MATH 235 Geogebra Directions and Applied Optimization Questions This assignment explores two real world problems that require applied optimization to solve

MATH 235 Geogebra Directions and Applied Optimization Questions This assignment explores two real world problems that require applied optimization to solve. These problems involve functions that are relatively complicated, meaning that their derivatives will be challenging to work with. Instead of finding critical points by hand, you will learn how to use Geogebra to assist you in these algebraic and calculus-related computations. Materials: lab and text attached, https://www.geogebra.org/classic MATH 235 Lab #6:
Applied Optimization
Directions:
Format: Use Geogebra to complete all problems. Solutions should be summited as a lab
report. A “lab report” means that more explanation should be provided compared to a
homework assignment. Your name should be located at the top of your report as well as Lab
#6. Then, you should write the problem number and solution. Your solution should include
all the work required to answer the question including explanations in complete sentences.
You may choose to typeset or hand-write your solutions, but note that there are places where
you will have to include graphics from Geogebra in your work. Problems should be in
numerical order. Do not provide several pages of hand written work with all Geogebra
graphics at the end. The report needs to be easy to follow.
Grading: This lab is worth 100 points. Details are provided below.
1a.) 5 pts
1b.) 15 pts
1c.) 5 pts
1d.) 10 pts
1e.) 10 pts
1f.) 5 pts
2a.) 5 pts
2b.) 15 pts
2c.) 5 pts
2d.) 10 pts
2e.) 10 pts
2f.) 5 pts
——————————————————————————————————————–Overview
This assignment explores two real world problems that require applied optimization to
solve. (The necessary concepts are discussed in Section 3.4.). These problems involve
functions that are relatively complicated, meaning that their derivatives will be challenging
to work with. Instead of finding critical points by hand, you will learn how to use Geogebra
to assist you in these algebraic and calculus-related computations.
——————————————————————————————————————–Problems:
1.) Two vertical poles of heights 60 ft and 80 ft stand on level ground, with their bases 100 ft
apart. A cable is stretched from the top of one pole to some point on the ground between the
poles, and then to the top of the other pole. You need to find the minimum length of cable
that could be used to do this by following the steps below:
a. Draw a diagram illustrating this situation. Introduce notation/variables and label the
diagram with your symbols.
b. Based on your diagram and labels in part (a), determine a formula that represents the
total length of the cable needed as a function of the distance from the base of the one of
poles to the point on the ground where the cable touches. Make sure to show your work
and state the formula you determine. You MUST explain all calculations and equations
used to create this formula. (You may want to check your answer with the professor
before continuing.)
MATH 235
1
c. Find any restrictions on your independent variable. Write one sentence that discusses
and justifies the domain of the function based on these restrictions.
d. What are the critical numbers of the function you found in part (b)? Use the Geogebra
directions in the box below to complete this problem. Write your critical numbers as
decimals rounded to 5 places and include units of measurement. State your final answer
by writing a couple of sentences explaining what Geogebra did and referring to the screen
shot of the work you completed in Geogebra. Note that I want a picture of your work in
the input lines, not necessarily the graph. The graph will only show up if you change the
numbers on the axes appropriately.
Geogebra Directions for Finding Critical Numbers
Step 1: Enter your expression from part (b) as a function, f(x), in Geogebra. This means,
in the input line, we type the following and press enter.
f(x)=(put your function here)
Step 2: In another input line, type the following command and press enter.
Roots(f ’(x), 0, 100)
This single command takes the derivative of your function, and then solves the equation
for where the derivative is zero on the interval [0,100]. (Root is another way of saying: the
solution to an equation that is set equal to zero.) The command will report the roots as
ordered pairs such as (3, 0). The first number in the ordered pair is the root (the number
that causes the derivative to equal zero). The second number in the ordered pair tells us
that the first derivative is zero at this point.
To change the number of decimal places, click on the Graphics View Toolbar
on
the upper right side of the screen. Then click on the gear icon
. Finally click on the
other gear icon that appears below the ×. Change the number of decimals as appropriate.
Include a screen shot of your Geogebra work for this part.
e. What is the minimum length of cable needed? Justify your answer using the calculus ideas
we have been discussing before this. Explain what you are doing using complete
sentences. Remember that you can have Geogebra calculate function values by typing f(a
number) into an input line.
f. Where should we locate the place where the cable touches the ground in order to achieve
the minimum length? Be specific.
MATH 235
2
2.) A pipeline for transporting oil will connect two points A and B that are 3 miles apart (as
the crow flies) and on opposite banks of a straight river one mile wide. (Note that A and B
must not be directly across from each other based on the information given so far.) Part of
the pipeline will run under water from point A to a point C on the other side of the river. Then
the pipeline will run above ground from point C to point B. If the cost per mile of running the
pipeline under water is three million dollars a mile, while the cost per mile of running it
above ground is two million dollars a mile, find the location of point C that will minimize
the cost (ignoring the slope of the river bed). Hint: Let D be the point directly across the river
from A so that C is between D and B.
a. Draw a diagram illustrating this situation. Introduce notation/variables and label the
diagram with your symbols.
b. Based on your diagram and labels in part (a), determine a formula that represents the
total cost of the pipeline. Make sure to show your work and state the formula you
determine. You MUST explain all calculations and equations used to create this formula.
(You may want to check your answer with the professor before continuing.)
c. Find any restrictions on your independent variable. Write one sentence that discusses
and justifies the domain of the function based on these restrictions.
d. What are the critical numbers of the function you found in part (b)? Use the Geogebra
directions in the previous problem to do this. Write your critical numbers as decimals
rounded to 5 places and include units of measurement. State your final answer by writing
a couple of sentences explaining what Geogebra did and referring to the screen shot of
the work you completed in Geogebra.
e. What is the minimum cost of the pipeline? Justify your answer using the calculus ideas
we have been discussing before this. Explain what you are doing using complete
sentences. Remember that you can have Geogebra calculate function values by typing f(a
number) into an input line.
f. Where should we locate point C in order to minimize the cost of the pipeline? Be specific.
MATH 235
3
ACTIVE CALCULUS
2018 Edition – UPDATED
Matthew Boelkins
David Austin
Steven Schlicker
Active Calculus
Active Calculus
Matthew Boelkins
Grand Valley State University
Contributing Authors
David Austin
Grand Valley State University
Steven Schlicker
Grand Valley State University
Production Editor
Mitchel T. Keller
Morningside College
July 2, 2019
Cover Photo: James Haefner Photography
Edition: 2018 Updated
Website: http://activecalculus.org
©2012–2019 Matthew Boelkins
Permission is granted to copy and (re)distribute this material in any format and/or adapt it
(even commercially) under the terms of the Creative Commons Attribution-ShareAlike 4.0
International License. The work may be used for free in any way by any party so long as attribution is given to the author(s) and if the material is modified, the resulting contributions
are distributed under the same license as this original. All trademarks™ are the registered®
marks of their respective owners. The graphic
that may appear in other locations in the text shows that the work is licensed with the Creative Commons and that the work may be used for free by any party so long as attribution
is given to the author(s) and if the material is modified, the resulting contributions are distributed under the same license as this original. Full details may be found by visiting
https://creativecommons.org/licenses/by-sa/4.0/ or sending a letter to Creative Commons,
444 Castro Street, Suite 900, Mountain View, California, 94041, USA.
vi
Acknowledgements
This text began as my sabbatical project in the winter semester of 2012, during which I wrote
most of the material for the first four chapters. For the sabbatical leave, I am indebted to
Grand Valley State University for its support of the project, as well as to my colleagues in
the Department of Mathematics and the College of Liberal Arts and Sciences for their endorsement of the project. I’m also grateful to the American Institute of Mathematics for their
support of free and open textbooks in general, and their support of this one in particular.
The beautiful full-color .eps graphics in the text are only possible because of David Austin
of GVSU and Bill Casselman of the University of British Columbia. Building on their longstanding efforts to develop tools for high quality mathematical graphics, David wrote a library of Python routines that employ Bill’s PiScript program; David’s routines are so easy
to use that even I could generate graphics like the professionals that he and Bill are. I am
deeply grateful to them both.
The current .html version of the text is possible only because of the amazing work of Rob
Beezer and his development of the original Mathbook XML, now known as PreTeXt. My
ability to take advantage of Rob’s work is due in large part to the support of the American
Institute of Mathematics, which funded me to attend a weeklong workshop in Mathbook
XML in San Jose, CA, in April 2016, as well as the support of the ongoing user group. A
subsequent workshop in June 2019 has offered further support and more improvements to
the text.
David Farmer’s conversion script saved me hundreds of hours of work by taking my original
LATEX source and converting it to PreTeXt; David remains a major source of ongoing support
and advocacy. Alex Jordan of Portland Community College has also been a tremendous
help, and it is through Alex’s fantastic work that live WeBWorK exercises are not only possible, but also included from the 2017 version forward. Mitch Keller of Morningside College
agreed in early 2018 to serve as the book’s production editor; his technical expertise has
contributed to many aspects of the book, including the presence of answers to activities and
non- WeBWorK exercises and other supporting material for instructors.
For the 2018 edition, Kathy Yoshiwara of the AIM Editorial Board read the entire text and
contributed editorial suggestions for every section. In short, she made the prose cleaner,
more direct, and simply better. I’m deeply thankful for her time, effort, and insights.
Over my more than 20 years at GVSU, many of my colleagues have shared with me ideas
and resources for teaching calculus. I am particularly indebted to David Austin, Will Dickinson, Paul Fishback, Jon Hodge, and Steve Schlicker for their contributions that improved my
teaching of and thinking about calculus, including materials that I have modified and used
over many different semesters with students. Parts of these ideas can be found throughout
this text. In addition, Will Dickinson and Steve Schlicker provided me access to a large number of their electronic notes and activities from teaching of differential and integral calculus,
and those ideas and materials have similarly impacted my work and writing in positive ways,
with some of their problems and approaches finding parallel presentation here.
In the summer of 2012, David and Steve each agreed to write a chapter to support the completion of the material on integral calculus. David is the lead author of Chapter 7 and Steve
the lead author of Chapter 8. Along with our colleague Ted Sundstrom, Steve has also contributed a large number of problem and activity solutions and answers. I’m especially grateful for how the work of these friends and colleagues has made the text so much better.
Shelly Smith of GVSU and Matt Delong of Marian University both provided extensive comments on the first few chapters of early drafts, feedback that was immensely helpful in improving the text. As more and more people use the text, I am grateful to everyone who
reads, edits, and uses this book, and hence contributes to its improvement through ongoing
discussion.
Finally, I am grateful for all that my students have taught me over the years. Their responses
and feedback have helped to shape me as a teacher, and I appreciate their willingness to
wholeheartedly engage in the activities and approaches I’ve tried in class, to let me know
how those affect their learning, and to help me learn and grow as an instructor. Early on,
they also provided useful editorial feedback on this text.
Any and all remaining errors or inconsistencies are mine. I will gladly take reader and user
feedback to correct them, along with other suggestions to improve the text.
viii
Contributors
A large and growing number of people have generously contributed to the development or
improvement of the text. Contributing authors David Austin and Steven Schlicker have each
written drafts of at least one full chapter of the text. Production editor Mitchel Keller has
been an indispensable source of technological support and editorial counsel.
The following contributing editors have offered feedback that includes information about
typographical errors or suggestions to improve the exposition.
David Austin
GVSU
Rene Ardila
GVSU
Patti Hunter
Westmont College
Mitchel Keller
Morningside College
Allan Bickle
GVSU
Sam Kolins
Lebanon Valley College
David Clark
GVSU
Will Dickinson
Dave Kung
St. Mary’s College of Maryland
GVSU
Nate Eldredge
University of Northern Colorado
Charles Fortin
Champlain Regional College
St-Lambert, Quebec, Canada
Marcia Frobish
GVSU
Teresa Gonske
University of Northwestern – St. Paul
Paul Latiolais
Portland State University
Hugh McGuire
GVSU
Martin Mohlenkamp
Ohio University
Ray Rosentrater
Westmont College
Luis Sanjuan
Conservatorio Profesional de Musica de Avila
Spain
Michael Santana
GVSU
Steven Schlicker
GVSU
Amy Stone
GVSU
Robert Talbert
GVSU
Greg Thull
GVSU
Michael Shulman
University of San Diego
Sue Van Hattum
Contra Costa College
Brian Stanley
Foothill Community College
Kathy Yoshiwara
x
AIM Editorial Board
Active Calculus: Our Goals
Several fundamental ideas in calculus are more than 2000 years old. As a formal subdiscipline of mathematics, calculus was first introduced and developed in the late 1600s, with
key independent contributions from Sir Isaac Newton and Gottfried Wilhelm Leibniz. The
subject has been understood rigorously since the work of Augustin Louis Cauchy and Karl
Weierstrass in the mid 1800s when the field of modern analysis was developed. As a body
of knowledge, calculus has been completely understood for at least 150 years. The discipline is one of our great human intellectual achievements: among many spectacular ideas,
calculus models how objects fall under the forces of gravity and wind resistance, explains
how to compute areas and volumes of interesting shapes, enables us to work rigorously
with infinitely small and infinitely large quantities, and connects the varying rates at which
quantities change to the total change in the quantities themselves.
While each author of a calculus textbook certainly offers their own creative perspective on
the subject, it is hardly the case that many of the ideas they present are new. Indeed, the
mathematics community broadly agrees on what the main ideas of calculus are, as well as
their justification and their importance. In the 21st century and the age of the internet, no one
should be required to purchase a calculus text to read, to use for a class, or to find a coherent
collection of problems to solve. Calculus belongs to humankind, not any individual author
or publishing company. Thus, a primary purpose of this work is to present a calculus text
that is free. See https://activecalculus.org for links to both the .html and .pdf versions of the
text. In addition, instructors who are looking for a calculus text should have the opportunity
to download the source files and make modifications that they see fit; thus this text is opensource. See GitHub for the source. Since August 2013, Active Calculus – Single Variable has
been endorsed by the American Institute of Mathematics and its Open Textbook Initiative.
In Active Calculus – Single Variable, we actively engage students in learning the subject through
an activity-driven approach in which the vast majority of the examples are generated by
students. Where many texts present a general theory followed by substantial collections of
worked examples, we instead pose problems or situations, consider possibilities, and then
ask students to investigate and explore. Following key activities or examples, the presentation normally includes some overall perspective and a brief synopsis of general trends or
properties, followed by formal statements of rules or theorems. While we often offer plausibility arguments for such results, rarely do we include formal proofs. It is not the intent of
this text for the instructor or author to demonstrate to students that the ideas of calculus are
coherent and true, but rather for students to encounter these ideas in a supportive, leading
manner that enables them to begin to understand calculus for themselves. This approach is
consistent with the scholarly consensus that calls for students to be interactively engaged in
class.
Moreover, this approach is consistent with the following goals:
• To have students engage in an active, inquiry-driven approach, where learners construct solutions and approaches to ideas, with appropriate support through questions
posed, hints, and guidance from the instructor and text.
• To build in students intuition for why the main ideas in calculus are natural and true.
Often we do this through consideration of the instantaneous position and velocity of
a moving object.
• To challenge students to acquire deep, personal understanding of calculus through
reading the text and completing preview activities on their own, working on activities
in small groups in class, and doing substantial exercises outside of class time.
• To strengthen students’ written and oral communicating skills by having them write
about and explain aloud the key ideas of calculus.
xii
Features of the Text
Instructors and students alike will find several consistent features in the presentation, including:
Motivating Questions At the start of each section, we list 2–3 motivating questions that provide motivation for why the following material is of interest to us. One overall goal of
each section is to answer each of the motivating questions.
Preview Activities Each section of the text begins with a short introduction, followed by a
preview activity. This brief reading and the preview activity are designed to foreshadow
the upcoming ideas in the remainder of the section; both the reading and preview
activity are intended to be accessible to students in advance of class, and to be completed
by students before the day on which a particular section is to be considered.
Activities A typical section in the text has at least three activities. These are designed to
engage students in an inquiry-based style that encourages them to construct solutions
to key examples on their own, working individually or in small groups.
Exercise…
Purchase answer to see full
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