Table Of ContentJuan J. Prieto-Valdés
BUSINESS CALCULUS
Notebook
2nd Edition
Practicing Calculus, Statistics, and Technology.
YOURS NAME: _____________________________________________________________
April 2021
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Business Calculus Notebook
About the author
Juan Jose Prieto-Valdes (Juan Prieto); An accomplished educator, committed to student success; under whose
guidance and supervision a large number of students have been prepared for careers, new programs, and a
broad group of projects has been implemented. Prieto-Valdes holds a Doctor of Philosophy Degree in Physical
& Mathematical Sciences. Over 20 years of educational experience teaching at undergraduate and graduate
levels, advisor in MSc and PhD Thesis and Dissertations, author of more than 100 academic papers with
internationally recognized results in materials research, environmental control technologies, math modeling,
and business applications. Dr. Prieto-Valdés practices fine art and literature hobbies, he owns a valuable number
of artistic copyrights, however, based in his own words: one of his preferred activities is to teach mathematics,
mostly, using real application examples.
Some words from the author
In this notebook is intended to improve business calculus learning conferring special attention to the today
data driven society. This is not a regular textbook, it was firstly prepared to discuss new trends in teaching
calculus at the international conference, and later was adopted to improve traditional business calculus course.
In addition to traditional calculus topics, a group of new challenging computer-aided exercises are included;
Microsoft Mathematics and Excel are introduced to solve calculus problems, as well as currently online
available software and engines as referred at the preface pages. Elements of the big data analytics are also
included and new calculus-statists basic overview for business application. A group of unpublished before
exercises served to teach how to optimize the profit, the material consumption, the traffic light timeline
intervals, the optimal route of evacuation, stocks, portfolio and other real case examples.
I structured this book as a working notebook, leaving some free spaces to complete the lecture contents and to
perform the homework practice. I have observed that when the students take their notes in the same place
where the subject is exposed, they had better organized their questions and thoughts, notably improving
learning process.
I have to confess that I spend a lot of time organizing the material, creating new problems and formulating real
live examples, all of them firstly published hereto.
Truly yours;
Juan José Prieto-Valdes
BUSINESS CALCULUS Notebook, 2nd Edition
Practicing Calculus, Statistics, and Technology.
ISBN: 9798729023608
All Rights Reserved.
The Cataloging-in-Publication Data was filled at the US Library of Congress
Included materials:
Business Calculus. © 2014, Juan J. Prieto-Valdes, Library of Congress, Registration: TXu001954964
Pre-Calculus. © 2016, Juan J. Prieto-Valdes, Library of Congress, Registration TXu2014602
Trigonometry. © 2015, Juan J. Prieto-Valdes, Library of Congress, Registration TXu0019744949
College Algebra. © 2013, Juan J. Prieto-Valdes, Library of Congress, Registration TXu001880424
Screenshots to show software application with reference to the original online mechanism.
Computer aided examples and projects developed by Prof. Prieto-Valdes.
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Juan J. Prieto-Valdés
TABLE OF CONTENTS
About the author ................................................................................................................................................... 2
Some words from the author ................................................................................................................................ 2
Preface ................................................................................................................................................................... 5
CHAPTER-1. LIMITS ......................................................................................................................... 7
1.1. The Idea of a Limit ..................................................................................................................................... 7
1.2. Formal Limit Definition ............................................................................................................................ 12
1.3. Determining Limits and Algebraic Properties of Limits ........................................................................... 14
Homework Topics 1.1, 1.2, and 1.3 ..................................................................................................................... 18
1.4. Continuity, Intermediate Value Property, and One-Side Limits. ............................................................. 22
1.5. Limits at the Infinity (Analyzing also One Side Limit) .............................................................................. 28
Homework Topic 1.4 and 1.5. .............................................................................................................................. 30
1.6. Example of Limit Calculation Using Technology ...................................................................................... 33
PROJECT-1. Find the Limit Using technology. ..................................................................................................... 34
PRACTICING QUIZ-1. Limits ................................................................................................................................. 41
CHAPTER-2. THE DERIVATIVE AND DIFFERENTIATION .................................................. 42
2.1. Definition of Derivative ................................................................................................................................ 42
Homework Topic 2.1 ............................................................................................................................................ 48
PROJECT-2. The Derivative Using Limit Definition .............................................................................................. 50
2.2. Techniques of differentiation. ................................................................................................................. 53
2.3. Product and Quotient Rules .................................................................................................................... 53
Homework Topic 2.2 and 2.3. .............................................................................................................................. 54
2.4. The Chain Rule ......................................................................................................................................... 60
2.5. Implicit Differentiation. ........................................................................................................................... 62
2.6. High Order Derivatives (second, third, …): .............................................................................................. 64
Homework Topics 2.4, 2.5, and 2.6. .................................................................................................................... 64
PRACTICING QUIZ-2. Derivatives ......................................................................................................................... 69
2.7. Real Case Examples (Marginal Analysis and Approximations) ................................................................ 70
Homework Topic 2.7. ........................................................................................................................................... 72
2.8. Increasing and Decreasing Functions (critical points) ............................................................................. 74
2.9. Rolle's Theorem and The Mean Value Theorem ..................................................................................... 76
Homework Topic 2.8 and 2.9. .............................................................................................................................. 78
2.10. Concavity and Point of Inflexion .......................................................................................................... 80
Homework Topic 2.10.......................................................................................................................................... 82
PRACTICING QUIZ-3. Application of Derivatives.................................................................................................. 87
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Business Calculus Notebook
CHAPTER-3. EXPONENTIAL and LOGARITHMIC FUNCTIONS. OPTIMIZATIONS. ....... 88
3.1. Remembering Exponential and Logarithmic Functions (an Algebra Topic): ............................................ 88
Homework Topic 3.1 ............................................................................................................................................ 94
3.2. Differentiation of Exponential Functions ............................................................................................... 104
3.3. Differentiation of Logarithmic Functions ............................................................................................... 106
Homework Topic 3.2 and 3.3 ............................................................................................................................. 107
3.4. L’ Hospital Rule / Revisiting Limits .......................................................................................................... 111
Homework Topic 3.4. ......................................................................................................................................... 113
3.5. Optimization and Business Applications Examples. ............................................................................... 114
Homework Topic 3.5. ......................................................................................................................................... 120
PRACTICING QUIZ-4. Optimization, Exponential and Logarithmic Functions. ................................................... 127
PROJECT-3. Quick Example of a Business Profit Optimization ........................................................................... 128
CHAPTER-4. INTEGRATION ....................................................................................................... 132
4.1. Indefinite Integration and Simple Differential Equation ........................................................................ 132
4.2. Integration by substitution ..................................................................................................................... 132
Homework Topic 4.1 and 4.2. ............................................................................................................................ 134
4.3. Riemann Integration / Riemann Sum. .................................................................................................... 138
Homework Topic 4.3 .......................................................................................................................................... 140
4.4. Fundamental Theorem of Calculus and Definite Integral ...................................................................... 140
4.5. Application of Integrals (Average Function Value). ................................................................................ 142
Homework Topic 4.4 and 4.5. ............................................................................................................................ 143
4.6. Application of Integrals (Area Under and Between Curves). ................................................................. 144
Homework Topic 4.6 .......................................................................................................................................... 148
PRACTICING QUIZ-5. Integrals and its applications ............................................................................................ 150
PROJECT – 4. Optimization of the Material Cost. ............................................................................................... 151
PROJECT – 5. Profit Optimization (Transport Marine Company) ....................................................................... 156
PROJECT – 6. Optimization of fictitious city evacuation. ................................................................................... 157
CHAPTER 5. INTRODUCTORY STATISTICS WITH CALCULUS. .......................................... 158
5.1. Continuous Random Variable, Probability, and Histograms. ................................................................. 158
5.2. Probability Density Functions: Uniform, Exponential, and Normal. ...................................................... 158
5.3. Uniform Density Function ...................................................................................................................... 162
5.4. Exponential Density Functions ............................................................................................................... 163
5.5. Normal Distribution ................................................................................................................................ 166
5.6. Mean and Median. ................................................................................................................................. 170
5.7. Variance and Standard Deviation ........................................................................................................... 174
5.8. Variance in different types of distributions: .......................................................................................... 180
Homework Chapter 5 ......................................................................................................................................... 182
PROJECT – 7. Portfolio Optimization .................................................................................................................. 183
REVIEW CHAPTER. ....................................................................................................................... 188
R-1. Some Elementary Algebra Concepts ......................................................................................... 188
R-2. Algebra Review .......................................................................................................................... 192
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Juan J. Prieto-Valdés
Preface
Learning depends on the interest in gain knowledge that learners have. If you use technology to formalize a method to
copy the answer of mathematical fitness, you will develop a brute skill to complete an assignment. However, using
technology to learn different method of solutions and systematic procedures, checking graphical representations and
physical meaning of the mathematical exercises will substantially help to develop your mathematical and analytical
expertise.
The following are the most popular technological recourses currently available for learning mathematics on Internet.
Wolfram Alpha and Calculus Course Assistant.
https://www.wolframalpha.com/ and/or https://products.wolframalpha.com/courseassistants/calculus.html
Using this application, you can evaluate any numeric expression or substitute a value for a variable. Plot
basic, parametric, or polar plots of the function(s) of your choice. Determine the limit of a function as it
approaches a specific value. Differentiate any function or implicit function. Find critical points and
inflection points of a function; identify the local and absolute extrema of a function. Integrate a function,
with or without limits, sum a function given a lower and upper bound. Find a closed form of a sequence or generate
terms for a specific sequence; among other math applications.
Microsoft Math. Add-in for Word Office.
https://www.microsoft.com/en-us/download/details.aspx?id=17786
Microsoft Mathematics Add-in for Microsoft Word makes it easy to plot graphs in 2D and 3D, solve
equations or inequalities, and simplify algebraic expressions in your Word documents and OneNote.
Using this add-in you can compute standard mathematical functions, such as roots and logarithms,
trigonometric functions, find sums and products of series, matrix and complex numbers operations,
solve equations and inequalities, factor polynomials or integers, simplify or expand algebraic expressions,
and more advanced operation.
Microsoft Excel
Microsoft Excel is a helpful and powerful program for data analysis and documentation. It is a
spreadsheet program, which contains a number of columns and rows, where each intersection of a
column and a row is a “cell.” Each cell contains one point of data or one piece of information. With
Excel you can import, export, and convert data. Use editing formula to perform diverse mathematical
calculations on your data, business application and analysis. You can use formatting to create your own
style. In addition, you can create charts and functional graphics with your data.
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Business Calculus Notebook
Desmos Graphing Calculator- DG
https://www.desmos.com/
Desmos is an advanced graphing calculator implemented as a web and mobile application. Users can
create accounts and save the graphs and plots that they have created. Any algebraic function/expression
can be graphically graphed and analyzed.
GeoGebra-GG
https://www.geogebra.org/
GeoGebra is an interactive mathematics software suit for learning science, technology, engineering, and
mathematics from primary school up to the university level. Constructions can be made with points,
vectors, segments, lines, polygons, conic sections, inequalities, implicit polynomials and functions.
Symbolab-Sy
https://www.symbolab.com/
Symbolab is an answer online engine/service that computes step-by-step solutions to mathematical
problems in a range of subjects. Symbolab can interpret a user-entered equation or symbolic problem and
find the solution if it exists and step-by-step solution.
Math Solver (Matrix Calculator)-MS
https://www.math10.com/scripts/matrices/determinant-matrix-calculators.html
Math Solver offers solving fraction, metric conversions, power and radical problems. Area
and volume of rectangles, circles, triangles, trapezoids, boxes, cylinders, cones, pyramids,
spheres. You can simplify and evaluate expressions, and more advanced problems
including multiplication, division, and matrix.
LaTeX Editor
http://www.codecogs.com/latex/eqneditor.php
http://www.numberempire.com/texequationeditor/equationeditor.php
LaTeX is a document preparation system widely used in academia for the communication and publication of scientific
documents in many fields, including mathematics, statistics, computer science, engineering, chemistry, physics,
economics, linguistics, quantitative psychology, philosophy, and political science.
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Juan J. Prieto-Valdés
CHAPTER-1. LIMITS
Read Chapter R (page 189) about basics on algebra before working with differentiation
Nowadays graphs of functions are easy to produce using computers; however, calculus persists as the perfect tool for
graphing, but most important, to explain and to predict the behavior of a function, extremely useful when solving real
problems in different disciplines; business, economics, biology, engineering, environmental, physical and chemical
sciences, among many other fields and applications.
The two basic operations in calculus are derivatives and integrals, which give proper name to the two primary parts of
the subject: differential and integral calculus. In order to understand these primaries calculus workings, is required to
understand the concept of a limit. This concept is probably the most difficult to understand at the beginning, it took
mathematicians 150 years to arrive at it; remaining today the most important and most useful to define continuity,
derivatives, and integrals.
1.1. The Idea of a Limit
In mathematics, a limit is the value that a function (𝑓(𝑥)) or sequence (𝑆(𝑥)) reach at some value of the variable 𝑥.
Imagine a person walking up; let us consider that his horizontal
position is measured by the value of 𝑥 (distance from the wall) and
his vertical position (the high) by the coordinate 𝑦. As he gets closer
and closer to 𝑐 horizontal position (𝑥 → 𝑐), he approaches L
vertical position (the high). It means that his altitude gets nearer and
nearer to L (except for a possible small error in accuracy) as +
position approach 𝑐 distance.
For example, suppose we set a particular accuracy goal for our
traveler: He must get the high of 𝐿 = 120 inches. He reports that
he can get 120 inches of L, when he is within 3-5 horizontal inches
away from c. does not matter if he is going up or dawn, he notices
that he is near to 120 inches in high when he is located around 𝑐
position. Previous informal statement can be formulated as follow:
The limit of a function 𝑓(𝑥) as 𝑥 approaches 𝑐 is a number 𝐿 with the following property: given any target distance
from 𝐿, there is a distance from 𝑐 within which the values of 𝑓(𝑥) remain within the target distance. This explicit
statement is quite close to the formal definition of the limit we will study in the next topic.
Examples of Limit Evaluation, an introductory view about the evaluation tactics:
1. Find the limit (𝒚 value) as 𝒙 approaches 𝟓 of the linear equation: 𝒚 = 𝟑𝒙+𝟕
We have to write this equation using limit notation, and then substitute the 𝑥 approaching given value. We will
follow the following rule: 𝑙𝑖𝑚[𝑓(𝑥)+𝑔(𝑥)] =𝑙𝑖𝑚𝑓(𝑥)+ 𝑙𝑖𝑚𝑔(𝑥)= 𝑓(𝑐)+𝑔(𝑐), hereto accepted as introductory
𝑥→c 𝑥→c 𝑥→c
method for solving given question (further, we will discuss the properties of the Limits operations)
𝑙𝑖𝑚 3𝑥+7 = 𝑙𝑖𝑚 3𝑥+𝑙𝑖𝑚 7 = 3(5)+7 = 22
𝑥→5 𝑥→5 𝑥→5
2. Find the limit (𝒚 value) as 𝒙 approaches 𝟐 of the quadratic function: 𝒚 = 𝟐𝒙𝟐 +𝟓𝐱+𝟐
We have to write this equation using limit notation, and then substitute the x (approaching) value.
𝑙𝑖𝑚 2𝑥2+5x+2 = 𝑙𝑖𝑚 2𝑥2+𝑙𝑖𝑚 5𝑥+𝑙𝑖𝑚 2 = 2(22)+5(2)+2 = 20
𝑥→2 𝑥→2 𝑥→2 𝑥→2
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Business Calculus Notebook
𝟐
3. Find the limit (𝒚 value) as 𝒙 approaches 𝟑 of the rational equation: 𝒚 =
𝟑+𝒙
We have to write this equation using limit notation, and then substitute the x (approaching) value.
𝑓(𝑥) 𝑙𝑖𝑚𝑓(𝑥) 𝑓(𝑐)
In this case, we will consider the following rule: 𝑙𝑖𝑚 = 𝑥→c = ; 𝑖𝑓 𝑙𝑖𝑚𝑔(𝑥) ≠ 0
𝑥→c𝑔(𝑥) 𝑙𝑖𝑚𝑔(𝑥) 𝑔(𝑐) 𝑥→c
𝑥→c
𝑙𝑖𝑚 2 𝑙𝑖𝑚 2
2 2 1
𝑙𝑖𝑚 = 𝑥→3 = 𝑥→3 = =
𝑥→3 3+𝑥 𝑙𝑖𝑚 (3+𝑥) 𝑙𝑖𝑚 3+𝑙𝑖𝑚 𝑥 3+3 3
𝑥→3 𝑥→3 𝑥→3
𝒙+𝟑
4. Prove that 𝒍𝒊𝒎 = 𝟏
𝒙→𝟒 𝟓+√𝒙
Formally, we have to decompose the limit question into a group of single elements; however, direct substitution
will produce the same answer:
(𝑙𝑖𝑚 𝑥+𝑙𝑖𝑚 3)/(𝑙𝑖𝑚 5+𝑙𝑖𝑚 √𝑥) = 1
𝑥→4 𝑥→4 𝑥→4 𝑥→4
4+3
At this moment we know that the direct substitution of 𝑥 = 4 will produce same result. In this case = 1. This
5+2
“direct” substitution works because there is not undefinition at the requested 𝑥 value. Further, we will see different
situation when having undefined position, as it is observed in the next example.
5. Find the limit as 𝒙 approaches 𝟏 of the rational expression using numeric and graphic
interpretation:
√𝑥−1 Notice that we cannot solve by substituting 𝑥 = 1 because
𝑦 = 𝑙𝑖𝑚
0
𝑥→1 𝑥−1 ⁄ is undefined, and therefore it is not continuity: However,
0
√1−1 0 we can solve the question by conducting a numeric
𝑙𝑖𝑚 = = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑.
𝑥→1 1−1 0 investigation, approaching 𝑥 = 1 from the left and from the
right using a very small deviation (see the table below):
(𝑥) value 0.8 0.9 0.999 1.0 1.001 1.1 1.2
√(𝑥)−1
𝑙𝑖𝑚 0.527864... 0.513167.. 0.500125... Und. 0.499875... 0.488088... 0.477226...
𝑥→1 (𝑥)−1
√𝑥−1
Resulting that: 𝑙𝑖𝑚 = 0.5, as from both sides it towards to 0.5.
𝑥→1 𝑥−1
We can graph it and analyze its value ( 𝑦 ) when 𝑥 → 1.
When 𝑥 = 1, the value will be undefined (there is a hole in the function),
however, from the left side and from the right side around 𝑥 = 1, the
function gets 𝑦 = 0.5.
6. Find the limit as 𝒙 approaches 𝟒 of the rational expression:
We cannot simplify by substituting 𝑥 = 4 because the expression is
𝑥3+𝑥2−16 𝑥−16 undefined at this value of 𝑥 , and therefore the function is not
𝑙𝑖𝑚 = 8
𝑥→4 𝑥2−3𝑥−4 continuous. We can get the result approaching 𝑥 = 4 from the left and
from the right using a 0.01, 0,001, and 0.001 deviation (see the table):
𝑥 value 3.9 3.99 3.999 4 4.001 4.01 4.1
𝑙𝑖𝑚 7.9 7.99 7.999 ___ 8.001 8.01 8.1
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Juan J. Prieto-Valdés
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Business Calculus Notebook
7. Estimate the value of the limit as 𝒙 approaches 𝟎 of the rational expression using numeric and
graphic interpretation:
√𝑥2+9−3 Notice that we cannot solve by substituting 𝑥 = 0 because
𝑙𝑖𝑚
𝑥→0 𝑥2 0⁄0is undefined, and therefore it is not continuity: However,
√9−3 0 we can solve the question by conducting a numeric
𝑙𝑖𝑚 = = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑.
𝑥→1 0 0 investigation, approaching 𝑥 = 0 from the left and from the
right using a very small deviation (see the table):
(𝑥) value -0.5 -0.1 -0.05 0.001 0.05 0.1 0.5
√𝑥2+9−3
𝑙𝑖𝑚 0.165525… 0.166620… 0.166655… 0.16666… 0.166655… 0.166620… 0.165525…
𝑥→1 𝑥2
√𝑥2+9−3
Resulting that: 𝑙𝑖𝑚 =
𝑥→0 𝑥2
1
0.166666… =
6
We can graph it and analyze its value
when 𝑥 → 0. There is a oscillating
process around 𝑥 = 0 as it can be
observed in the right graph.
8. Estimate the value of the limit when the function exists in one side, using graphic
interpretation:
𝑙𝑖𝑚√𝑥2−1 = 0
𝑥→1+
Notice that we 0 ≤ 𝑥 ≤ 1 the solution will be imaginary.
Therefore, the limit only will exist when approaching 𝑥 = 1
from the positive side. Further, we will discuss such type of
example; Now it is included for illustration about the one
side limit.
9. Estimate the value of the limit when the function exists in one side, use graphic interpretation:
ln(𝑥)
𝑙𝑖𝑚 = −∞
𝑥→0 𝑥
Notice that we 𝑥 approach 0 from the negative side, the
function does not exist. However, when approaching 0
from the positive side the function towards to negative
infinite.
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