Lab 6 calc 1 file
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Towson University *
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Course
273
Subject
Mathematics
Date
Feb 20, 2024
Type
Pages
2
Uploaded by PrivateTeam19514
MATH 273.003
Lab Report #6
1
Name:
1. (3 Points)
Use the
SageMath
demo above Exercise 1 in the online version of this lab to
answer the following questions.
(a)
Use the plot of
f
0
(
x
) on the right to identify the critical points of
f
(
x
).
(b)
Find all values of
x
at which
f
(
x
) has a local extremum. Identify each extremum as
a local maxmimum or a local minimum.
(c)
Find all values of
x
at which
f
(
x
) has an absolute maximum on the interval [0
,
4]
and find all values of
x
at which
f
(
x
) has an absolute minimum on that interval.
2. (3 Points)
Consider the function
f
(
x
) =
-
2 cos(
x
)
-
(cos(
x
))
2
+ 2 sin(
x
)
on the interval [1
,
10]. Use the
SageMath
evaluation cell below Exercise 2 in the online
version of the lab to answer the following questions.
(a)
Define
df(x)
to be the derivative of
f
(
x
). Then, plot the derivative of
f
(
x
) on the
given interval. Attach a printout of the plot to your lab report.
(b)
Observe that
f
0
(
x
) has three roots on the given interval.
Use the
find_root()
command to identify each of these roots. Attach a printout of your code and solutions
to this lab report. Use the
print()
command to display all three roots at once.
(c)
What are the critical points of
f
(
x
) on the interval [1
,
10]?
MATH 273.003
Lab Report #6
2
(d)
Find the absolute maximum and the absolute minimum of
f
(
x
) on the interval [1
,
10].
3. (4 Points)
In this exercise, we investigate the distance between Earth and Mars over the
five-year period from January 1, 2033 (Day 12055) to January 1, 2038 (Day 13881).
(a)
Copy the equations provided above Exercise 3 in the online version of the lab into
the
SageMath
evaluation cell below Exercise 3 to define a function
dist(t)
that
measures the distance between Earth and Mars on day
t
. Then plot the Earth-to-
Mars distance function over the given five-year period. Attach a printout of the plot
to your lab report.
(b)
How many local extrema do you see on this graph?
How many of them are local
maxima and how many local minima?
(c)
Define
ddist(t)
to be the derivative of dist(
t
). Then plot
ddist(t)
over the given
five-year period and attach a printout of this plot to your report.
(d)
Use the
find_root()
command to identify each root of
ddist(t)
shown in the plot
from part (c). Write down the critical points of dist(
t
) in the given five-year period.
(e)
When were the two planets closest during this five-year period?
How close were
they? When were they furthest? How far apart were they?
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