Calculus Lab Week 2 Write-up
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School
San Francisco State University *
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Course
226
Subject
Mathematics
Date
Feb 20, 2024
Type
Pages
11
Uploaded by yinkukchow
Yin Kuk Chow
Griffin Wetzel
Ayelen Higbee
McKenna Kelly
Zachary Leyden
Math 226-07
8 February 2024
Calculus Lab Week 2 Write-up
1.
Demonstrate and Explain
(a) Questions from the Lab 2A Rocket Launch
i. (Part 2) Sketch a graph of h(t) and describe how the shape of the graph reflects
changes in the astronauts' speed over time.
The shape of the graph reflects that the speed of the astronaut increases rapidly over time,
as the height of the rocket also increases rapidly over time.
ii. (Part 4) Show which graph you chose for v(t) and explain what aspects of that
graph lead you to believe that is the best fit.
The average speed of the rocket over several different 10 seconds time intervals is:
v(10) = (1520-0)/(20-0) = 76 feet per second
v(20) = (3330-390)/(30-10) = 147 feet per second
v(30) = (5760-1520)/(40-20) = 212 feet per second
v(40) = (8750-3330)/(50-30) = 271 feet per second
v(50) = (12240-5760)/(60-40) = 324 feet per second
With the method of elimination, we eliminated the top graph because the speed of the
rocket should not be anything other than 0 when t=0. That left us with the middle and the
bottom graph. According to the calculation above, v(50) = 324 feet per second.
Therefore, we can eliminate the middle graph. And this is the graph my team selected:
iii. (Part 5) If you could zoom in very carefully on the velocity graph you choose in
part d, you will be able to see that v(10) = 77. This means that 10 seconds after
launch, the rocket is traveling at a speed of 77 ft/sec. You might expect that this
means that the rocket travels 77 feet per second, so over the course of the next ten
seconds, it should travel 770 feet, however, we can see that this is not actually true.
How many feet does the rocket actually travel during the seconds between t = 10
seconds and t = 20 seconds? Explain why this number is not 770 feet.
The velocity of the rocket is at a speed of 77 feet per second, but it only means the rocket
is traveling at that speed at that particular time when t=10, in other words it is its
instantaneous rate of change and it does not indicate its travel distance. In order to
calculate its travel distance, we need to look at h(t) instead of v(t). The distance traveled
by the rocket at t=10 is 390 feet, and the distance traveled by the rocket at t=20 is 1520
feet. Therefore, we can calculate that the rocket actually traveled 1130 feet (1520-390)
between t = 10 seconds and t = 20 seconds.
(b) Questions from the Lab 2B Limits at a Point
i. (Part 1a) Explain how to determine which of the three graphs matches each of the
functions of g(x), h(x), and k(x).
; graph (a)
This is the only graph that shows a linear equation that matches f(x).
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;
graph (b)
There is an asymptote at x=3, this is the only graph that matches g(x).
; graph (c)
It is not defined when x=3, hence it is a hole on the line of the graph. The only graph with
a hole is graph (c).
; graph (d)
This is the only graph that matches k(x).
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ii. (Part 1f) Explain using the graphs, tables and equations if the limits in Part f exist
and if so, what they are.
The table below shows the value of g(x), h(x) and k(x) respectively as x approaches to 3.
x
2.5
2.9
2.99
2.999
3.001
3.01
3.1
3.5
g(x)
0
-8.8
-98.98
-998.998
1000.998
100.98
10.8
2
h(x)
3
2.2
2.02
2.002
1.998
1.98
1.8
1
k(x)
1
0.2
0.02
0.002
1.998
1.98
1.8
1
The graphs below shows the value of g(x), h(x) and k(x) respectively as x approach 3.
From the equations for g(x), h(x) and k(x),
g(3), h(3) and k(3) are all undefined. Therefore, we could not simply evaluate g(3), h(3)
and k(3) to identify its limit when x approaches 3.
From the table and the graphs above, we can indicated that:
For g(x), when x<3, the value is approaching
-1000
when x gets close to 3; when x>3, the
value is approaching
1000
when x gets close to 3.
For h(x), when x<3, the value is approaching
2
when x gets close to 3; when x>3, the
value is approaching
2
when x gets close to 3.
For k(x), when x<3, the value is approaching
0
when x gets close to 3; when x>3, the
value is approaching
2
when x gets close to 3.
Therefore, we can conclude that:
=
DNE
=
2
=
DNE
iii. (Part 2b & 2c) Show using table and graphs how to determine the limits in Part
2b & 2c.
Part 2b
The following table shows the value of the function as x approaches 3.
x
2.9
2.99
2.999
3.001
3.01
3.1
f(x)
5.3574
5.5260
5.5433
5.5471
5.5644
5.7419
Here is the graph of the function
From the table and the graphs above, we can indicated that:
When x<3, the value is approaching
5.54
when x gets close to 3; when x>3, the value is
approaching
5.54
when x gets close to 3.
Therefore, we can conclude that:
=
5.54
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Part 2c
The following table shows the value of the function as x approaches 0.
x
-0.1
-0.01
-0.001
0.001
0.01
0.1
f(x)
0.99833
0.99998
0.99999
0.99999
0.99998
0.99833
Here is the graph of the function
From the table and the graphs above, we can indicated that:
When x<0, the value is approaching
1
when x gets close to 3; when x>0, the value is
approaching
1
when x gets close to 3.
Therefore, we can conclude that:
=
1
2.
Summarize:
In the “Rocket Launch” , the big ideas are average rate of change and instantaneous rate
of change. It is important to understand the difference between the two. Average rate of
change represents the total change in one variable in relation to the total change of
another variable. Instantaneous rate of change measures the specific rate of change at a
particular value. In the “Limits at a Point”, the limits are the method by which the rate of
change of a function is calculated. It is defined as the value that the function approaches
as it goes to an x value.
3.
Reflect:
For the lab this week, I learnt from the mistakes of the lab last week and tried to engage
more in the group. I become more proactive and the impact is positive. The
communication in our group improved significantly and the efficiency increased as well.
Our group was always able to complete the task in the lab early. The only thing I need to
improve my learning experience is that I should try to fact-check the answers from my
peers, so that I will have a much better understanding of the concepts and have a better
learning experience.