2022W1_MATH_100C_ALL_2022W1.OWDAYLFG3E04.Webwork-Assignment-9-8
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University of British Columbia *
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Course
math 100
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
10
Uploaded by HighnessBravery13778
Lea Grandin
2022W1
MATH
100C
ALL
2022W1
Assignment Webwork-Assignment-9 due 11/24/2022 at 11:59pm PST
Problem 1.
(1 point)
Each row of the spreadsheet below contains the position of a parti-
cle at a given time. The times are stored in column B, and the cor-
responding positions in column A. Column C should contain the
average rate of change between two consecutive rows. To com-
pute these rates of change, you will write a formula in cell C1 and
copy it to cells C2, C3, C4, and C5.
Assume all cells not shown below are blank
position
time
average rate of change
A
B
C
1
0
0
?
2
0
1
↓
3
4
3
↓
4
5
4
↓
5
6
7
↓
6
8
8
a) What formula should you write in C1?
•
(
A
2-
A
1)/(
B
2-
B
1)
•
=(A2-A1)/(B2-B1)
•
=(B2-B1)/(A2-A1)
•
=(
A
2-
A
1)/(
B
2-
B
1)
•
(B2-B1)/(A2-A1)
•
(
B
2-
B
1)/(
A
2-
A
1)
•
(A2-A1)/(B2-B1)
•
=(
B
2-
B
1)/(
A
2-
A
1)
b) Suppose you copy the formula from C1 to cell C6. What num-
ber will be displayed when you press the enter key?
Answer(s) submitted:
•
•
(incorrect)
1
Problem 2.
(1 point)
In the spreadsheet pictured, an arrow indicates the contents of a
cell are copied down its column.
In the spreadsheet below, column A holds different values of
h
,
and column B computes
f
(
0
+
h
)
-
f
(
0
)
h
for the function
f
(
x
) =
|
x
|
.
(Writing
=abs(A1)
in a spreadsheet will compute the absolute
value of the number in cell A1.)
A
B
1
1
=abs(A1)/A1
2
=-A1/10
↓
3
↓
↓
4
↓
↓
5
↓
↓
6
↓
↓
a) What number shows up in cell A5?
b)
f
0
(
0
)
does not exist. Select below the best explanation of a way
you could suspect that from the spreadsheet.
•
The numbers in column B have different signs
•
The values in column B are not converging to one number
•
The numbers in column B have the same absolute value
•
The values in column A are converging to 0
•
The values in column A are not converging to 0
Answer(s) submitted:
•
•
(incorrect)
Problem 3.
(1 point)
Is each of the following functions a solution to the differential
equation
y
00
+
3
y
0
-
10
y
=
0?
?
1.
y
=
-
8
e
3
x
?
2.
y
=
4
e
-
4
x
?
3.
y
=
7
e
2
x
Answer(s) submitted:
•
•
•
(incorrect)
Problem 4.
(1 point)
Find all values of
r
so that the function
y
=
x
r
solves the differen-
tial equation
x
2
y
00
+
2
xy
0
-
12
y
=
0
.
Hint: Plug
y
=
x
r
into the equation and find values of
r
that satisfy
the resulting equation.
r
=
If there are more than one answer, use commas to separate the
answers.
Answer(s) submitted:
•
(incorrect)
Problem 5.
(1 point)
Verify that every member of the family of functions
y
=
ln
x
+
C
x
is a solution of the differential equation
x
2
y
0
+
xy
=
1
.
Answer the
following questions.
1.
Find a solution of the differential equation that satisfies the
initial condition
y
(
3
) =
10
.
Answer:
y
=
2.
Find a solution of the differential equation that satisfies the
initial condition
y
(
10
) =
3
.
Answer:
y
=
Answer(s) submitted:
•
•
(incorrect)
Problem 6.
(1 point)
Let
y
00
-
49
y
=
0.
Find all values of
r
such that
y
=
ke
rx
satisfies the differential
equation.
If there is more than one correct answer, enter your
answers as a comma separated list.
r
=
help (numbers)
Answer(s) submitted:
•
(incorrect)
2
Problem 7.
(1 point)
This problem is an example of a
differential equation
: an equation
that relates a function to one or more of its derivatives. You can
solve this problem by doing some educated guessing. (”educated”
means ”remember what we did in the past.”)
Suppose
f
is the function that satisfies
f
0
(
x
) =
-
f
2
(
x
)
for all
x
in its domain, and
f
(
1
) =
1
.
Then
f
(
x
) =
.
Hint: try some of your familiar functions: parabolas, hyperbolas,
exponential, trig, power functions.
Answer(s) submitted:
•
(incorrect)
Problem 8.
(1 point)
Check by differentiation that
y
=
3cos3
t
+
5sin3
t
is a solution to
y
00
+
9
y
=
0 by finding the terms in the sum:
y
00
=
9
y
=
So
y
00
+
9
y
=
Answer(s) submitted:
•
•
•
(incorrect)
Problem 9.
(1 point)
For each of the following differential equations, determine if the
proposed function is a solution.
a)
Let
C
and
k
be constants.
Given the differential equation
f
0
(
t
) =
k f
(
t
)
, the function
f
(
t
) =
Ce
kt
is
?
b) Given the differential equation
dy
dt
=
2
y
t
, the function
y
(
t
) =
3
t
2
is
?
c)
Given the differential equation
dy
dx
=
-
3
y
,
the function
y
=
2
e
-
3
x
is
?
d) Given the differential equation
f
0
(
t
) =
1
-
f
(
t
)
, the function
f
(
t
) =
-
e
-
t
is
?
Answer(s) submitted:
•
•
•
•
(incorrect)
Problem 10.
(1 point)
In the figure below, which of the following is a solution to the
differential equation
dy
dt
=
0
.
5
y
with initial value
y
(
0
) =
3? [?/I/II/III/IV]
Note: pay attention to axis labels.
Answer(s) submitted:
•
(incorrect)
3
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Problem 11.
(1 point)
Consider the slope field shown.
(a) For the solution that satisfies
y
(
0
) =
0, sketch the
solution curve and estimate the following:
y
(
1
)
≈
and
y
(
-
1
)
≈
(b) For the solution that satisfies
y
(
0
) =
1, sketch the
solution curve and estimate the following:
y
(
1
)
≈
and
y
(
-
1
)
≈
(c) For the solution that satisfies
y
(
0
) =
-
1, sketch the
solution curve and estimate the following:
y
(
1
)
≈
and
y
(
-
1
)
≈
Answer(s) submitted:
•
•
•
•
•
•
(incorrect)
Problem 12.
(1 point)
Consider the slope field shown.
(a) For the solution that satisfies
y
(
0
) =
0, sketch the
solution curve and estimate the following:
y
(
1
)
≈
and
y
(
-
1
)
≈
(b) For the solution that satisfies
y
(
0
) =
1, sketch the
solution curve and estimate the following:
y
(
0
.
5
)
≈
and
y
(
-
1
)
≈
(c) For the solution that satisfies
y
(
0
) =
-
1, sketch the
solution curve and estimate the following:
y
(
1
)
≈
and
y
(
-
1
)
≈
Answer(s) submitted:
•
•
•
•
•
•
(incorrect)
4
Problem 13.
(1 point)
Consider the differntial equation
dy
dt
= (
3
+
y
)
2
.
Part A). Sketch the phase portrait in the space below.
To place a symbol on the line click a symbol button then click a
point on the line.
To remove a symbol from the line click the ”delete” button then
click the symbol on the line.
1
Part B). What happens to solutions with initial conditions
y
(
0
)
>
-
3 as
t
increases?
• ?
• They tend to zero.
• They tend to equilibrium.
• They grow without bound.
• They decrease without bound.
If you entered ’they tend to equilibrium’, what is the value of the
equilibrium? Enter ’NA’ if you chose another answer. The equi-
librium point is
.
Part C). Describe the behavior of solutions with initial conditions
y
(
0
)
<
-
3 as
t
increases.
• ?
• They tend to zero.
• They tend to equilibrium.
• They grow without bound.
• They decrease without bound.
If you entered ’they tend to equilibrium’, what is the value of the
equilibrium? Enter ’NA’ if you chose another answer. The equi-
librium point is
.
Answer(s) submitted:
•
•
•
•
•
(incorrect)
Problem 14.
(1 point)
Use the symbols provided below to sketch the phase line for the
differential equation
x
0
= (
x
+
2
)(
x
-
1
)
.
To place a symbol on the line click a symbol button then click a
point on the line.
To remove a symbol from the line click the ”delete” button then
click the symbol on the line.
1
Answer(s) submitted:
•
(incorrect)
5
Problem 15.
(1 point)
The slope field for the equation
dP
dt
=
0
.
1
P
(
10
-
P
)
,
for
P
≥
0, is shown below.
On a print out of this slope field, sketch the solutions that pass
through
(
0
,
0
)
;
(
1
,
4
)
;
(
4
,
1
)
;
(
-
4
.
5
,
1
)
;
(
-
2
,
12
)
; and
(
-
2
,
10
)
.
For which positive values of
P
are the solutions increasing?
Increasing for:
(Give your answer as an interval or list of intervals, e.g., if P
is increasing between 1 and 5 and between 7 and infinity, enter
(1,5),(7,Inf)
.)
For what positive values of
P
are the solutions decreasing?
Decreasing for:
(Again, give your answer as an interval or list of intervals, e.g., if
P is decreasing between 1 and 5 and between 7 and infinity, enter
(1,5),(7,Inf)
.)
What is the equation of the solution to this differential equation
that passes through (0,0)?
P
=
If the solution passes through a value of
P
>
0, what is lim
t
→
∞
P
(
t
)
?
lim
t
→
∞
P
(
t
) =
Answer(s) submitted:
•
•
•
•
(incorrect)
Problem 16.
(1 point)
The graph of the function
f
(
y
)
is
(the horizontal axis is
y
, the vertical axis is
f
(
y
)
.)
Consider the differential equation
y
0
(
t
) =
f
(
y
)
.
List the steady state solutions to this differential equation in in-
creasing order.
y
=
,
y
=
,
y
=
,
y
=
HINT:
All of your answers should be integers.
Answer(s) submitted:
•
•
•
•
(incorrect)
6
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Problem 17.
(1 point)
Match the following four differential equations to their corre-
sponding slope field diagram below.
i) The differential equation
dy
dx
=
2
y
x
has slope field diagram ?
(0.625 points.)
ii) The differential equation
dy
dx
=
y
(
2
-
y
)
has slope field diagram
?
(0.625 points.)
iii) The differential equation
dy
dx
=
2
y
has slope field diagram ?
(0.625 points.)
iv) The differential equation
dy
dx
=
y
+
x
has slope field diagram
?
(0.625 points.)
A)
B)
7
C)
D)
Answer(s) submitted:
•
•
•
•
(incorrect)
Problem 18.
(1 point)
Numerically approximate the solution to the differential equation
dB
dt
=
0
.
04
B
with initial value
B
=
1400 when
t
=
0. Compute each value using
linear approximation (this method is also called Euler’s method).
A.
Δ
t
=
1 and 1 step:
B
(
1
)
≈
B.
Δ
t
=
0
.
5 and 2 steps:
B
(
1
)
≈
C.
Δ
t
=
0
.
25 and 4 steps:
B
(
1
)
≈
Remark: on an exam, you may be asked to do a few simple itera-
tions of Euler’s method by hand, and you may also be asked about
using a spreadsheet, so make sure you know how to do it both
ways. For this question, Euler’s Method leads to a nice pattern.
If you simplify every step, you may find an easy way to do it by
hand.
Answer(s) submitted:
•
•
•
(incorrect)
8
Problem 19.
(1 point)
Consider the initial value problem
dy
dt
=
y
+
5
t
y
(
0
) =
4
.
In this question we will use a spreadsheet to approximate the value
y
(
1
)
. The picture below shows a spreadsheet being used to imple-
ment a linear approximation scheme (ie.
Euler’s method) with
intial value
y
0
=
y
(
0
) =
4 and a step size of 0.1 to make such an
approximation in 10 steps. A downwards arrow indicates the con-
tents from the cell above will be dragged down.
A
B
C
1
4
0
?
2
?
?
↓
3
↓
↓
↓
4
↓
↓
↓
5
↓
↓
↓
6
↓
↓
↓
7
↓
↓
↓
8
↓
↓
↓
9
↓
↓
↓
10
↓
↓
↓
a) Which entry should go in cell C1? (1 point.)
•
A. =B1+0.1
•
B. =A1+0.1*5*B1
•
C. =B1+1
•
D. =A1+ 4*B1
•
E. =A1+ B1
•
F. =A1+5* B1
b) Which entry should go in cell A2? (1 point.)
•
A. =0.1
•
B. B1+0.1*C1
•
C. =A1+5 *B1
•
D. =A1+0.1*C1
•
E. =A1+0.1*B1
•
F. =A1+0.1
c) Which entry should go in cell B2? (1 point.)
•
A. =C1+0.1
•
B. =A1+5 *C1
•
C. =A1+5 *B1
•
D. =0
•
E. =B1+0.1
•
F. =1
Now, using a spreadsheet, or otherwise, make the following ap-
proximations of y(0) for decreasing step sizes
Δ
t
. Give your an-
swers correct to at least 3 decial places.
d) Using Euler’s method with 1 step (
Δ
t
=
1), starting at
y
(
0
) =
4
we find
y
(
1
)
≈
(0.5 points.)
e) Using Euler’s method with 2 steps (
Δ
t
=
1
2
), starting at
y
(
0
) =
4
we find
y
(
1
)
≈
(0.5 points.)
f) Using Euler’s method with 10 steps (
Δ
t
=
1
10
), starting at
y
(
0
) =
4 we find
y
(
1
)
≈
(1 point.)
Answer(s) submitted:
•
•
•
•
•
•
(incorrect)
9
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Problem 20.
(1 point)
Consider the differential equation
dy
dx
=
3
x
,
with initial condition
y
(
0
) =
4.
A.
Use linear approximation (Euler’s method) with two steps to
estimate
y
when
x
=
1:
y
(
1
)
≈
(Be sure
not
to round your calculations at each step!)
Now use four steps:
y
(
1
)
≈
(Be sure
not
to round your calculations at each step!)
B.
What is the solution to this differential equation (with the given
initial condition)?
y
=
(Try using your knowledge of derivatives to guess the solution,
and then if needed, you can use Wolfram Alpha to look it up.
Hint
: this is asking for a function that has a derivative of
3
x. What
function do you have to start with to get that derivative?)
C.
What is the magnitude of the error in the two Euler approxima-
tions you found?
Magnitude of error in Euler with 2 steps =
Magnitude of error in Euler with 4 steps =
Note that in this example, the right hand side of the differential
equation is a function of the independent variable
x
instead of the
dependent variable
y
. Nonetheless, you can apply Euler’s method.
Answer(s) submitted:
•
•
•
•
•
(incorrect)
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10