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2: Basic Skills for Calculus - Mathematics


In this chapter, we will look at several basic skills and topics that will be used often in calculus: linear functions, solving inequalities, function domains, graphs and graphing, and completing the square. Completing the square will show up in integral calculus when you need to have your function in a particular form.

Thumbnail: Functions map one number into another. (CC-BY 4.0; OpenStax)


Knowledge & Skills Gained

In his message to the students in the College of Arts and Sciences, Dean Boocker explains the importance of "making knowledge matter." We believe that means helping you develop useful, real-world skills alongside the sense of fulfillment and enrichment that studying math can provide.

We also believe in making you aware of the knowledge and skills you're developing along the way, so that you can capitalize on your strengths in the marketplace, graduate school and in life.

Knowledge & Skills Gained as a Math Major:

In addition to the specific knowledge acquired in each course, all math majors learn that:

  • Mathematics is a universal language
  • Mathematics is the art and science of problem solving
  • Math is all around us, from the simplistic to the complex
  • Mathematics is essential for solving real-world problems
  • Calculus is the mathematics of change
  • Logic is the basis for all mathematical reasoning
  • Proofs are the essence of mathematics
  • Adept at solving quantitative problems
  • Ability to understand both concrete and abstract problems
  • Proficient in communicating mathematical ideas
  • Detail-oriented
  • Ability to make critical observations
  • Accurately organize, analyze, and interpret data
  • Extract important information and patterns
  • Assess and solve complex problems
  • Able to work independently and on a team

Announcements


MATH 1226 Course Page

MATH 1226 is a four-credit second-semester calculus course that is included in the Pathways curriculum for Quantitative and Computational Thinking. Topics of study include techniques and applications of integration, Trapezoidal and Simpson’s rules, improper integrals, sequences and series, power series, parametric curves and polar coordinates. Software-based techniques will be emphasized.

Text: Calculus: Early Transcendentals by Stewart (9 th edition) with WebAssign access

Prerequisites: MATH 1225 (minimum grade of C–)

Download the Complete Syllabus with Problem Assignments (PDF)

Syllabus: Topics & Chapters

Unit 1: Topics & Chapters

Section Topic
5.5 The Substitution Rule
5.2 The Definite Integral
6.1 Areas Between Curves
6.2 Volumes
6.4 Work
6.5 Average Value of a Function
7.1 Integration by Parts
7.2 Trigonometric Integrals
7.3 Trigonometric Substitution

Unit 2: Topics & Chapters

Section Topic
7.4 Integration by Partial Fractions
7.5 Strategies for Integration
7.7 Approximate Integration
6.1 Income Inequality and the Gini Index
8.3 Centers of Mass
4.4 Indeterminate Forms
7.8 Improper Integrals
8.5 Probability

Unit 3: Topics & Chapters

Section Topic
11.1 Sequences
11.2 Series
11.3 The Integral Test
11.4 The Comparison Tests
11.5 Alternating Series and Absolute Convergence
11.6 The Ratio and Root Tests
11.7 Strategies for Testing Series
11.8 Power Series
11.9 Representations of Functions as Power Series
11.10 Taylor and Maclaurin Series
11.11 Applications of Taylor Polynomials

Unit 4: Topics & Chapters

Section Topic
10.1 Curves defined by Parametric Equations
10.2 Calculus of Parametric Curves
10.3 Polar Coordinates

Final Exam

The final exam is a Common Time Exam.

The exam consists of two parts:

  1. Common Exam
    • This is multiple choice exam taken by all sections of MATH 1226. Samples of Common Time Final Exams given in previous years are available (koofers).
  2. Free Response Exam
    • Your instructor will give you information on what to expect for the second portion of the exam.

Note: Both portions of this exam will be administered virtually.

Check the timetable or your instructor's Canvas course site for the date and time of the common final exam.

Instructors & Sections

See the Timetable of classes for information on current offerings of MATH 1226

Basic Skills Review

The Department of Mathematics encourages each student enrolled in MATH 1226 to take, at the beginning of the term, an online Basic Skills Review covering very basic concepts from MATH 1225.

Why a Basic Skills Review?

The primary goal of the MATH 1226 Basic Skills Review is to get you quickly and purposefully active in doing mathematics you have already seen in MATH 1225 (or an equivalent course). You may have forgotten some of the material over summer/holiday breaks. The review will provide a quick and timely reminder of those basic skills from MATH 1225 (or an equivalent course) that are essential for success in MATH 1226.

Not passing the Basic Skills Review may indicate that you are better suited to take MATH 1225 this semester please consult your instructor or advisor if you do not pass the Basic Skills Review.

Tell me more about the Basic Skills Review

The MATH 1226 Basic Skills Review is comprised of 6 questions covering differentiation (product rule, quotient rule, trig functions, chain rule) and basic integration, including u-substitution. A score of at least 5 correct is considered passing.

How any questions are on the Basic Skills Review?

The MATH 1226 Basic Skills Review is an online assessment, comprised of 6 questions. A score of 5 is considered passing.

What is covered on the Basic Skills Review?

The Basic Skills Review covers differentiation and basic integration, including u-substitution.

How many times can the Basic Skills Review be taken?

A student may make three total attempts at the Basic Skills Review.

When should I take the Basic Skills Review?

Students in MATH 1226 are urged to take the MATH 1226 Basic Skills Review during the first few days of classes so that they will have maximum flexibility in making course schedule adjustments should that become necessary.

The Basic Skills Review will become available to all students enrolled in MATH 1226 on the first day of the semester.

Check your instructor's Canvas course site for more information on accessing the Basic Skills Review.

What if I don't earn a passing score?

Not passing the Basic Skills Review may indicate that you are better suited to take MATH 1225 this semester please consult your instructor or advisor if you do not pass the Basic Skills Review.

How can I access the Basic Skills Review?

The MATH 1226 Basic Skills Review will become available to all students enrolled in MATH 1226 on the first day of the semester.

Check your instructor's Canvas course site for more information on accessing the Basic Skills Review.

Honor System Information

The Undergraduate Honor Code pledge that each member of the university community agrees to abide by states:

“As a Hokie, I will conduct myself with honor and integrity at all times. I will not lie, cheat, or steal, nor will I accept the actions of those who do.”

Students enrolled in this course are responsible for abiding by the Honor Code. A student who has doubts about how the Honor Code applies to any assignment is responsible for obtaining specific guidance from the course instructor before submitting the assignment for evaluation. Ignorance of the rules does not excuse any member of the University community from the requirements and expectations of the Honor Code.


Is calculus necessary?

"Your short article on why we teach calculus is marred with flaws. Why should we continue to teach something just because it has a long tradition? And not having a thing to replace it is your most valid excuse? Also, since when is calculus a part of everyday culture? It sounds like you just want a reason to defend your profession."

Its a valid point. As a teacher one is of course biased and in general, everybody overestimates and naturally hypes his or her own work or profession.
There is a general principle however: if one wants to replace something, one has to constructively build an alternative which works and demonstrate that it can work on a large scale. Just calling for a replacement is cheap. In the case of calculus, it is not only the results which have an excellent track record (major industries mine calculus today), but also calculus as a "tool to sharpen thinking and problem solving skills" and prepare for other fields.
As replacement of calculus, both discrete math or statistics comes in mind. I myself know that both need solid calculus skills to be used effectively. Whoever calls for using stats as a replacement of calculus does not know stats. Whoever calls for discrete math as a replacement does not understand discrete math. Both fields really only shine if one knows calculus. I love discrete math, (work on it) and statistics (example [PDF]) and even discrete or alternate versions of calculus like here. Unfortunately, in many of todays implementations of discrete math or statistics, the replacement is used as an alternative, the implementation is an excuse to stop practicing harder problems or acquire more sophisticated problem solving skills or to learn harder subjects. Many discrete math courses are a race to the bottom, the reason being the lack of a clear goal to reach. Calculus has the fortune to have a clear goal: the fundamental theorem of calculus (both in single and multivariable calculus), as well as established levels of sophistication like integration skills, knowledge about series and the ability to solve differential equations. Yes, these skills can be hard to reach, but it is worth it. If acquired, the usual discrete math or stats curricula are more rewarding.

Maybe we have to look at history also to see what worked and where things were successful. The biggest advancements in discrete math, physics, statistics,philosophy or computer science were done by people who knew calculus well: Euler invented graph theory and was a master in calculus, Leibniz invented determinants and a computing device and a master in calculus, Newton figured out the laws of gravity, and was a master in calculus, Kepler figured out the laws with which the planets move and was a master in calculus, von Neumann invented modern computers, game theory and was a master in calculus, Archimedes invented countless of machines and was a master in calculus. Kolmogorov wrote the first textbook in probability and was the first put the subject on a solid foundation and was a master in calculus. Riemann dove into the deep mysteries of the prime numbers and was a master of calculus.


Student Learning Outcomes/Learning Objectives

1. Students will feel a sense of accomplishment in their increasing ability to use mathematics to solve problems of interest to them or useful in their chosen fields. Students will attain more positive attitudes based on increasing confidence in their abilities to learn mathematics.

2. Students will learn to understand material using standard mathematical terminology and notation when presented either verbally or in writing.

3. Students will improve their skills in describing what they are doing as they solve problems using standard mathematical terminology and notation.

Student Learning Outcomes: Upon successful completion of this course, a student will be able to do:

I.Concepts and skills associated with whole numbers

1.write the standard form of a whole number

2.round whole numbers and use rounding to estimate values involving whole number arithmetic

3.perform the four basic arithmetic operations (+, -, x and ÷) on whole numbers

4.solve application problems involving the four basic operations on whole numbers

5.identify the order relation between two whole numbers

6.simplify exponential expressions with whole number exponents

7.use the order of operations to simplify expressions.

8.prime factor whole numbers

9.find the least common multiple of two or more whole numbers

II.Concepts and skills associated with fractions

1.perform the four basic arithmetic operations on fractions

2.solve application problems involving the four basic operations on fractions

3.simplify fractions to lowest terms

4.convert between mixed numbers and improper fractions

5.use the order of operations to simplify expressions with fractions, exponents, grouping symbols,

6.identify the order relation between two fractions

III.Concepts and skills associated with decimals

1.write the standard form of a decimal

2.round decimals and use rounding to estimate values involving decimal arithmetic

3.perform the four basic arithmetic operations on decimals

4.solve application problems involving the four basic operations on decimals

5.convert between fractions and decimals

6.use the order of operations with decimals, exponents, grouping symbols, arithmetic operations.

7.identify the order relation between two decimals or between a decimal and a fraction

IV.Concepts and skills associated with integers and rational numbers

1.perform the four basic arithmetic operations on rational numbers

2.use the order of operations with rational numbers, exponents, arithmetic operations

3.solve application problems involving the four basic operations on rational numbers

4.identify the order relation between two rational numbers

V.Concepts and skills associated with ratios, proportions and percents

1.convert between fractions and percents and between decimals and percents

3.find the missing number in a proportion

4.solve ratio and proportion application problems

5.solve application problems involving percents

VI.Concepts and skills involving linear equations in one variable

1.solve linear equations in one variable involving integers, decimals and fractions

2.solve application problems that yield linear equations

VII.Concepts and skills associated with polynomials

1.identify terms of a polynomial, and classify polynomials by number of terms

2.use the exponent laws to simplify algebraic expressions involving whole number exponents

3.use the order of operations to evaluate variable expressions and formulas

5.add and subtract polynomials

6.multiply monomials by polynomials

VIII.Use statistics to collect and interpret data

1.determine the mean, median, and mode

2.interpret graphs (pictographs, circle graphs, bar graphs and line graphs) and analyze data

IX.Concepts and skills associated with geometry

1. know the appropriate vocabulary/facts about angles, triangles, rectangles, squares, and circles

2. find perimeters of rectilinear figures

3. use standard formulas to find perimeters and areas of triangles, rectangles, squares and circles


Miscellaneous math materials

Mathematics Benchmarks, Grades K-12
These are from Charles A. Dana Center at The University of Texas at Austin. The benchmarks describe the content and skills necessary for students on any given grade (K-6), or by strands (K-6 and 7-12). You can use these to have an idea of what topics to cover on any grade.

Parent Roadmaps to the Common Core Standards- Mathematics
These parent roadmaps provide guidance to parents about what children will be learning in each grade, including three-year snapshots showing how selected standards progress from year to year.


College of Arts and Sciences

Students completing a degree in the College of Arts and Sciences must complete three separate categories under the Quantitative and Logical Skills requirement. Bachelor of Arts and Bachelor of Science degrees have different courses to fulfill the Quantitative and Logical Skills requirement. This page includes the math requirements for each major in the ASC and the Quantitative and Logical Skills requirements for BA and BS students.

Note: Honors students in the College of Arts and Sciences may be expected to complete higher-level course work to satisfy the math requirement. Honors students should refer to the Honors Guides to the GE for honors math requirements.

Fulfill one of the following:

  • Achieve a standardized test score as specified by the State of Ohio 1
  • Attain Math Placement Test Score R or higher
  • Complete Math 1060 2 or Math 1075. Math 1060 2 and Math 1075 are remedial and do not count toward the 121 hour minimum requirement for the Bachelor of Arts or the Bachelor of Arts in Journalism.

1 If students earn an ACT Math Subscore of 22 or higher an SAT Math score of 520 or higher a score of 108EA (Elementary Algebra) or 69 CLM (College Level Math) on the College Board's Accuplacer tests or an Algebra Scale Score of 52 on the ACT's Compass math placement test, and the test was taken within two years of enrollment, then by State of Ohio law you are not required to take remedial math (courses numbered 1075 and below) regardless of the score on the Math Placement Test. However, students are strongly encouraged to take the math course tested into, especially if students plan to continue taking a sequence of math courses.

2 Math 1060 is typically offered on the regional campuses. Math 1060 is a terminal math course designed to fulfill the Quantitative and Logical Skills: Basic Computational Skills category of the GE. Students may follow up with Math 1116 to satisfy the Math and Logical Analysis category of the GE. Students who wish to take math courses higher than Math 1116 should take Math 1075 rather than Math 1060.

Choose one course from the following list:

    or any course at the 1200-level or above 3 2001, 3802 : any course at the level of Math 1116 or above 3 1500, 1501, 2500 : any course 3 except one of those listed in the Data Analysis category

3 except Math 1125, Math 1126, and courses numbered XX93 or XX94

Choose one course from the following list:

The course may also count in your major, if it is at the 2000-level or above and approved by the student's advisor.

    Data Analysis for Agribusiness and Applied Economics Data Analysis and Interpretation for Decision Making Methods of Astronomical Observation & Data Analysis Quantitative Analysis
  • Community Leadership 3537 Data Analysis in the Applied Sciences 2245 Introductory Data Analysis The Analysis and Display of Data Natural Resources Data Analysis Mapping Our World Data Analysis & Interpretation for Decision Making The Analysis and Display of Data Analyzing the Sounds of Language Probability, Data, and Decision Making Experimental Physics Instrumentation and Data Analysis Lab Techniques of Political Analysis 3549 Statistics in Sociology 2051 Analyzing the Sounds of Language 1350, 1430, 1450, 2450, 2480, 3450, 3460, 3470, 4202, 5301, 5302

Fulfill one of the following:

  • ​Attain Math Placement Test Score R or higher
  • Achieve a standardized test score as specified by the State of Ohio 1
  • Complete Math 1075. Math 1075 are remedial and do not count toward the 121 hour minimum requirement for the Bachelor of Science.
  • Achieve a standardized test score as specified by the State of Ohio 1

1 If students earn an ACT Math Subscore of 22 or higher an SAT Math score of 520 or higher a score of 108EA (Elementary Algebra) or 69 CLM (College Level Math) on the College Board's Accuplacer tests or an Algebra Scale Score of 52 on the ACT's Compass math placement test, and the test was taken within two years of enrollment, then by State of Ohio law you are not required to take remedial math (courses numbered 1075 and below) regardless of the score on the Math Placement Test. However, students are strongly encouraged to take the math course tested into, especially if students plan to continue taking a sequence of math courses.

​Complete Math 1151, Calculus I, or equivalent. Note: students that do not place into Math 1151 must complete prerequisite coursework.


How to improve basic math skills

There are four main ways you can improve your basic math skills:

1. Use workbooks

Math workbooks come with many sample problems you can solve to give yourself practice. They will also typically provide some instructions and advice on how to complete the problem, along with answers in the back of the book so you can check to see if you were right. If there&aposs a particular topic in math that you find challenging, look for a math workbook that focuses primarily on this topic.

2. Take a class

Many community colleges offer basic math classes, or you can see if there are other adult education classes near you. Another possibility is finding a basic math course online and completing it from home. Math classes provide the advantage of more detailed instruction and the ability to ask questions if you are confused about a topic.

3. Ask for help

If you know someone who has strong math skills, you could ask them for assistance. Let them know which areas you find challenging and see if they have any advice. Friends, family members and coworkers can provide a new perspective or perhaps explain things in more relatable terms, which will help to grow your understanding of the topic.

You can also hire a tutor to give you one-on-one attention, either in person or online. This tutor can then provide you with example problems to help strengthen your skills or answer any specific questions you may have.

4. Practice

The best way to improve your basic math skills is simply practicing. Using your skills consistently can ensure you maintain your proficiency. Try to avoid using a calculator for every problem you come across or asking someone else to do things for you. Seize every opportunity you can to use your basic math skills, and they will grow stronger over time.


Mathematics Learning Goals and Objectives

1. Learning Goal: Mathematics majors will develop computational skills in first-year calculus needed for more advanced calculus-based courses.

  1. evaluate derivatives for complexly constructed elementary functions
  2. evaluate definite and indefinite integrals and
  3. evaluate limits using algebraic, geometric, analytic techniques.

2. Learning Goal: Mathematics majors will learn and retain basic knowledge in the core branches of mathematics.

Objectives: Students will, during their senior year:

  1. demonstrate proficiency in calculus
  2. demonstrate proficiency in linear algebra and
  3. demonstrate proficiency in algebra.

3. Learning Goal: Mathematics majors will be able to learn and explain mathematics on their own.

  1. read a mathematics journal article and explain it, orally or in writing, to an audience of math majors and
  2. after graduation, be able to master new mathematics necessary for their employment.

4. Learning Goal: Mathematics majors will be able to read and construct rigorous proofs.

  1. construct clearly written proofs which use correct terminology and cite previous theorems
  2. construct proofs using mathematical induction
  3. construct proofs by contradiction and
  4. judge whether a proof is sound, and identify errors in a faulty proof.

5. Learning Goal: Mathematics majors will be able to obtain employment in their area of mathematical interest or gain admittance to a graduate program in mathematics.

  1. seek admission to graduate schools in mathematics will succeed in gaining admission, and perform adequately in these programs
  2. seek entry-level employment in math-related fields will obtain it
  3. specialize in actuarial science will obtain entry-level work as actuaries, if they seek it
  4. specialize in secondary education will demonstrate proficiency in mathematics needed to obtain Initial Certification in New York State or
  5. seek jobs in secondary or elementary education will obtain jobs at the appropriate grade level.

6. Learning Goal: Master’s students will recognize connections between different branches of mathematics.

Learning Objectives: Students will:

  1. correctly incorporate specific examples from one branch of mathematics into their study of another branch of mathematics (e.g., Lp-spaces as an example in linear algebra) and
  2. identify and explain cases in which major results of one branch of mathematics rely nontrivially on results from another branch (e.g., the application of linear algebra to solving systems of differential equations).

7. Learning Goal: Graduating master’s degree students will be able to obtain employment in their area of mathematical interest or gain admittance to a doctoral program in mathematics.


Perfect Your Mathematical Skills (Pre-Calculus Course)

This course is carefully designed to explain various topics of Basic Math, Algebra 1 & 2, Pre-Calculus.

It has 104 lectures spanning 15+ hours of on-demand videos that are divided into 17 sections. Each topic is explained extensively - by solving multiple questions along with the student during the lectures. The students are also provided and encouraged to solve practice questions & quizzes provided at the end of each topic.

This course will give you firm understanding of the fundamentals and is designed in a way that a person with little or no previous knowledge can also understand very well.

Topics covered in the course:

Polygons: Sides, Angles and Diagonals

Here's what some students say about the course:

"This is a good course to brush up your math skills. Even for beginners, it is very good. There are many points in this course which I learned are very helpful. The instructor's knowledge is superb and knows how to keep the students engaged. " - Ismeet Singh Saluja

"I have a big phobia of math. Taking this course has helped me ease the fear of math. Thus far the instructor is demystifying the subject for me" - Isabel Quezada

"This short course is like a refreshing mathematical knowledge. It is also a test of my knowledge. I got all quizzes correct. Thank you, Ruchi Chhabra" - Rabin Hada

"Great teaching style. Very clear explanations. Easy to follow" - Jackie Miller

"This is a comprehensive course in mathematical skills. I enjoyed this course!" - Tan Duong

With this course you'll also get:

Full lifetime access to Perfect Your Mathematical Skills

Complete support for any question, clarification or difficulty you might face on the topic

Udemy Certificate of Completion available for download

30-day money-back guarantee

Feel free to contact me for any questions or clarifications you might have.