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Business Calculus Online Help w/ Videos and Practice Problems!

// Last Updated: October 06, 2021 – Watch Video //

The following video provides an outline of all the topics you would expect to see in a typical college-level Business Calculus class.

This online course contains:

• Full Lectures – Designed so you’ll learn faster and see results in the classroom more quickly.
• 450+ HD Video Library – No more wasted hours searching youtube.
• Available 24/7 – Never worry about missing a class again.
• Practice Problems – Check your knowledge along the way.
• Chapter Tests – Ensure you’re ready for your in-class assessments.

Discrete Math Course Topics

If you need help in business calculus, you’re in the right place. This course covers everything you need to tackle those questions that keep you up at night.

The following sections provide links to the complete lessons on all course topics.

7 Videos 77 Examples

• Finding Limits Graphically
• Limit Rules
• Indeterminate Forms
• Squeeze Theorem
• Limits at Infinity
• Limits Review
• Limits and Continuity

Derivatives

24 Videos 105 Examples

• Definition Of Derivative
• Product Rule
• Quotient Rule
• Derivative Rules
• Derivative Of Exponential Function
• Derivatives Of Logarithmic Functions
• Derivative of Inverse Functions
• Implicit Differentiation
• Higher Order Derivatives
• Logarithmic Differentiation
• Average Rate Of Change Calculus
• L’Hopitals Rule
• Equation Of Tangent Line
• Linear Approximation
• Continuity And Differentiability
• Derivatives Using Charts
• Limit Definition Of Derivative
• Newton’s Method
• Related Rates
• Functions of Several Variables
• Partial Derivatives
• Chapter Test

Application of Derivatives

12 Videos 143 Examples

• Absolute Extrema
• Rolles Theorem
• Mean Value Theorem
• First Derivative Test
• Second Derivative Test
• Curve Sketching
• Derivative Graph
• Particle Motion
• Optimization
• Demand Function
• Elasticity of Demand

17 Videos 105 Examples

• Riemann Sum
• Sigma Notation
• Integration Rules (Indefinite Integrals)
• Fundamental Theorem of Calculus (Definite Integrals)
• Mean Value Theorem for Integrals
• U Substitution
• Separable Differential Equations
• Exponential Growth and Decay
• Euler’s Method
• Logistic Equations
• Area Between Curves
• Improper Integrals
• Integration By Parts
• Extrema and Saddle Points
• Lagrange Multipliers
• Equation of Tangent Planes

Common Questions

Business calculus vs calculus.

There are two key differences between them —  business calculus does not cover trigonometry or theory . They both cover differential and integral calculus topics, but each with a different emphasis.

Business calculus will focus on such topics as compound interest and marginal analysis and how they apply to business aspects. Therefore, the emphasis will be on method and application.

Whereas Calculus will work with varying types of functions, especially trigonometric functions, and how and why they can be applied to the physical world.

Is Business Calculus Hard?

No. Business Calculus is designed for non-engineering majors. Therefore, it is focused more on method and application rather than theory.

However, it is a college level class — consequently, it will still be challenging and require time, energy, and perseverance. The key to achieving success is your effort .

Why Do Business Majors Need Calculus?

Because a career in business is made stronger with calculus. In fact, financial analysis and estimations require a firm knowledge of calculus. Being able to gauge wait times, optimize pricing, minimize or maximize revenue and profit all utilize calculus. Thus, every business major should take calculus to better understand their field of study.

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Math Courses

Math 1071q — calculus for business and economics (fall 2022).

• Instructors

Course Coordinator (Storrs): Katie Hall

About the course

Welcome! This course is an introduction to calculus and applications designed primarily for students majoring in business, economics, or the life and social sciences. We will cover essential concepts of differential and integral calculus.

OpenStax Calculus Volume 1: UConn Custom Business Calculus Edition. Available here: https://www2.math.uconn.edu/ClassHomePages/Math1071/Textbook

Course Objectives

By the end of the semester, students should be able to: 

• Use mathematical models to represent various business and economics related functions, including cost, revenue, profit, and demand. 
• Understand and work with exponential and logarithmic expressions, especially as related to interest and other exponential growth problems. 
• Understand the idea of a limits and limits at infinity. Evaluate various types of limits and define concepts like asymptotes, continuity and derivatives in terms of limits. 
• Understand the derivative as the instantaneous rate of change and also the slope of the tangent line to a curve. Use the tangent line to approximate functions.
• Evaluate the derivative of functions using standard derivative rules. 
• Understand elasticity of demand. 
• Use the first and second derivative of a function, as well as its properties like domain, asymptotes and symmetry, to understand the overall shape of a function and draw its graph. 
• Setup and solve optimization and related rates problems in various contexts, using derivatives. 
• Evaluate both indefinite and definite integrals, using basic antiderivative rules, including substitution. This includes developing an understanding of the Fundamental Theorem of Calculus. 
• Use integrals/antiderivatives to solve problems, like finding cost from marginal cost, net change from a rate of change, or the area under or between curves.

Masking/Social Distancing Policy

Please be sure to follow the current UConn guidelines about masking and social distancing. In times when social distancing is not required inside classrooms, please be respectful of the wishes of others who prefer to maintain social distancing. Even when not required, all students and faculty are always welcome to wear masks.

Calculators

Graphing calculators: TI 82, 83, 84, 84 plus, 85 or 86 may be used. Models TI-89 and above  (including TI-Nspire) are not permitted on the exams.  See this link   (from the University of Arizona) for more detailed description of what is and is not allowed (including Casio brand calculators).  At your instructors discretion, you may also use the Desmos.com Test Mode app, a free calculator app.

Online Homework

Worksheets, quizzes and exams, worksheets on huskyct.

There are 10 worksheets designed to give you an idea of what Exam questions will look like. They are due each Friday.  Worksheets that are submitted will be graded for correctness, completeness and/or clarity.  Refer to your individual instructor for additional details.    

Midterm Exams

There will be three midterm exams on 9/19, 10/24, 11/14 (Monday of Weeks 4, 9 and 12). They will consist of a mix of multiple choice and free response questions.  Exams will be given during class, in person. In the event of make-up exams, exams may be proctored using either WebEx or Lockdown Browser with   Respondos   Monitor or might be in-person. This is at your instructor’s discretion. 

There will be a final exam during the final exam week consisting of both multiple choice and free response questions. Students who score higher on the final than their lowest midterm will automatically have their final exam grade replace their lowest midterm grade.  All students, including students with Dean’s approval for rescheduled final exams are required to take the final exam in-person.

Extra Credit

Some extra credit questions may be available on each exam.  Additionally, your instructor might offer extra credit opportunities throughout the semester.  All extra credit opportunities will be made available to the entire class. At no time will extra credit assignments be made available for individual students.

Grading Scale: 

Due dates and late policy.

All due dates for online homework will be listed in HuskyCT . Due dates for Worksheets are given in class. Deadlines are based on Eastern Time unless otherwise specified. The instructor reserves the right to change dates accordingly as the semester progresses.  All changes will be communicated in an appropriate manner.

Worksheets are due in class on Friday (Ask your instructor if there are online submission options). Late worksheets will be accepted, with possible penalty depending on the individual’s circumstances, but all worksheets must be submitted by Monday at noon. Solutions to worksheets are posted on Mondays at noon and therefore no late worksheets will be accepted after this time.

If you miss an exam, you must contact your course instructor within 48 hours to discuss the possibility of a make-up. Generally, makeups are only offered in the case of extenuating circumstances or with a large penalty.  Make-ups might be required to be taken with proctoring software.

Feedback and Grades

We will make every effort to provide feedback and grades within a week of your submissions . To keep track of your performance in the course, refer to My Grades in HuskyCT.

Weekly Time Commitment

You should expect to dedicate 9 to 12  hours a week to this course. This expectation is based on the various course activities, assignments, and assessments and the University of Connecticut’s policy regarding credit hours . (More information related to hours per week per credit can be accessed at the Online Student website ) .

• Academic Integrity

“A fundamental tenet of all educational institutions is academic honesty. Academic work depends upon respect for and acknowledgement of the research and ideas of others. Misrepresenting someone else’s work as one’s own is a serious offense in any academic setting and it will not be condoned.” A student who knowingly assists another student in committing an act of academic misconduct shall be equally accountable for the violation, and both shall be subject to the sanctions and other remedies. See the  Uconn Student Code, Appendix A.

All students shall act in accordance with the   Student Code   at the University of Connecticut, which states that: “Academic misconduct is dishonest or unethical academic behavior that includes, but is not limited to, misrepresenting mastery in an academic area (e.g., cheating), failing to properly credit information, research, or ideas to their rightful originators or representing such information, research, or ideas as your own (e.g., plagiarism).” In particular, this means that any work you submit in this course should be your own.

It is expected that you will struggle with various aspects of this course, and you are encouraged to seek help from your instructor, your peers, the Q Center, and other sources in understanding the concepts and computations.   However, you are expected to turn in work that reflects your own understanding of the topics and ideas. Therefore, your work should not bear resemblance to that of any other student in the course or to any other sources used, and any ideas used for which any other party had a share in developing should be cited as such.

For example, it is a good idea to look at examples in the text, notes, or online for problems similar to the one you are stuck on, and looking for ways to adapt the ideas and methods to your current problem.   In the interest of both your learning and academic honesty, you may NOT search for solutions to the specific problem you are stuck on.   In all cases, you must write up a solution that is completely in your own words and honestly reflects your own understanding of the ideas.

While you may look online for help, as clearly stated on each Worksheet,   y ou may not view solutions to specific Worksheet, Exam Review problems or Exam problems (or small variations to these problems) on paid sites like Chegg or Bartleby.   You may still use (with proper citations) class discussion boards, Office Hours, and other free outside resources like Khan Academy and YouTube. This means that if you use these allowed resources, even just for a push in the right direction, you should mention that you used them and then still write the solution in our own words to reflect your own understanding.

Consequences of academic misconduct include, but are not limited to, a zero on the assignment or exam and/or a grade of F in the course. If you are unsure that what you are doing to complete the work of this course is acceptable, contact the instructor for helpful tips and advice on how to protect your work and ensure that you are not violating the academic integrity policies of the instructor, the course, or the university.

How to Succeed in this Course

All students can succeed in this course and we are here to help you along the way.  Please do not hesitate to ask questions during class or attend office hours. 

Success in this course program depends heavily on your personal health and well-being.  Recognize that stress is an expected part of the college experience, and it often can be compounded by unexpected setbacks or life changes outside the classroom. We strongly encourage you to reframe challenges as an unavoidable pathway to success.  Reflect on your role in taking care of yourself throughout the semester, before the demands of exams and projects reach their peak.  Please feel free to reach out to us about any difficulty you may be having that may impact your performance in your courses or campus life as soon as it occurs and before it becomes too overwhelming.  In addition to your academic advisor, we strongly encourage you to contact the many other support services on campus that stand ready to assist you.        

Here are some helpful links:   Dean of Students Office , Academic Achievement Center , Writing  Center , Quantitative Learning Center , Center for Students with Disabilities , Title IX Office , Student Health and Wellness — Mental Health   

*This statement was adapted from one prepared by CLAS.  We feel it very much represents our thoughts and feelings, and so are including it in a largely unedited way. (With this citation!)

Husky Study Groups 

Are you interested in forming a study group with other students in the class?  There is a study group application in Nexus that can help you get started. Check out this video wand go   here ( https://nexus.uconn.edu/secure_per/studygroups/index.php ) for more information.

Resources for Students Experiencing Distress

The University of Connecticut is committed to supporting students in their mental health, their psychological and social well-being, and their connection to their academic experience and overall wellness. The university believes that academic, personal, and professional development can flourish only when each member of our community is assured equitable access to mental health services. The university aims to make access to mental health attainable while fostering a community reflecting equity and diversity and understands that good mental health may lead to personal and professional growth, greater self-awareness, increased social engagement, enhanced academic success, and campus and community involvement. 

Students who feel they may benefit from speaking with a mental health professional can find support and resources through the Student Health and Wellness-Mental Health (SHaW-MH) office. Through SHaW-MH, students can make an appointment with a mental health professional and engage in confidential conversations or seek recommendations or referrals for any mental health or psychological concern. 

Mental health services are included as part of the university’s student health insurance plan and also partially funded through university fees. If you do not have UConn’s student health insurance plan, most major insurance plans are also accepted. Students can visit the Student Health and Wellness-Mental Health located in Storrs on the main campus in the Arjona Building, 4th Floor, or contact the office at (860) 486-4705, or https://studenthealth.uconn.edu / for services or questions .

Accommodations for Illness or Extended Absences 

Please stay home if you are feeling ill and please go home if you are in class and start to feel ill.  If illness prevents you from attending class, it is your responsibility to notify your instructor as soon as possible. You do not need to disclose the nature of your illness, however, you will need to work with your instructor to determine how you will complete coursework during your absence.

If life circumstances are affecting your ability to focus on courses and your UConn experience, students can email the Dean of Students at [email protected] to request support.  Regional campus students should email the Student Services staff at their home campus to request support and faculty notification.  

COVID-19 Specific Information: People with COVID-19 have had a wide range of symptoms reported – ranging from mild symptoms to severe illness. These symptoms may appear 2-14 days after exposure to the virus and can include:

• Shortness of breath or difficulty breathing
• Repeated shaking with chills
• Muscle pain
• Sore throat
• New loss of taste or smell

Additional information including what to do if you test positive or you are informed through contract tracing that you were in contact with someone who tested positive, and answers to other important questions can be found here: https://studenthealth.uconn.edu/updates-events/coronavirus/

Students with Disabilities and Special accommodations

The University of Connecticut is committed to protecting the rights of individuals with disabilities and assuring that the learning environment is accessible.  If you anticipate or experience physical or academic barriers based on disability or pregnancy, please let me know immediately so that we can discuss options. Students who require accommodations should contact the Center for Students with Disabilities, Wilbur Cross Building Room 204, (860) 486-2020 or http://csd.uconn.edu/ .

Student Athletes and Students with Disabilities should inform your instructor of your commitments as an athlete, any special needs that you have, etc. within the first three weeks of the semester. You will be expected to bring in a letter from the Athletics Department or the Center for Students with Disabilities.

The University Senate passed a motion on about religious observances which stipulated that Students anticipating such a conflict should inform their instructor in writing within the first three weeks of the semester, and prior to the anticipated absence, and should take the initiative to work out with the instructor a schedule for making up missed work. For conflicts with final examinations, students should, as usual, contact the Dean of Students.

Note: We know that this transition to online classes has the chance to increase the number of students who need accommodations and the form those accommodations will take. We will work with all students to the best of our ability to ensure their specific needs are met the best we can. Please contact your instructor to discuss any needs to you.

Student Responsibilities and Resources 

As a member of the University of Connecticut student community, you are held to certain standards and academic policies. In addition, there are numerous resources available to help you succeed in your academic work. Review these important standards, policies and resources , which include:

• Resources on Avoiding Cheating and Plagiarism
• Copyrighted Materials
• Credit Hours and Workload
• Netiquette and Communication
• Adding or Dropping a Course
• Academic Calendar
• Policy Against Discrimination, Harassment and Inappropriate Romantic Relationships
• Sexual Assault Reporting Policy

Software/Technical Requirements (with Accessibility and Privacy Information)

The software/technical requirements for this course include:

• HuskyCT/Blackboard ( HuskyCT/ Blackboard Accessibility Statement , HuskyCT/ Blackboard Privacy Policy )
• Adobe Acrobat Reader ( Adobe Reader Accessibility Statement , Adobe Reader Privacy Policy )
• Google Apps ( Google Apps Accessibility , Google for Education Privacy Policy )
• Dedicated access to high-speed internet with a minimum speed of 1.5 Mbps (4 Mbps or higher is recommended).

For information on managing your privacy at the University of Connecticut, visit the University’s Privacy page .

NOTE: This course has NOT been designed for use with mobile devices.

Technical and Academic Help provides a guide to technical and academic assistance.

This course uses the learning management platform, HuskyCT . If you have difficulty accessing HuskyCT, you have access to the in person/live person support options available during regular business hours through the Help Center .  You also have 24×7 Course Support including access to live chat, phone, and support documents.

Student Technology Training

Student technology training is now available in a new HuskyCT short course created by students for students. It will prepare you to use the IT systems and services that you will use throughout your time at UConn, whether learning online or on-campus.  It is available at https://lms.uconn.edu/ultra/courses/_80016_1/cl/outline .

Minimum Technical Skills

To be successful in this course, you will need the following technical skills:

• Use electronic mail with attachments.
• Save files in commonly used word processing program formats.
• Scan handwritten work to PDFs.
• Copy and paste text, graphics or hyperlinks.
• Work within two or more browser windows simultaneously.
• Open and access PDF files.

University students are expected to demonstrate competency in Computer Technology. Explore the Computer Technology Competencies page for more information..

Evaluation of Course Experience

Students will be given an opportunity to provide feedback on their course experience and instruction using the University’s standard procedures, which are administered by the Office of Institutional Research and Effectiveness (OIRE).

The University of Connecticut is dedicated to supporting and enhancing teaching effectiveness and student learning using a variety of methods. The Student Evaluation of Teaching (SET) is just one tool used to help faculty enhance their teaching. The SET is used for both formative (self-improvement) and summative (evaluation) purposes.

Additional informal formative surveys and other feedback instruments may be administered within the course.

The course coordinator:  Professor Katherine Hall([email protected]). Or your instructor.

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MSCS Courses

College of liberal arts and sciences, math 165: calculus for business, call to action links heading link copy link.

• Document icon Undergraduate Catalog Listing
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Course prerequisite(s).

• Grade of C or better in MATH 110 (College Algebra)
• Appropriate performance on the math placement test

Course Description

Math 165 is a calculus course intended for those studying business, economics, or other related business majors. The following topics are presented with applications in the business world: functions, graphs, limits, exponential and logarithmic functions, differentiation, integration, techniques and applications of integration, partial derivatives, optimization, and the calculus of several variables. Each textbook section has an accompanying homework set to help the student better understand the material.

Credit Awarded

5 hours (some exceptions noted below)

Prior credit for MATH 170 or MATH 180 will be lost with subsequent completion of MATH 165.

Course Materials

None required

Gradarius is an online-learning tool for calculus.  A “Complete” account is required for this course.  It can be purchased by following the link provided in the blackboard page for this course.

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Follow mscs.uic.edu.

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MATH 110: Business Calculus

Prerequisites & Bulletin Description

Course Objectives

The principal aim of Business Calculus is for students to grasp the concept of rate of change in the context of business and financial applications. Students should understand average and instantaneous rates of change and their relations to the notion of marginal analysis of cost, revenue and profit functions. They should be able to explain their understanding in their own words and solve problems requiring marginal analysis of simple production cost and supply/demand models. In solving these problems they should make appropriate use of differentiation formulas for powers, roots, exponentials and logarithms and of basic differentiation rules. Students should understand simple optimization problems and be able to solve them using differential calculus. Optionally, instructors may include the use of spreadsheets for graphical and numerical analysis of simple financial models. 

Evaluation of Students

Instructors design their own assessment schemes, which should include graded homework, midterm examinations and a final exam. The final exammust include problems covering the following measurable student learning outcomes (MSLOs): [Analyzing rates] Use rates of change in solving business problems;[Differentiation] Differentiate functions using differentiation formulas; Optimization] Analyze and solve optimization problems using the differential calculus; Writing about mathematical topics is a component ofthis course, and students should be engaged in some graded writing assignments during the semester. Application problems in exams should require written interpretations of the results. These interpretations are to be expressed in complete sentences with correct grammar, proper diction and adequate punctuation. Appropriate units must be attached to all numerical quantities. 

Course Outline

• Review of algebra concepts and introduction to average rate of change. Includes review of polynomials, exponential and logarithmic functions. 
• Conceptual development of instantaneous rates of change with applications to economics and business. 
• Formal differentiation. 
• Applications mostly to business and economics including optimization. 

Textbooks & Software

S. Waner and S. Costenoble,  Math for Business Analysis  , Thomson, 2008. (This is a custom edition for MATH 110, SF State). 

Submitted by: Sergei Ovchinnikov   Date: May 29, 2008

• Email: [email protected]
• Telephone: (415) 338-2251

Office Hours

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Section 2.2: The Derivative

Definition of the derivative.

When working with linear functions, we could find the slope of a line to determine the rate at which the function is changing. For an arbitrary function, we can determine the average rate of change of the function. This is the slope of the secant line through those two points on the graph.

Average Rate of Change

Average Rate of Change = \( \dfrac{\Delta \text{output}}{\Delta \text{input}} = \dfrac{f(b) - f(a)}{b-a}\)

= the slope of the secant line through the two points \(\left(a, f(a)\right)\) and \(\left(b, f(b)\right)\)

Suppose the total cost of producing \(x\) items is given by \(TC(x) = 200+30x-0.1x^2\). Determine the average rate of change of cost when increasing production from 25 units to 100 units.

The cost when producing 25 units is \(TC(25) = 200+30(25)-0.1(25)^2 = \$887.50\)

The cost when producing 100 units is \(TC(100) = 200+30(100)-0.1(100)^2 = \$2200\)

The cost increased by $2200-$887.50 = $1312.50 while production increased by 100 – 25 = 75 items. The average rate of change is: \[\frac{TC(100) - TC(25)}{100 - 25} = \frac{2200-887.50}{100-25} = \frac{1312.50}{75} = 17.50 \text{dollars per unit}\]

This tells us that on average the cost increases by $17.50 for each unit produced.

Thinking about the last example, suppose instead we asked the question "How fast are costs increasing when production is 25 units?" Notice this is a different kind of question. The question in the example asked for the rate of change over an interval, as production increased from one value to another. This question is again asking for a rate of change, but an instantaneous rate of change , at a particular moment.

Suppose the total cost of producing \(x\) items is given by \(TC(x) = 200+30x-0.1x^2\). Estimate the instantaneous rate of change when 25 items are being produced.

We can see in the previous graph that the secant slope on the interval \(25\le x \le 100\) is not a particularly good estimate of the instantaneous rate of change since the cost function seems steeper at \(x = 25\) then over the secant slope. We could improve the estimate by choosing a smaller interval.

Approach 1: Using smaller intervals

Using the interval \(25\le x \le 30\), the average rate of change would be: \[\frac{TC(30) - TC(25)}{30 - 25} = \frac{1010-887.50}{30-25} = \frac{122.50}{5} = 24.50 \text{dollars per unit}\]

Using an interval on the other side, \(20\le x \le 25\), the average rate of change would be: \[\frac{TC(25) - TC(20)}{25 - 20} = \frac{887.50-760}{25-20} = \frac{127.5}{5} = 25.50 \text{dollars per unit}\]

We would expect the instantaneous rate of change to be somewhere between these two values. Averaging them, we get an estimate of $25 per unit for the instantaneous rate of change.

Approach 2: Estimating from the graph.

This approach is more commonly used when we only have the graph of a function, and don’t have a formula to evaluate, but we will illustrate it here using the same function.

To estimate the instantaneous rate of change, we can now calculate the slope of this drawn-in line. Looking at the graph, the tangent line appears to pass approximately through (30, 1000) and (70, 2000) which would give a slope of \(\frac{2000-1000}{70-30}=\frac{1000}{40}=25\) dollars per unit.

Next we will explore the same ideas using a function defined in a table, and in another context.

Suppose we drop a tomato from the top of a 100 foot building and time its fall.

• How long did it take for the tomato to drop 100 feet?
• How far did the tomato fall during the first second?
• How far did the tomato fall during the last second?
• How far did the tomato fall between \(t =0.5\) and \(t = 1\)?
• What was the average velocity of the tomato during its fall?
• What was the average velocity between \( t=1\) and \(t=2\) seconds?
• How fast was the tomato falling 1 second after it was dropped?

Some of these questions are pretty easy to answer, while some are more complex.

• From the table, it took 2.5 seconds for the tomato to drop 100 feet
• During the first second, the tomato fell 100 – 84 = 16 feet
• During the last second, the tomato fell 64 – 0 = 64 feet
• Between \(t =0.5\) and \(t = 1\) the tomato fell 96 – 84 = 12 feet
• Velocity is similar to speed, and is a rate of change. We can calculate average velocity the same way we did average rate of change earlier. \[\text{Average velocity}=\frac{\text{distance fallen}}{\text{total time}}=\frac{\Delta\text{position}}{\Delta\text{time}}=\frac{-100 \text{ ft}}{2.5 \text{ s}}=-40 \text{ ft/s}\]
• Between \( t=1\) and \(t=2\) seconds, \[\text{Average velocity}=\frac{\Delta\text{position}}{\Delta\text{time}}=\frac{36\text{ ft}- 84\text{ ft}}{2\text{ s} - 1\text{ s}}=\frac{-48 \text{ ft}}{1 \text{ s}}=-48 \text{ ft/s}\]

This question is significantly different from the previous two questions about average velocity. Here we want the instantaneous velocity , the velocity at an instant in time. Unfortunately the tomato is not equipped with a speedometer so we will have to give an approximate answer. Like in our Approach 1 in the previous example, we will estimate it using secant slopes.

One crude approximation of the instantaneous velocity after 1 second is simply the average velocity during the entire fall, -40 ft/s . But the tomato fell slowly at the beginning and rapidly near the end so the "-40 ft/s" estimate may or may not be a good answer.

We can get a better approximation of the instantaneous velocity at \(t=1\) by calculating the average velocities over a short time interval near \(t = 1\). The average velocity between \(t = 0.5\) and \(t = 1\) is \(\dfrac{-12\text{ feet}}{0.5\text{ s}} = -24\text{ ft/s}\), and the average velocity between \(t = 1\) and \(t = 1.5\) is \(\dfrac{-20\text{ feet}}{0.5\text{ s}} = -40\text{ ft/s}\) so we can be reasonably sure that the instantaneous velocity is between -24 ft/s and -40 ft/s. The average, –32 ft/s, would be a good estimate for the instantaneous velocity.

In general, the shorter the time interval over which we calculate the average velocity, the better the average velocity will approximate the instantaneous velocity. The average velocity over a time interval is \( \dfrac{\Delta\text{position}}{\Delta\text{time}} \), which is the slope of the secant line through two points on the graph of height versus time. The instantaneous velocity at a particular time and height is the slope of the tangent line to the graph at the point given by that time and height.

Average vs Instantaneous Velocity

Average velocity = \( \dfrac{\Delta\text{position}}{\Delta\text{time}} \) = slope of the secant line through 2 points.

Instantaneous velocity = slope of the line tangent to the graph.

Suppose we set up a machine to count the number of bacteria growing on a Petri plate. At first there are few bacteria so the population grows slowly. Then there are more bacteria to divide so the population grows more quickly. Later, there are more bacteria and less room and nutrients available for the expanding population, so the population grows slowly again. Finally, the bacteria have used up most of the nutrients, and the population declines as bacteria die.

Use the population graph to estimate the answer to the questions below.

• What is the bacteria population at time \(t = 3\) days?
• What is the population increment from \(t = 3\) to \(t =10\) days?
• What is the rate of population growth from \(t = 3\) to \(t = 10\) days?
• What is the rate of population growth on the third day, at \(t = 3\) ?
• From the graph, at \(t = 3\), the population is about 0.5 thousand, or 500 bacteria.
• At \(t = 10\), the population is about 4.5 thousand, so the increment is about 4000 bacteria.

The rate of growth from \(t = 3\) to \(t = 10\) is the average rate of change in population during that time: \[ \begin{align*} \text{average change in population } & = \frac{\text{change in population}}{\text{change in time}}\\ & = \frac{\Delta\text{population}}{\Delta\text{time}} \\ & = \frac{4000\text{ bacteria}}{7\text{ days}} \\ & \approx 570\text{ bacteria/day}. \end{align*} \]

This is the slope of the secant line through the two points (3, 500) and (10, 4500).

This question is asking for the instantaneous rate of population change, the slope of the line which is tangent to the population curve at (3, 500). If we sketch a line approximately tangent to the curve at (3, 500) and pick two points near the ends of the tangent line segment , we can estimate that instantaneous rate of population growth is approximately 320 bacteria/day .

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In the previous examples, we noticed that as the interval got smaller and smaller, the secant line got closer to the tangent line and its slope got closer to the slope of the tangent line. That’s good news – we know how to find the slope of a secant line.

Let's explore further this idea of finding the tangent slope based on the secant slope.

Find the slope of the line \(L\) in the graph below which is tangent to \(f(x) = x^2\) at the point (2,4).

We could estimate the slope of \(L\) from the graph, but we won't. Instead, we will use the idea that secant lines over tiny intervals approximate the tangent line.

We can see that the line through (2,4) and (3,9) on the graph of \(f\) is an approximation of the slope of the tangent line, and we can calculate that slope exactly: \(m = \frac{\Delta y}{\Delta x} = \frac{9-4}{3-2} = 5\). But \(m = 5\) is only an estimate of the slope of the tangent line and not a very good estimate. It's too big. We can get a better estimate by picking a second point on the graph of \(f\) which is closer to (2,4) – the point (2,4) is fixed and it must be one of the points we use.

From the second figure, we can see that the slope of the line through the points (2,4) and (2.5,6.25) is a better approximation of the slope of the tangent line at (2,4): \(m = \frac{\Delta y}{\Delta x} = \frac{6.25 - 4}{2.5 - 2} = \frac{2.25}{0.5} = 4.5 \), a better estimate, but still an approximation. We can continue picking points closer and closer to (2,4) on the graph of \(f\), and then calculating the slopes of the lines through each of these points and the point (2,4):

The only thing special about the x–values we picked is that they are numbers which are close, and very close, to \(x = 2\). Someone else might have picked other nearby values for \(x\). As the points we pick get closer and closer to the point (2,4) on the graph of \( y = x^2\), the slopes of the lines through the points and (2,4) are better approximations of the slope of the tangent line, and these slopes are getting closer and closer to 4.

We can bypass much of the calculating by not picking the points one at a time: let's look at a general point near (2,4). Define \( x = 2 + h\) so \(h\) is the increment from 2 to \(x\). If \(h\) is small, then \(x = 2 + h\) is close to 2 and the point \((2+h, f(2+h) ) = \left(2+h, (2+h)^2\right) \) is close to (2,4). The slope \(m\) of the line through the points (2,4) and \(\left(2+h, (2+h)^2\right)\) is a good approximation of the slope of the tangent line at the point (2,4):

The value \( m = 4 + h \) is the slope of the secant line through the two points (2,4) and \(\left( 2+h, (2+h)^2 \right)\). As \(h\) gets smaller and smaller, this slope approaches the slope of the tangent line to the graph of \(f\) at (2,4).

More formally, we could write: \[\text{Slope of the tangent line} = \dfrac{\Delta y}{\Delta x} = \lim\limits_{h\to 0} (4+h). \]

We can easily evaluate this limit using direct substitution, finding that as the interval \(h\) shrinks towards 0, the secant slope approaches the tangent slope, 4.

Use the applet below to explore this. You can drag the base point on the graph to explore the behavior at different locations on the graph. Once setting the base point, use the slider to see how the secant lines approach the tangent line as \(h\) approaches zero.

Finding tangent slopes and finding the instantaneous rate of change are the same problem. In each problem we wanted to know how rapidly something was changing at an instant in time , and the answer turned out to be finding the slope of a tangent line , which we approximated with the slope of a secant line . This idea is the key to defining the slope of a curve.

The Derivative

We can view the derivative in different ways. Here are a three of them:

• The derivative of a function \(f\) at a point (x, f(x)) is the instantaneous rate of change.
• The derivative is the slope of the tangent line to the graph of \(f\) at the point \((x, f(x))\).
• The derivative is the slope of the curve \(f(x)\) at the point \((x, f(x))\).

A function is called differentiable at \((x, f(x))\) if its derivative exists at \((x, f(x))\).

Notation for the Derivative

The derivative of \(y = f(x)\) with respect to \(x\) is written as \[f'(x)\] (read aloud as "\(f\) prime of \(x\)"), or \[y'\] (read aloud as "why prime") or \[\frac{dy}{dx}\] (read aloud as "dee why dee ex"), or \[\frac{df}{dx}.\]

The notation that resembles a fraction is called Leibniz notation . It displays not only the name of the function (\(f\) or \(y\)), but also the name of the variable (in this case, \(x\)). It looks like a fraction because the derivative is a slope. In fact, this is simply \( \frac{\Delta y}{\Delta x} \) written in Roman letters instead of Greek letters.

We find the derivative of a function, or take the derivative of a function, or differentiate a function.

We use an adaptation of the \( \frac{df}{dx} \) notation to mean "find the derivative of \(f(x)\):" \[\frac{d}{dx}\left[f(x)\right]=\frac{df}{dx}.\] [The book uses parentheses instead of brackets–both are acceptable forms of the notation.]

Formal Algebraic Definition

\[f'(x)=\lim\limits_{h\to 0} \dfrac{f(x+h)-f(x)}{h}\]

Practical Definition

The derivative can be approximated by looking at an average rate of change, or the slope of a secant line, over a very tiny interval. The tinier the interval, the closer this is to the true instantaneous rate of change, slope of the tangent line, or slope of the curve.

Looking Ahead

We will have methods for computing exact values of derivatives from formulas soon. If the function is given to you as a table or graph, you will still need to approximate this way.

This is the foundation for the rest of this chapter. It’s remarkable that such a simple idea (the slope of a tangent line) and such a simple definition (for the derivative \( f'(x) \)) will lead to so many important ideas and applications.

Find the slope of the tangent line to \( f(x)=\frac{1}{x} \) at \(x = 3\).

The slope of the tangent line is the value of the derivative \(f'(3)\). \( f(3)=\frac{1}{3}\) and \( f(3+h)=\frac{1}{3+h} \), so, using the formal limit definition of the derivative, \[ f'(3)=\lim\limits_{h\to 0}\frac{f(3+h)-f(3)}{h}=\lim\limits_{h\to 0}\frac{\frac{1}{3+h}-\frac{1}{3}}{h}. \]

We can simplify by giving the fractions a common denominator: \[ \begin{align*} \lim\limits_{h\to 0}\frac{\frac{1}{3+h}-\frac{1}{3}}{h} & = \lim\limits_{h\to 0}\frac{\frac{1}{3+h}\cdot\frac{3}{3}-\frac{1}{3}\cdot\frac{3+h}{3+h}}{h} \\ & = \lim\limits_{h\to 0}\frac{\frac{3}{9+3h}-\frac{3+h}{9+3h}}{h} \\ & = \lim\limits_{h\to 0}\frac{\frac{3-(3+h)}{9+3h}}{h} \\ & = \lim\limits_{h\to 0}\frac{\frac{3-3-h}{9+3h}}{h} \\ & = \lim\limits_{h\to 0}\frac{\frac{-h}{9+3h}}{h} \\ & = \lim\limits_{h\to 0}\frac{-h}{9+3h}\cdot\frac{1}{h} \\ & = \lim\limits_{h\to 0}\frac{-1}{9+3h} \\ \end{align*} \] and the evaluate using direct substitution: \[\lim\limits_{h\to 0}\frac{-1}{9+3h}=\frac{-1}{9+3(0)}=-\frac{1}{9}.\]

Thus, the slope of the tangent line to \( f(x)=\frac{1}{x} \) at \(x = 3\) is \( -\frac{1}{9} \).

The Derivative as a Function

We now know how to find (or at least approximate) the derivative of a function for any \(x\)-value; this means we can think of the derivative as a function, too. The inputs are the same \(x\)’s; the output is the value of the derivative at that \(x\) value.

Below is the graph of a function \( y=f(x) \). We can use the information in the graph to fill in a table showing values of \( f'(x): \)

At various values of \(x\), draw your best guess at the tangent line and measure its slope. You might have to extend your lines so you can read some points. In general, your estimate of the slope will be better if you choose points that are easy to read and far away from each other. Here are estimates for a few values of \(x\) (parts of the tangent lines used are shown above in the graph):

We can estimate the values of \(f'(x)\) at some non-integer values of \(x\), too: \(f'(0.5) \approx 0.5\) and \(f'(1.3) \approx -0.3\).

We can even think about entire intervals. For example, if \(0 \lt x \lt 1\), then \(f(x)\) is increasing, all the slopes are positive, and so \(f'(x)\) is positive.

The values of \(f'(x)\) definitely depend on the values of \(x\), and \(f'(x)\) is a function of \(x\). We can use the results in the table to help sketch the graph of \(f'(x)\).

To get a better feel for this, explore the applet below. The top graph is the graph of the original function \(g(x)\). The bottom graph shows the slopes of \(g(x)\), so is a graph of the derivative, \(g'(x)\). Drag the point a and notice how the slope of the tangent line corresponds to the value of the derivative \(g'(x)\).

Shown is the graph of the height \(h(t)\) of a rocket at time \(t\).

Sketch the graph of the velocity of the rocket at time \(t\). (Velocity is the derivative of the height function, so it is the slope of the tangent to the graph of position or height.)

We can estimate the slope of the function at several points. The lower graph below shows the velocity of the rocket. This is \(v(t) = h'(t)\).

In some applications, we need to know where the graph of a function f(x) has horizontal tangent lines (slopes = 0).

Below is the graph of \(y = g(x)\). At what values of \(x\) does the graph of \(g(x)\) have horizontal tangent lines?

The tangent lines to the graph of \(g(x)\) are horizontal (slope = 0) when \(x\approx -1, 1, 2.5, \text{ and } 5\).

We can also find derivative functions algebraically using limits.

Find \( \frac{d}{dx}\left( 2x^2-4x-1 \right) \).

Setting up the derivative using a limit, \[ f'(x)=\lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}.\]

We will start by simplifying \( f(x+h) \) by expanding: \[ \begin{align*} f(x+h) & = 2(x+h)^2-4(x+h)-1 \\ & = 2(x^2+2xh+h^2)-4(x+h)-1 \\ & = 2x^2+4xh+2h^2-4x-4h-1 \end{align*} \]

Now finding the limit: \[ \begin{align*} f'(x) & = \lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h} \\ & = \lim\limits_{h\to 0} \frac{(2x^2+4xh+2h^2-4x-4h-1)-(2x^2-4x-1)}{h} \\ & = \lim\limits_{h\to 0} \frac{2x^2+4xh+2h^2-4x-4h-1-2x^2+4x+1}{h} \qquad \text{(Substitute in the formulas.)} \\ & = \lim\limits_{h\to 0} \frac{4xh+2h^2-4h}{h} \qquad \text{(Now simplify.)}\\ & = \lim\limits_{h\to 0} \frac{h(4x+2h-4)}{h} \qquad \text{(Factor out the \( h \), then cancel.)} \\ & = \lim\limits_{h\to 0} (4x+2h-4) \end{align*} \] We can find the limit of this expression by direct substitution: \[ f'(x)=\lim\limits_{h\to 0} (4x+2h-4)=4x-4\]

Notice that the derivative depends on \(x\), and that this formula will tell us the slope of the tangent line to \(f\) at any value \(x\). For example, if we wanted to know the tangent slope of \(f\) at \(x = 3\), we would simply evaluate: \( f'(3)=4(3)-4=8 \).

A formula for the derivative function is very powerful, but as you can see, calculating the derivative using the limit definition is very time consuming. In the next section, we will identify some patterns that will allow us to start building a set of rules for finding derivatives without needing the limit definition.

Interpreting the Derivative

So far we have emphasized the derivative as the slope of the line tangent to a graph. That interpretation is very visual and useful when examining the graph of a function, and we will continue to use it. Derivatives, however, are used in a wide variety of fields and applications, and some of these fields use other interpretations. The following are a few interpretations of the derivative that are commonly used.

Rate of Change: \(f '(x)\) is the rate of change of the function at \(x\). If the units for \(x\) are years and the units for \(f(x)\) are people, then the units for \( \frac{df}{dx} \) are \(\frac{\text{people}}{\text{year}}\), a rate of change in population.

Slope: \(f '(x)\) is the slope of the line tangent to the graph of \(f\) at the point \(( x, f(x) )\) .

Velocity: If \(f(x)\) is the position of an object at time \(x\), then \(f '(x)\) is the velocity of the object at time \(x\). If the units for \(x\) are hours and \(f(x)\) is distance measured in miles, then the units for \(f '(x) = \frac{df}{dx}\) are \( \frac{\text{miles}}{\text{hour}} \), miles per hour, which is a measure of velocity.

Acceleration: If \(f(x)\) is the velocity of an object at time \(x\), then \(f '(x)\) is the acceleration of the object at time \(x\). If the units are for \(x\) are hours and \(f(x)\) has the units \( \frac{\text{miles}}{\text{hour}} \), then the units for the acceleration \(f '(x) = \frac{df}{dx}\) are \( \frac{\text{miles/hour}}{\text{hour}} =\frac{\text{miles}}{\text{hour}^2} \), miles per hour per hour.

Marginal Cost, Marginal Revenue, and Marginal Profit: We'll explore these terms in more depth later in the section. Basically, the marginal cost is approximately the additional cost of making one more object once we have already made \(x\) objects. If the units for \(x\) are bicycles and the units for \(f(x)\) are dollars, then the units for \(f '(x) = \frac{df}{dx}\) are \( \frac{\text{dollars}}{\text{ bicycle}} \), the cost per bicycle.

In business contexts, the word " marginal " usually means the derivative or rate of change of some quantity.

One of the strengths of calculus is that it provides a unity and economy of ideas among diverse applications. The vocabulary and problems may be different, but the ideas and even the notations of calculus are still useful.

Suppose the demand curve for widgets was given by \( D(p)=\frac{1}{p} \), where \(D\) is the quantity of widgets, in thousands, at a price of \(p\) dollars. Interpret the derivative of \(D\) at \(p = \)$3.

Note that we calculated \( D'(3) \) earlier to be \( D'(3)=-\frac{1}{9}\approx -0.111 \).

Since \(D\) has units thousands of widgets and the units for \(p\) is dollars of price, the units for \(D'\) will be \( \frac{\text{thousands of widgets}}{\text{dollar of price}} \). In other words, it shows how the demand will change as the price increases.

Specifically, \( D'(3)\approx -0.111 \) tells us that when the price is $3, the demand will decrease by about 0.111 thousand items for every dollar the price increases.

(Note: The screen shots in the following video are from an earlier version of the book, so some of the section numbers or titles may not look the same. However, much of the content is the same, and the comments still apply.)

Applied Calculus

Business calculus, content overview.

Business Calculus by Dale Hoffman, Shana Calloway, and David Lippman is a derivative work based on Dale Hoffman’s Contemporary Calculus. The course covers one semester of Business Calculus for college students and assumes students have had College Algebra. Students will learn to apply calculus in economic and business settings, like maximizing profit or minimizing average cost, finding elasticity of demand, or finding the present value of a continuous income stream.

This course package is delivered either in Lumen OHM, or can be imported into your LMS (Canvas, Blackboard, D2L, Moodle).  This course also contains video examples created by James Sousa ( mathispower4u.com ).

Topic Overview

This course is delivered in 4 modules including a review  of functions (no trig) and the following topics:

Module 1: Review

• A review of the basic toolkit of functions including transformations, and compositions

Module 2: The Derivative

• Instantaneous rates of change
• Limits and continuity
• Applications of Rates
• Formulas for differentiation
• Second derivatives
• Optimizations
• Applied Optimization
• Tangent line approximation, elasticity of demand
• Implicit differentiation, related rates

Module 3: The Integral

• Definite integrals
• The fundamental theorem of calculus
• Formulas for integration
• Substitution
• Integration by parts and using tables
• Applications of integrals – area, volume and average value
• Applications to business
• Differential equations

Module 4: Functions of Two Variables

• Calculus of functions of two variables
• Optimization

Length: One semester

Delivery:  This course has been taught online and face to face

Online Content

Each section has an algorithmic problem set delivered by Lumen OHM, a set of supporting videos and text.

One section

Online Practice Problems

Online homework problem set.

Online Assessment Features

Many questions in the Lumen OHM libraries are randomized, algorithmic questions. Students get immediate feedback after they submit an answer. Question types include:

• Entering an integer, fraction, decimal
• Reading information from a graph
• Multiple choice
• Free-writing (instructor graded)

Because of the open license on the libraries of questions in Lumen OHM, you are also free to edit and create your own questions with the question writing tools.

Course Review Access

To view the fully integrated content housed in Lumen OHM, you can access the course as a guest student here .  Simply enter guest  in the username field – no password required.

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1.E: Succeeding in Business Mathematics (Exercises)

• Last updated
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• Page ID 38349

• Jean-Paul Olivier
• Red River College of Applied Arts, Science, & Technology

1.3: Where do we go from here?

In questions 1–8, identify where you would look in this textbook if you needed to find the indicated information.

• The definition of a mathematical concept or term.
• An explanation of a particular calculator function.
• Guided solutions and examples to illustrate a mathematical concept.
• An illustration of how to use the calculator to solve a particular scenario.
• Step-by-step procedures for solving a mathematical problem.
• More practice questions.
• Detailed explanations of how to use the Excel spreadsheet templates.
• A full discussion of a mathematical concept.
• What does the acronym PUPP stand for? Why is it important?
• You just completed the "Mechanics" questions for a particular section. Do you need to do any more questions, or is that enough for you to pass the course?
• Every chapter includes Review Exercises at the end. Why do you think it is important to complete these exercises?

Applications

• Think about your current job. List five activities that you perform that involve mathematics.
• Talk to family members. Note their individual occupations and ask them about what work activities they perform every day that involve mathematics.

For questions 14–17, gather a few of your fellow students to discuss business mathematics.

• Identify five specific activities or actions that you need to perform to succeed in your business math course.
• Consider how you will study for a math test. Develop three specific study strategies.
• Many students find it beneficial to work in study groups. List three ways that a study group can benefit you in your business math course.
• For those students who may experience high levels of stress or anxiety about mathematics, identify three coping strategies.

Go online and enter the phrase "business math" or "business math help" into a search engine. Look at the table of contents for this book and note any websites that may help you study, practice, or review each chapter.

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Is Business Calculus Hard? Unraveling the Complexity for Beginners

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Core Calculus Principles

Key business applications, tackling calculus problems.

Yes, Business Calculus can be challenging for many students, but it is typically less intricate than the more theory-intensive calculus courses required for science and engineering majors.

In my experience, Business Calculus focuses more on practical application and less on the theoretical aspects. It is designed for those pursuing a business degree , emphasizing methods relevant to business scenarios, like maximizing profit and minimizing costs .

As an undergraduate subject, it brings to light essential concepts such as derivatives and integration at a basic level, with a particular emphasis on functions related to economics .

The difficulty of the course can vary depending on your background and affinity for mathematics . Yet, it’s essential to approach it with the understanding that, like any college-level class, it requires dedication and effort to grasp the concepts thoroughly.

Understanding Business Calculus Concepts

In this section, I’ll guide you through the fundamental concepts of Business Calculus and how they apply to real – world business scenarios.

To grasp Business Calculus , a solid understanding of basic calculus principles is essential. I’ve found that focusing on the core topics like limits , derivatives , integrals , and their applications helps establish a strong foundation.

It’s crucial to be comfortable with the notion of a limit , expressed as $\lim_{x \to c} f(x)$, because it forms the basis for differentiation and integration —two cornerstone operations in calculus.

Additionally, mastering differentiation and integration techniques, such as the chain rule for derivatives $\frac{df(g(x))}{dx} = f'(g(x))g'(x)$ and basic integration methods , paves the way for solving more complex problems. Understanding functions and their behaviors is also integral to Business Calculus, as many problems revolve around analyzing function models.

Business Calculus directly addresses the quantifiable aspects of business through concepts like marginal analysis and elasticity of demand . The marginal analysis utilizes derivatives to determine the additional cost or revenue derived from increasing output by one unit, represented by $\frac{dC}{dQ}) or (\frac{dR}{dQ}$, where (C) is cost, (R) is revenue, and (Q) is quantity.

The elasticity of demand , which measures how the quantity demanded of a good responds to a price change, relies on the concept of implicit differentiation . It helps businesses understand their consumers and make pricing decisions that can lead to optimal profit .

When working through calculus problems, consistent practice and review are my best strategies for success. Homework assignments often mirror the types of questions that appear on quizzes and the final exam . These problems typically include real-world business scenarios involving cost , revenue , and profit maximization.

My advice for students is to ensure a strong grasp of algebra and, to a lesser extent, trigonometry , as these subjects form the basis of Business Calculus problems. Lastly, students shouldn’t hesitate to seek explanations from their professors and utilize resources like textbooks and practice exams for additional examples and clarification.

In my experience, the perception of difficulty in business calculus can vary greatly among students. Some find the emphasis on real-world application quite helpful. Instead of abstract computations, I’ve noticed that understanding comes easier when problems are framed in a familiar context.

I recognize that integration represented as $\int f(x) ,dx$, can intimidate newcomers. However, with business calculus typically requiring fewer integration methods, the learning curve isn’t as steep. While challenging , mastering just a couple of techniques is usually sufficient.

My peers often express relief that advanced topics like trigonometry are not a major component, and the number ‘e’ , an important constant approximately equal to 2.71828, is often as complex as numbers get here. For those worried about tackling hard mathematics, business calculus may come as a pleasant surprise.

Personally, I found that a steady study routine and practical application of concepts were the keys to overcoming initial hurdles. I encourage students to approach business calculus with an open mind, as it can be quite manageable with the right mindset and resources.

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Math 1476: Business Calculus II

MATH 1476 Business Calculus II Information for Students:  2002-2003

Text : Applied Calculus with Linear Programming A Special Edition by Barnett & Ziegler, Pearson Custom Publishing. (An errata sheet is available. Ask your instructor.)

Calculator : Students need either a scientific or business calculator. If you cannot purchase one, they are available from the library. Graphing calculators are fine, but their use may be restricted on the graphing test.

COURSE DESCRIPTION

MATH 1476 BUSINESS CALCULUS AND APPLICATIONS II (4-4-0). A course treating multivariable calculus and its applications for business students, as well as selected other business applications. Topics include functions of several variables and their derivatives, partial differentiation, optimization problems and LaGrange multipliers, special methods of integration, differential equations, probability and calculus, Taylor polynomials and infinite series, and topics in matrix theory and linear programming. Prerequisites: MATH 1425 or MATH 2413 or the equivalent. (MTH 1684)

INSTRUCTIONAL METHODOLOGY :  This course is taught in the classroom primarily as a lecture/discussion course. This class will also have a computer lab component

Syllabus/Calendar/Suggested Testing Schedul e : Please note:  schedule changes may occur during the semester. Any changes will be announced in class

1. Business applications will be emphasized throughout the course.

2. Instructors may introduce supplemental materials as needed to enhance and clarify topics covered in the text.

3. Incomplete grades (I) will be given only in very rare circumstances. Generally, to receive a grade of I, a student must have taken all examinations, be passing, and have a personal tragedy occur after the last date to withdraw which prevents course completion.

. COURSE RATIONALE:  This is the second course in a two-course business calculus sequence. The course covers more multivariable calculus, differential equations, probability, numerical techniques and linear programming. The course stresses applications in business and economics, and is intended to give students the appropriate conceptual and computational mathematical background for future study in business.

TESTING CENTER POLICY:  ACC Testing Center policies can be found at: http://www.austincc.edu/testctr/ .  Deadlines for all tests will be announced in class.  Any tests taken after the announced deadline are considered late.

STUDENT SERVICES:  The web address for student services is:  http://www3.austincc.edu/evpcss/rss/Default.htm . The ACC student handbook can be found at:  http://www.austincc.edu/handbook .

INSTRUCTIONAL SERVICES:  The web address is:  http://www3.austincc.edu/evpcss/newsemester/ , then click on ?Campus Based Student Support Overview?.

Course-Specific Support Services

Sometimes sections of MATH 0163(1-0-2) are offered . The lab is designed for students currently registered in Business Calculus and Applications I, MATH 1425. It offers individualized and group setting to provide additional practice and explanation. This course is not for college-level credit. Repeatable up to two credit hours. Students should check the course schedule for possible offerings of the lab class.

ACC main campuses have Learning Labs which offer free first-come first-serve tutoring in mathematics courses, but not all tutors can help with this class. Check in advance . The locations, contact information and hours of availability of the Learning Labs are posted at: http://www2.austincc.edu/rvslab/labhours.htm

Each ACC campus offers support services for students with documented physical or psychological disabilities.  Students with disabilities must request reasonable accommodations through the Office of Students with Disabilities on the campus where they expect to take the majority of their classes.  Students are encouraged to do this three weeks before the start of the semester.

Students who are requesting accommodation must provide the instructor with a letter of accommodation from the Office of Students with Disabilities (OSD) at the beginning of the semester.   Accommodations can only be made after the instructor receives the letter of accommodation from OSD.

Statement on Academic Freedom:  Institutions of higher education are conducted for the common good.  The common good depends upon a search for truth and upon free expression.  In this course the professor and students shall strive to protect free inquiry and the open exchange of facts, ideas, and opinions.  Students are free to take exception to views offered in this course and to reserve judgment about debatable issues. Grades will not be affected by personal views.  With this freedom comes the responsibility of civility and a respect for a diversity of ideas and opinions.  This means that students must take turns speaking, listen to others speak without interruption, and refrain from name-calling or other personal attacks.

Time required and outside help : To do homework and study requires two or three times as much time outside of class as the time you spend in class in order to succeed in this course. Free tutoring is available in the Learning Labs (see above) and your instructor has office hours and can give some extra help, of course.

Additional information about ACC's mathematics curriculum and faculty is available on the Internet at http://www.austincc.edu/math/

Errata in Business Calculus text, updated 5/15/00

p.8, same example, bottom of the page, should read f(1)= 5 instead of f(-1)=5.

p.88, In the blue box, the marginal average cost and the marginal average revenue should have bars over the C and R.  The marginal average profit is ok.

p. 88, in the green box, there should be bars over the C's in the Marginal Average Cost Equation and there should be bars over the R's in the Marginal Average Revenue Equation.

page 112, the first derivative test graphic: The second sentence should read "Construct a sign chart for  f'(x) .".

p. 133, #43, change the -4x^3 term to +4x^3 if you'd like the answer in the back and the solution in the Student Solutions Manual to be correspond correctly,   f(x) = -x 4 + 4x 3 + 3x + 7  instead of  f(x) = -x 4 - 4x 3 + 3x + 7

page 342, Problem 67.the exponent on "e" should be -0.1 not  -.01 the answer key has problem worked with exponent  -0.1

page 357, Example 2, the shading in the graph (green) should include the sliver on the lower left.

page 373, five lines below Problem 10, the word "equilibrium" is misspelled in text

Page 497 #10     Chapter Six Review:  The answer in the back of the book assumes a FOURTH point which is not given in the problem, namely, (8,3) .

page 538   example 13,  third line of equations     exponent on e should be "0.08" , not  "0.008",  you will note exponent is correct everywhere else in the problem

There are a number of errors in Chapter 8, Sections 1 and 2, on Taylor polynomials and series, one bad one in the green box on p. 581 (should  be an "a" not a "0" in derivative.  Also on p. 594, the Problem 6 title is one line too high, which is confusing.  And the calculations in the  third line of Example 7, p. 595,  are all messed up.

page 736, green box, line with integral, should be  F(x)  where the type is blotched

page 756.  On the top of the page, in a green box entitled "Uniform Probability Density Function" under the heading "cumulative probability distribution" it reads  F(x) = 0  if  x<b.  It should read  F(x) = 0 if x<a.

Pg 850, Exercise 21, Second constraint equation (one with "less than 3"), First variable should be "x-sub-two".

p. 922, 15.  Solution is  y' = 3x^2y/(1+y)

p. 923, 1(A)  Delete "finger" at end of line (Who knows where that came from???)