2023W1_MATH_100C_ALL_2023W1.RMS560AP9L04.WW11 (1)
pdf
keyboard_arrow_up
School
University of British Columbia *
*We aren’t endorsed by this school
Course
100
Subject
Mathematics
Date
Feb 20, 2024
Type
Pages
9
Uploaded by CorporalStrawElk30
Sayra Arij
2023W1
MATH
100C
ALL
2023W1
Assignment WW11 due 12/01/2023 at 11:59pm PST
Problem 1.
(1 point)
Please complete the following survey:
https://ubc.ca1.qualtrics.com/jfe/form/SV
3
OXCXlvLKVeJ jro
This survey will help the Department of Mathematics build
a complete picture of students’ attitudes and perceptions in
first-year mathematics courses at UBC. It will only take 5-10
minutes to complete, and there are no wrong answers.
The survey answers will be kept confidential; only the re-
searchers collecting the data will see them.
• ?
•
I clicked on the link!
• I did not click on the link.
Answer(s) submitted:
•
I clicked on the link!
submitted: (correct)
recorded: (correct)
Problem 2.
(1 point)
() Use Newton’s method to approximate a root of the equation
3
x
3
+
7
x
2
+
2
=
0
as follows:
Let
x
1
=
-
2
be the initial approximation.
The second approximation
x
2
is
and the third approximation
x
3
is
Answer(s) submitted:
•
-
11
4
•
-
2363
946
submitted: (correct)
recorded: (correct)
Problem 3.
(1 point)
() Use Newton’s Method with the function
f
(
x
) =
x
2
-
2
and
initial value
x
0
=
1
to calculate
x
1
,
x
2
,
x
3
.
x
1
=
x
2
=
x
3
=
Answer(s) submitted:
•
3
2
•
17
12
•
577
408
submitted: (correct)
recorded: (correct)
Problem 4.
(1 point)
() Use Newton’s method to find the roots of the equation:
log
(
x
) =
4
-
x
. Start with
x
0
=
3
.
Report your answers to 4 decimal places.
x
1
=
x
2
=
x
3
=
CAUTION: Carry all decimal places when preforming calcu-
lations.
Answer(s) submitted:
•
3
-
log
(
3
)
-
1
4
3
•
3
-
log
(
3
)
-
1
4
3
-
log
3
-
log
(
3
)
-
1
4
3
+
3
-
log
(
3
)
-
1
4
3
-
4
1
3
-
log
(
3
)
-
1
4
3
+
1
•
2
.
92627
-
log
(
2
.
92627
)+
2
.
92627
-
4
1
2
.
92627
+
1
submitted: (correct)
recorded: (correct)
Problem 5.
(1 point)
() In this question we will use Newton’s Method to approx-
imate the
x
-value of the intersection between
f
(
x
) =
1
6
x
and
g
(
x
) =
x
3
1
+
3
x
3
closest to
x
=
2
. Start your approximation with
x
0
=
2
and apply two iterations of Newton’s Method to the
function
F
(
x
) =
f
(
x
)
-
g
(
x
)
. Give your answers correct to at
least 4 decimal places. (Remember that WeBWorK can un-
derstand calculator-ready numbers, and make liberal use of
copy-paste!)
x
1
=
x
2
=
Answer(s) submitted:
•
no response
•
no response
submitted: (incorrect)
recorded: (incorrect)
1
Problem 6.
(1 point)
() Give the first 5 values found using Newton’s Method with
the given function and initial value.
f
(
x
) =
x
3
-
3
x
2
+
x
+
3
,
x
0
=
1
x
1
=
x
2
=
x
3
=
x
4
=
x
5
=
Why does Newton’s Method fail in finding the root?
•
The function has no roots
•
The derivative becomes 0
•
The approximations will not converge
•
More iterations are needed before the root can be
found
Answer(s) submitted:
•
2
•
1
•
2
•
1
•
2
•
The approximations ... not converge
submitted: (correct)
recorded: (correct)
Problem 7.
(1 point)
() In this question we will use a spreadsheet to approximate
the root of
f
(
x
) =
x
5
+
3
x
3
-
6
, using Newton’s method with
x
0
=
1
. The picture below shows a spreadsheet being used to
perform the calculation.
A downwards arrow indicates the
contents from the cell above will be copied down.
A
B
C
1
?
?
?
2
=A1-B1/C1
↓
↓
3
↓
↓
↓
4
↓
↓
↓
a) Which entry should go in cell A1?
•
0
•
=5*B1
4
+
9
*
B
1
2
=
B
1
5
+
3
*
B
1
3
-
6
•
=5*A1
4
+
9
*
A
1
2
1
•
=A1
5
+
3
*
A
1
3
-
6
b) Which entry should go in cell B1?
•
=5*B1
4
+
9
*
B
1
2
=
B
1
5
+
3
*
B
1
3
-
6
•
=A1
5
+
3
*
A
1
3
-
6
0
•
1
•
=5*A1
4
+
9
*
A
1
2
c) Which entry should go in cell C1?
•
=A1
5
+
3
*
A
1
3
-
6
=
5
*
B
1
4
+
9
*
B
1
2
•
1
•
=5*A1
4
+
9
*
A
1
2
0
•
=B1
5
+
3
*
B
1
3
-
6
2
Using a spreadsheet, find the following values.
d)
x
1
=
(Give your answer to at least 3 decimal
places.)
e)
x
2
=
(Give your answer to at least 3 decimal
places.)
Answer(s) submitted:
•
Choice 5
•
Choice 3
•
Choice 4
•
8
7
•
1
.
122
submitted: (correct)
recorded: (correct)
Problem 8.
(1 point)
() In this question, we will approximate
5
√
3126
with a rational
number.
a)To use the linear approximation
f
(
x
)
≈
f
(
x
0
)+
f
0
(
x
0
)(
x
-
x
0
)
there is an obvious choice of function
f
(
x
)
and point
x
0
. What
are they?
f
(
x
)
=
x
0
=
b) Use a linear approximation of
f
(
x
)
at
x
=
x
0
to find a ratio-
nal number approximating
5
√
3126
. Give your answer correct
to at least 4 decimal places.
5
√
3126
≈
c) Newton’s Method finds roots (zeroes) of functions. In or-
der to use Newton’s Method to approximate
5
√
3126
, we need
a function
g
(
x
)
such that
g
5
√
3126
=
0
and
g
(
x
)
has rational
coefficients. There is an obvious choice for
g
(
x
)
.
(For example:
to approximate
√
50
, the obvious choice is
g
(
x
) =
x
2
-
50
; to approximate
7
√
10
, the obvious choice is
g
(
x
) =
x
7
-
10
.)
What is the obvious choice of
g
(
x
)
to approximate
5
√
3126
?
g
(
x
)
=
d) Which integer
x
0
makes
g
(
x
0
)
as close as possible to 0?
x
0
=
e) Use two iterations of Newton’s Method (using
g
(
x
)
and
x
0
found above) to approximate
5
√
3126
. Give your answers cor-
rect to at least 4 decimal places.
x
1
=
x
2
=
Answer(s) submitted:
•
√
x
•
no response
•
no response
•
no response
•
no response
3
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
•
no response
•
no response
submitted: (incorrect)
recorded: (incorrect)
Problem 9.
(1 point)
() Suppose
f
(
x
,
y
) =
xy
2
-
2
. Compute the following values:
f
(
-
3
,
-
1
)
=
f
(
-
1
,
-
3
)
=
f
(
0
,
0
)
=
f
(
5
,
-
1
)
=
Answer(s) submitted:
• -
5
• -
11
• -
2
•
3
submitted: (correct)
recorded: (correct)
Problem 10.
(1 point)
() Consider the concentration
C
(in mg/litre) of a drug in the
blood as a function of the amount of drug injected,
x
, in mg,
and the time since injection,
t
, in hours. For
0
≤
x
≤
5
mg and
t
≥
0
hours, we have
C
=
f
(
x
,
t
) =
20
xte
-
5
t
.
Calculate the following:
f
(
1
,
5
) =
Give a practical interpretation of your answer.
Choose the
best option.
f
(
1
,
5
)
is:
•
A. the half-life of a 1 mg dose in the blood 5 hours after
injection.
•
B. the amount of a 5 mg dose in the blood 1 hours be-
fore injection.
•
C. the change in concentration of a 1 mg dose in the
blood 5 hours after injection.
•
D. the concentration in the blood 1 hours after injec-
tion of a 5 mg dose.
•
E. the concentration in the blood 5 hours after injec-
tion of a 1 mg dose.
•
F. the change in concentration of a 5 mg dose in the
blood 1 hours after injection.
Answer(s) submitted:
•
100
e
-
25
•
E
submitted: (correct)
recorded: (correct)
4
Problem 11.
(1 point)
() Consider the concentration
C
of a drug in the blood as a
function of
x
, the amount of the drug given, and
t
, the time
since the injection. For
0
≤
x
≤
3
and
t
≥
0
, we have
C
=
f
(
x
,
t
) =
xte
-
4
t
.
The units of
C
are milligrams per litre, the units of
x
are mil-
ligrams, and the units of
t
are hours.
Sketch the following two single-variable functions on a sepa-
rate page. Pay attention to the domain given at the start of the
problem.
(a)
f
(
2
,
t
)
(b)
f
(
x
,
1
.
5
)
Using your graph in (a), where is
f
(
2
,
t
)
...
a maximum?
t
=
a minimum?
t
=
Using your graph in (b), where is
f
(
x
,
1
.
5
)
...
a maximum?
x
=
a minimum?
x
=
Give a practical interpretation of the function
f
(
2
,
t
)
. Choose
the best answer.
•
A. the concentration of the drug in the blood 2 hours
after injection, as a function of the amount injected
•
B. the time after injecting 2 mg, as a function of con-
centration
•
C. the time after injection until the concentration in
the blood is 2 mg/L, as a function of the amount in-
jected
•
D. the initial injection necessary for concentration to
be 2 mg/L, as a function of time
•
E. the concentration of the drug in the blood resulting
from a 2 mg injection as a function of time
Give a practical interpretation of the function
f
(
x
,
1
.
5
)
.
Choose the best answer.
•
A. the concentration of the drug in the blood 1.5 hours
after injection, as a function of the amount injected
•
B. the concentration of the drug in the blood resulting
from a 1.5 mg injection as a function of time
•
C. the time after injection until the concentration in
the blood is 1.5 mg/L, as a function of the amount in-
jected
•
D. the initial injection necessary for concentration to
be 1.5 mg/L, as a function of time
•
E. the time after injecting 1.5 mg, as a function of con-
centration
Answer(s) submitted:
•
no response
•
no response
•
no response
•
no response
•
no response
•
no response
submitted: (incorrect)
recorded: (incorrect)
Problem 12.
(1 point)
You like pizza and you like cola. Which of the graphs in the
figure below represents your happiness
h
as a function of how
many pizzas
p
and how much cola
c
you have if
(a) There is such a thing as too many pizzas and too much
cola? figure [?/1/2/3/4]
(b) There is such a thing as too much cola but no such thing as
too many pizzas? figure [?/1/2/3/4]
1.
2.
3.
4.
Answer(s) submitted:
•
1
•
3
submitted: (correct)
recorded: (correct)
5
Problem 13.
(1 point)
() Suppose
f
(
x
,
y
)
is a differentiable function.
Match the
names of its derivatives in Newtonian and Leibnitz notation.
?
1.
∂
2
f
∂
y
∂
x
?
2.
∂
f
∂
x
?
3.
∂
2
f
∂
x
∂
y
?
4.
∂
2
f
∂
x
2
?
5.
∂
2
f
∂
y
2
?
6.
∂
f
∂
y
A.
f
xx
B.
f
yy
C.
f
y
D.
f
xy
E.
f
x
F.
f
yx
Answer(s) submitted:
•
D
•
E
•
F
•
A
•
B
•
C
submitted: (correct)
recorded: (correct)
Problem 14.
(1 point)
() Compute the partial derivative:
f
(
x
,
y
) =
cos
(
x
4
-
5
y
)
f
y
(
0
,
π
) =
Answer(s) submitted:
•
5sin
(
-
5
π
)
submitted: (correct)
recorded: (correct)
Problem 15.
(1 point)
() Let
z
=
√
x
+
5
y
. Then:
The rate of change in
z
at
(
5
,
1
)
as we change
x
but hold
y
fixed
is
, and
The rate of change in
z
at
(
5
,
1
)
as we change
y
but hold
x
fixed
is
.
Answer(s) submitted:
•
√
10
20
•
√
10
4
submitted: (correct)
recorded: (correct)
Problem 16.
(1 point)
() Find all the first and second order partial derivatives of
f
(
x
,
y
) =
-
8sin
(
2
x
+
y
)
-
1cos
(
x
-
y
)
.
A.
∂
f
∂
x
=
f
x
=
B.
∂
f
∂
y
=
f
y
=
C.
∂
2
f
∂
x
2
=
f
xx
=
D.
∂
2
f
∂
y
2
=
f
yy
=
E.
∂
2
f
∂
x
∂
y
=
f
yx
=
F.
∂
2
f
∂
y
∂
x
=
f
xy
=
Answer(s) submitted:
•
no response
•
no response
•
no response
•
no response
•
no response
•
no response
submitted: (incorrect)
recorded: (incorrect)
Problem 17.
(1 point)
() Given
u
(
x
,
t
) =
1
t
3
e
x
2
2
t
, compute:
u
xx
=
Answer(s) submitted:
•
no response
submitted: (incorrect)
recorded: (incorrect)
6
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
Problem 18.
(1 point)
() Find the first partial derivatives of
f
(
x
,
y
) =
3
x
-
3
y
3
x
+
3
y
at the point
(
x
,
y
) = (
4
,
1
)
.
∂
f
∂
x
(
4
,
1
) =
∂
f
∂
y
(
4
,
1
) =
Answer(s) submitted:
•
no response
•
no response
submitted: (incorrect)
recorded: (incorrect)
Problem 19.
(1 point)
() Find the partial derivatives of the function
f
(
x
,
y
) =
xye
-
4
y
f
x
(
x
,
y
) =
f
y
(
x
,
y
) =
f
xy
(
x
,
y
) =
f
yx
(
x
,
y
) =
Answer(s) submitted:
•
no response
•
no response
•
no response
•
no response
submitted: (incorrect)
recorded: (incorrect)
Problem 20.
(1 point)
() A one-meter long bar is heated unevenly, with temperature
in
◦
C at a distance
x
meters from one end at time
t
given by
H
(
x
,
t
) =
130
e
-
0
.
05
t
sin
(
π
x
)
0
≤
x
≤
1
. On a sheet of paper, sketch a graph of
H
against
x
for
t
=
0
and
t
=
1
. Think about why your graphs make sense.
(a) Calculate each of:
H
x
(
0
.
2
,
t
) =
H
x
(
0
.
8
,
t
) =
.
(Be sure that you can say in words what the practical interpreta-
tion (in terms of temperature) of these two partial derivatives is,
and why each has the sign that it does.)
(b) Calculate:
H
t
(
x
,
t
) =
.
(Again, be sure that you understand why it has the sign that it
does, and what its interpretation in terms of temperature is.)
Answer(s) submitted:
•
π
·
130
e
-
0
.
05
t
cos
(
π
·
0
.
2
)
•
π
·
130
e
-
0
.
05
t
cos
(
π
·
0
.
8
)
• -
6
.
5
e
-
0
.
05
t
sin
(
π
x
)
submitted: (correct)
recorded: (correct)
Problem 21.
(1 point)
() Suppose the
f
(
x
,
y
)
is a differentiable function and that
its partial derivatives have the values
f
x
(
1
,
-
3
) =
2
and
f
y
(
1
,
-
3
) =
-
5
. Given that
f
(
1
,
-
3
) =
0
, use this information
to estimate the following values:
f
(
2
,
-
3
)
≈
f
(
1
,
-
2
)
≈
f
(
2
,
-
2
)
≈
Note this is analogous to finding the tangent line approxima-
tion to a function of one variable. In fancy terms, it is the first
Taylor approximation.
Answer(s) submitted:
•
no response
•
no response
•
no response
submitted: (incorrect)
recorded: (incorrect)
7
Problem 22.
(1 point)
() Match each function with its graph.
?
1.
f
(
x
,
y
) =
-
4
e
-
x
2
-
y
2
?
2.
f
(
x
,
y
) =
x
+
2
y
+
3
?
3.
f
(
x
,
y
) =
-
y
2
?
4.
f
(
x
,
y
) =
x
3
-
sin
(
y
)
?
5.
f
(
x
,
y
) =
1
x
2
+
y
2
(You can drag the images to rotate them.)
A
B
C
D
E
Answer(s) submitted:
•
no response
•
no response
•
no response
•
no response
•
no response
submitted: (incorrect)
recorded: (incorrect)
Problem 23.
(1 point)
() Match each function with its graph.
?
1.
f
(
x
,
y
) =
3
1
+
x
2
+
y
2
?
2.
f
(
x
,
y
) =
3
-
p
x
2
+
y
2
?
3.
f
(
x
,
y
) =
3
-
x
2
-
y
2
?
4.
f
(
x
,
y
) =
3cos
x
2
+
y
2
4
(You can drag the images to rotate them.)
A
B
C
D
Answer(s) submitted:
•
no response
•
no response
•
no response
•
no response
submitted: (incorrect)
recorded: (incorrect)
8
Problem 24.
(1 point)
() Consider the function
f
(
x
,
y
) =
3
xy
2
+
2
x
3
6
.
Which graph below corresponds to the following traces:
1. The trace for
y
=
0
.
75
2. The trace for
x
=
-
0
.
4
3. The trace for
x
=
1
.
35
4. The trace for
y
=
-
1
.
8
A
B
C
D
Answer(s) submitted:
•
no response
•
no response
•
no response
•
no response
submitted: (incorrect)
recorded: (incorrect)
Problem 25.
(1 point)
() The graphs below are traces for a function at
x
=
1
and
y
=
-
1
.
Trace for
x
=
1
Trace for
y
=
-
1
If
f
(
x
,
y
) =
ax
2
+
by
then
a
=
and
b
=
.
Answer(s) submitted:
•
no response
•
no response
submitted: (incorrect)
recorded: (incorrect)
Generated by ©WeBWorK, http://webwork.maa.org, Mathematical Association of America
9
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help