqrt(8+5*xˆ4)]*5*4*xˆ3
Problem 7.
(1 point)
Find the derivative of
h
(
z
) =
b
a
+
z
2
2
Assume that
a
and
b
are constants.
h
0
(
z
) =
Answer(s) submitted:
•
2(b/(a+zˆ2))*(-b(a+zˆ2)ˆ-2)(2z)
(correct)
Correct Answers:
•
-1*2*bˆ2*2*z*(a+zˆ2)ˆ(-1*2-1)
Problem 8.
(1 point)
Find the derivative of
z
(
x
) =
9
√
4
x
+
8
z
0
(
x
) =
Answer(s) submitted:
•
(4ˆx*ln(4))/9(4ˆx +8)ˆ(-8/9)
(correct)
Correct Answers:
•
1/9*(4ˆx+8)ˆ[(1-9)/9]*ln(4)*4ˆx
Problem 9.
(1 point)
Let
f
(
x
) =
sin
1
x
.
f
0
(
x
) =
.
Let
g
(
x
) =
1
sin
x
.
g
0
(
x
) =
.
Answer(s) submitted:
•
-(cos(1/x))/xˆ2
•
-cot(x)csc(x)
(correct)
Correct Answers:
•
-cos(1/x)/(x*x)
•
-cos(x)/sin(x)**2
Problem 10.
(1 point)
Let
f
(
x
) =
5cos
(
sin
(
x
8
))
f
0
(
x
) =
Answer(s) submitted:
•
-40xˆ7 *sin(sin(xˆ8))*cos(xˆ8)
(correct)
Correct Answers:
•
-5*8*xˆ(8-1)*sin(sin(xˆ8))*cos(xˆ8)
Problem 11.
(1 point)
If
f
(
x
) =
3ln
(
ln
(
x
))
, find
f
0
(
x
)
.
f
0
(
x
) =
Answer(s) submitted:
•
3/(ln(x)*x)
(correct)
Correct Answers:
•
3/[x*ln(x)]
2
Problem 12.
(1 point)
Use logarithmic differentiation to find the derivative of the func-
tion.
y
=
x
5
/
x
y
0
=
Answer(s) submitted:
•
5xˆ((5-2x)/x) - 5xˆ((5-2x)/x) *ln(x)
(correct)
Correct Answers:
•
5*xˆ(5/x)*[1-ln(x)]/(xˆ2)
Problem 13.
(1 point)
Differentiate
y
=
p
arctan
(
x
)
.
y
0
=
Answer(s) submitted:
•
1/((2sqrt(arctan(x))) +2xˆ2sqrt(arctan(x)))
(correct)
Correct Answers:
•
1/(2*[atan(x)]ˆ0.5*(1+xˆ2))
Problem 14.
(1 point)
Use implicit differentiation to find an equation of the tangent line
to the curve at the given point.
x
2
+
y
2
= (
2
x
2
+
2
y
2
-
x
)
2
,
(
0
,
1
/
2
)
(cardioid)
y
=
Answer(s) submitted:
•
x+1/2
(correct)
Correct Answers:
•
x+1/2
Problem 15.
(1 point)
If ln
(
x
2
-
8
y
) =
x
-
y
-
2 and
y
(
3
) =
1, find
y
0
(
3
)
by implicit
differentiation.
y
0
(
3
) =
An equation of the tangent line to the curve at the point
(
3
,
1
)
is
y
=
.
Answer(s) submitted:
•
5/7
•
(5/7)(x-3)+1
(correct)
Correct Answers:
•
5/7
•
5/7*(x-3)+1
Problem 16.
(1 point)
Find the slope of the tangent line to the ellipse
x
2
9
+
y
2
25
=
1 at the
point
(
x
,
y
)
.
slope =
Are there any points where the slope is not defined? (Enter them
as comma-separated ordered-pairs, e.g., (1,3), (-2,5). Enter
none
if there are no such points.)
slope is undefined at
Answer(s) submitted:
•
-25x/(9y)
•
(-3,0), (3,0)
(correct)
Correct Answers:
•
-1*25*x/(9*y)
•
(-3,0), (3,0)
Problem 17.
(1 point)
The radius
r
of a sphere is expanding at a rate of 12 inches per
minute. Determine the rate at which the volume is changing with
respect to time when
r
=
16in.
The volume is changing at a rate of
in
3
/
min.
Answer(s) submitted:
•
12288pi
(correct)
Correct Answers:
•
38603.9
3
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Problem 18.
(1 point)
The length of a rectangle is increasing at a rate of 8cm/s and its
width is increasing at a rate of 3cm/s. When the length is 20cm
and the width is 15cm
,
how fast is the area of the rectangle in-
creasing?
Answer (in cm
2
/
s):
Answer(s) submitted:
•
180
(correct)
Correct Answers:
•
8*15+3*20
Problem 19.
(1 point)
Consider the growth of a cell, assumed spherical in shape. Sup-
pose that the radius,
r
, of the cell increases at a constant rate per
unit time. (Call the constant
k
.) Your answers will involve both
r
and
k
.
(a)
At what rate does the volume,
V
, increase ?
dV
dt
=
(b)
At what rate does the surface area,
S
, increase ?
dS
dt
=
(c)
At what rate does the ratio of surface area to volume
S
/
V
change?
d
dt
(
S
V
) =
Does the ratio
S
/
V
decrease or increase as the cell grows?
[?/Increase/Decrease]
[Remark: express your answers in terms of the radius of the cell
r
and its growth rate
k
.]
Answer(s) submitted:
•
4pi*rˆ2*k
•
8pi*r*k
•
-3k/rˆ2
•
Decrease
(correct)
Correct Answers:
•
4*pi*r**2*k
•
8*pi*r*k
•
-3*k/(r**2)
•
Decrease
Problem 20.
(1 point)
In 1905 a Bohemian farmer accidentally allowed several muskrats
to escape an enclosure. Their population grew and spread, occu-
pying increasingly larger areas throughout Europe. In a classical
paper in ecology, it was shown by the scientist Skellam (1951)
that the square root of the occupied area increased at a constant
rate,
k
. Determine the rate of change of the distance (from the site
of release) that the muskrats had spread. For simplicity, you may
assume that the expanding area of occupation is circular.
r
0
(
t
) =
Answer(s) submitted:
•
k/sqrt(pi)
(correct)
Correct Answers:
•
pi**(-0.5)*k
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4