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0: Pre-Calculus Refresher


0: Pre-Calculus Refresher

College of Engineering PREP-ME 2.0

PROGRAM DESCRIPTION

In advancing UIC’s mission of access to excellence and success, the College of Engineering offers the PREP-ME (Preparation for Majoring in Engineering) Workshop. PREP-ME makes the mathematics in engineering come alive, and previews the panorama of mathematics problems and core engineering concepts that UIC engineers will encounter in their undergraduate careers. Our student-centered format and emphasis on personalized instruction create a positive, motivating, welcoming classroom and program environment for students, plus a culturally focused and enriching interaction with corporate sponsors during the five-week session. To familiarize students with UIC’s institutional structure and campus personnel of the College of Engineering, technical and academic resources are introduced throughout the summer session. We strongly encourage all engineering freshmen to take advantage of this opportunity and get an early start on success at UIC.

Each summer, the Preparation for Majoring in Engineering (PREP-ME) program creates a positive, motivating, welcoming classroom environment for incoming freshmen while priming them for the challenges of engineering. Participants in this five to six-week session receive a refresher in pre-calculus, learn fundamental concepts in engineering, and interact with corporate partners. They become familiar with the faculty and staff who will support them during their time at UIC Engineering, as well as with the technical and academic resources they will need to excel in their classes. Preparation for Majoring in Engineering is offered at no cost to all incoming College of Engineering freshmen. Find additional information and registration procedures on the website of UIC Summer College, of which this program is a part.

PROGRAM DATES

PROGRAM DAYS AND TIMES

Monday - Thursday
9:00 a.m. – 12:00 p.m.

PROGRAM LOCATION

Virtual Remote Engagement

WHO SHOULD REGISTER?

All freshmen admitted to the College of Engineering should take a placement test prior to applying to PREP-ME. All incoming first-year, College of Engineering students are eligible to sign up.


PRECALCULUS

What is a rational number? Which numbers have rational square roots? The decimal representation of irrationals. What is a real number?

What is a function? The domain and the range.
Functional notation. The argument.
A function of a function.

The graph of a function . Coördinate pairs of a function. The height of the curve at x .

The constant function. The identity function.
The absolute value function. The parabola.
The square root function. The cubic function.
The reciprocal function.

Variables versus constants.
Definition of a polynomial in x .
The degree of a term and of a polynomial.
The leading coefficient .
The general form of a polynomial.
Domain and range.

A polynomial equation . The roots of a polynomial.
The x - and y-intercepts of a graph.
The relationship between the roots and the x -intercepts.

Definition of the slope. Positive and negative slope. A straight line has only one slope.
"Same slope" and "parallel." Perpendicular lines.
The slope and one point specify a straight line.

The equation of the first degree. The graph of a first degree equation: a straight line.
The slope-intercept form, and its proof.

Quadratic equation: Solution by factoring.
A double root. Quadratic inequalities.
The sum and product of the roots.

Solving a quadratic equation by completing the square . The quadratic formula.

The factor theorem. The fundamental theorem of algebra. The integer root theorem. Conjugate pairs.

Concave upward, concave downward.

Reflection about the x -axis. Reflection about the y -axis. Reflection through the origin.

Symmetry with respect to the y -axis. Symmetry with respect to the origin. Test for symmetry.
Odd and even functions.

Definition of a translation.
The equation of a circle.
The vertex of a parabola.
Vertical stretches and shrinks.

Singularities. The reciprocal function.
Horizontal and vertical asymptotes.

Definition of inverses. Constructing the inverse.
The graph of an inverse function.

The system of common logarithms.
The system of natural logarithms.
The three laws of logarithms.
Change of base.

Inverse relations.
Exponential and logarithmic equations.
Creating one logarithm from a sum.

The Fundamental Principle of Counting.
Factorial representations.
A binomial distribution.


LOGARITHMS

W HEN WE ARE GIVEN the base 2, for example, and exponent 3, then we can evaluate 2 3 .

Inversely, if we are given the base 2 and its power 8 --

-- then what is the exponent that will produce 8?

That exponent is called a logarithm . We call the exponent 3 the logarithm of 8 with base 2 . We write

We write the base 2 as a subscript.

3 is the exponent to which 2 must be raised to produce 8.

A logarithm is an exponent.

"The logarithm of 10,000 with base 10 is 4."

4 is the exponent to which 10 must be raised to produce 10,000.

"10 4 = 10,000" is called the exponential form .

"log1010,000 = 4" is called the logarithmic form .

That base with that exponent produces x .

Example 1. Write in exponential form: log232 = 5.

Example 2. Write in logarithmic form: 4 𕒶 = 1
16
.
Answer. log4 1
16
= 𕒶.

Problem 1. Which numbers have negative logarithms?

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!

Answer . 8 to what exponent produces 1? 8 0 = 1.

We can observe that, in any base, the logarithm of 1 is 0.

Answer . 5 with what exponent will produce 5? 5 1 = 5. Therefore,

In any base, the logarithm of the base itself is 1.

Answer . 2 raised to what exponent will produce 2 m ? m , obviously.

The following is an important formal rule, valid for any base b :

This rule embodies the very meaning of a logarithm. x -- on the right -- is the exponent to which the base b must be raised to produce b x .

The rule also shows that the exponential function b x is the inverse of the function log b x . We will see this in the following Topic.

Example 6 . Evaluate log3 1
9
.
Answer. 1
9
is equal to 3 with what exponent? 1
9
= 3 𕒶
log3 1
9
= log33 𕒶 = 𕒶.

Answer . . 25 = ¼ = 2 2 . Therefore,

Problem 2. Write each of the following in logarithmic form .

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

a) b n = x log b x = n b) 2 3 = 8 log 2 8 = 3
c) 10 2 = 100 log 10 100 = 2 d) 5 𕒶 = 1/25. log 5 1/25 = 𕒶.

Problem 3. Write each of the following in exponential form .

a) log b x = n b n = x b) log232 = 5 2 5 = 32
c) 2 = log864 8 2 = 64 d) log61/36 = 𕒶 6 𕒶 = 1/36

Problem 4. Evaluate the following.

a) log216 = 4 b) log416 = 2
c) log5125 = 3 d) log81 = 0
e) log88 = 1 f) log101 = 0

Problem 5. What number is n ?

a) log10 n = 3 1000 b) 5 = log2 n 32
c) log2 n = 0 1 d) 1 = log10 n 10
e) log n 1
16
= 𕒶 4 f) log n 1
5
= 𕒵 5
g) log2 1
32
= n 𕒹 h) log2 1
2
= n 𕒵

Problem 7. Evaluate the following.

a) log9 1
9
= log 9 9 𕒵 = 𕒵
b) log9 1
81
= 𕒶 c) log2 1
4
= 𕒶
d) log2 1
8
= 𕒷 e) log2 1
16
= 𕒸
f) log10 .01 = 𕒶 g) log10 .001 = 𕒷
h) log6 = 1/3 i) log b = 3/4

The system of common logarithms has 10 as its base. When the base is not indicated,

then the system of common logarithms -- base 10 -- is implied.

Here are the powers of 10 and their logarithms:

Powers of 10: 1
1000
1
100
1
10
1 10 100 1000 10,000
Logarithms: 𕒷 𕒶 𕒵 0 1 2 3 4

Logarithms replace a geometric series with an arithmetic series .

Problem 8. log 10 n = ? n . The base is 10.

Problem 9. log 58 = 1.7634. Therefore, 10 1.7634 = ?

58. 1.7634 is the common logarithm of 58. When 10 is raised to that exponent, 58 is produced.

Problem 10. log (log x ) = 1. What number is x ?

log a = 1, implies a = 10, which is the base.

Therefore, log (log x ) = 1 implies log x = 10. Since 10 is the base,

The system of natural logarithms has the number called e as its base. ( e is named after the 18th century Swiss mathematician, Leonhard Euler.) e is the base used in calculus. It is called the "natural" base because of certain technical considerations.

e x has the simplest derivative. Lesson 14 of An Approach to Calculus .)

e can be calculated from the following series expressed with factorials:

e is an irrational number its decimal value is approximately

To indicate the natural logarithm of a number we write "ln."

Problem 11. What number is ln e ?

ln e = 1. The logarithm of the base itself is always 1. e is the base.

Problem 12. Write in exponential form (Example 1): y = ln x .

1 . log b xy = log b x + log b y

" The logarithm of a product is equal to the sum
of the logarithms of each factor. "

" The logarithm of a quotient is equal to the logarithm of the numerator
minus the logarithm of the denominator. "

" The logarithm of x with a rational exponent is equal to
the exponent times the logarithm."

Example 9. Apply the laws of logarithms to log abc 2
d 3
.

Answer. According to the first two laws,

log abc 2
d 3
= log ( abc 2 ) − log d 3
= log a + log b + log c 2 − log d 3
= log a + log b + 2 log c − 3 log d ,

The Answer above shows the complete theoretical steps. In practice, however, it is not necessary to write the line

log abc 2
d 3
= log ( abc 2 ) − log d 3 .

The student should be able to go immediately to the next line --

log abc 2
d 3
= log a + log b + log c 2 − log d 3

-- if not to the very last line

log abc 2
d 3
= log a + log b + 2 log c − 3 log d .
Example 10. Apply the laws of logarithms to log
z 5
.
Answer. log
z 5
= log x + log − log z 5

log
z 5
= log x + ½ log y − 5 log z .

Example 11. Use the laws of logarithms to rewrite ln .

ln = ln (sin x ln x ) ½
= ½ ln (sin x ln x ), 3rd Law
= ½ (ln sin x + ln ln x ), 1st Law.

Note that the factors sin x ln x are the arguments of the logarithm function.

Example 12. Solve this equation for x :

log 3 2 x + 5 = 1
Solution . According to the 3rd Law, we may write
(2 x + 5)log 3 = 1.
Now, log 3 is simply a number. Therefore, on distributing log 3,
2 x · log 3 + 5 log 3 = 1
2 x · log 3 = 1 − 5 log 3
x = 1 − 5 log 3
2 log 3

By this technique, we can solve equations in which the unknown appears in the exponent.

Problem 13. Use the laws of logarithms to rewrite the following.

a) log ab
c
= log a + log b − log c
b) log ab 2
c 4
= log a + 2 log b − 4 log c
c) log
z
= 1/3 log x + 1/2 log y − log z

d) ln (sin 2 x ln x ) = ln sin 2 x + ln ln x = 2 ln sin x + ln ln x

e) ln = ½ ln (cos x · x 1/3 ln x )
= ½ (ln cos x + 1/3 ln x + ln ln x )
f) ln ( a 2 x − 1 b 5 x + 1 ) = ln a 2 x − 1 + ln b 5 x + 1
= (2 x − 1) ln a + (5 x + 1) ln b

ln 2 3 x + 1 = 5.
(3 x + 1) ln 2 = 5
3 x ln 2 + ln 2 = 5
3 x ln 2 = 5 − ln 2
x = 5 − ln 2
3 ln 2
Problem 15. Prove: −ln x = ln 1
x
.
−ln x = (𕒵)ln x = ln x 𕒵 , Third law
= ln 1
x

Proof of the laws of logarithms

The laws of logarithms will be valid for any base. We will prove them for base e , that is, for y = ln x .

The function y = ln x is defined for all positive real numbers x . Therefore there are real numbers p and q such that

Therefore, according to the rules of exponents,

ln ab = ln e p + q = p + q = ln a + ln b .

Which is what we wanted to prove.

In a similar manner we can prove the 2nd law. Here is the 3rd:

There is a real number p such that

And the rules of exponents are valid for all rational numbers n (Lesson 29 of Algebra an irrational number is the limit of a sequence of rational numbers). Therefore,

ln a n = ln e pn = pn = np = n ln a .

That is what we wanted to prove.

Say that we know the values of logarithms of base 10, but not, for example, in base 2. Then we can convert a logarithm in base 10 to one in base 2 -- or any other base -- by realizing that the values will be proportional.

Each value in base 2 will differ from the value in base 10 by the same constant k .

Now, to find that constant, we know that

Therefore, on putting x = 2 above:

By knowing the values of logarithms in base 10, we can in this way calculate their values in base 2.

In general, then, if we know the values in base a , then the constant of proportionality in changing to base b , is the reciprocal of its log in base a .

Problem 16. Write the rule for changing to base 8 from base e .

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Preparation

We recommend taking an online readiness test to practice for the MPE.

You are allowed to take the exam multiple times, but you have to wait until the new window of examination is offered.

Recommended refresher books

Just-In-Time Algebra and Trigonometry by Mueller & Brent. Addison-Wesley ISBN 0-201-41951-3

Schaum's Outline of College Algebra by Spiegel. McGraw Hill ISBN 0-07-060266-2. (More basic than the other Schaum's Outline book.)

Schaum's Outline of Precalculus by Fred Safier. McGraw Hill ISBN 0-07-057261-5. (More advanced than the other Schaum's Outline book.)


What Do You Learn in Precalculus?

Precalculus is a structured entry to advanced studies leading to calculus. The course builds on concepts learned in previous high school math courses, particularly Algebra 2. Theoretical mathematics principles are used to emphasize problem solving and mental mathematics.

Some of the chapters included in Time4Learning’s online precalculus course for high school students are:

  • Functions and Graphs
  • Lines and Rates of Change
  • Sequences and Series
  • Polynomial and Rational Functions
  • Exponential and Logarithmic Functions
  • Analytic Geometry
  • Linear Algebra and Matrices
  • Probability and Statistics

Math Placement

The SOAR (Stanford Online Academic Resources) program provides an opportunity for incoming first-year Stanford students to consolidate and bolster important skills that will help them succeed in their first-year classes and beyond. SOAR consists of two separate classes, one in Math and one in Writing. These online classes are offered at no cost to students, and each one carries one unit of credit. They may be taken individually or together. These classes meet three times per week for five weeks in late summer (7/26 to 8/27). The classes are small, with personalized curriculum, and strive for a friendly learning community. Our goal is to cultivate positive learning experiences that will help students prepare for future mathematics and writing courses at Stanford. Full details can be found at our website https://soarsummer.stanford.edu/

Placement Diagnostic

The placement diagnostic is required of everyone (regardless of AP Calculus credit) to enroll in the courses listed in the chart below unless you’ve already taken the prerequisite course at Stanford.

The placement diagnostic will recommend the initial math course in which to enroll, and this recommendation is purely advisory. The diagnostic will help identify areas you may want to review. For additional information regarding the courses recommended in the diagnostic, please see the Introductory Math Courses page.

The Precalculus Refresher in Canvas, consists of videos and associated exercises that are available for use at any time. It goes through the core skills in algebra, graphing, functions, and trigonometry that are necessary for learning and using calculus. It is recommended for anyone who needs to brush up on some of these skills or has background gaps. It is best to do this enough in advance so that you have worked through all relevant parts prior to enrollment in a calculus course here.

Course Placement Diagnostic or Prerequisite Course
Math 19
Math 20 or Math 19
Math 21 or Math 20
Math 51 or Math 21

Please note:

The diagnostic gives you the most useful feedback if taken in the quarter before you enroll in your first Math department course at Stanford (if at the level of Math 51 or below). Completing the diagnostic between August 1, 2020 and July 9, 2021 gives you permission to enroll in Math 19, 20, 21, and 51 for the 2020-21 academic year as your first Math department course here. Therefore, if you are planning to not take a first Math department course here at the level of Math 51 or below until the 2021-22 academic year or later then you will need to retake the diagnostic at that point.

Please note that after you have completed a first Math department course here at the level of Math 51 or below, you never need to retake the diagnostic for enrollment in another Math department course. For example, you can take Math 21 this year and put off Math 51 until next year or later without retaking the diagnostic.

Please also note that the outcome of this diagnostic has no effect on AP calculus credit at Stanford. However, choosing to enroll in a course for which you’ve received Transfer or Advanced Placement credit is subject to the Registrar’s policy on the duplication of credit.

Sample Calculus Course Progressions

The following table lists course plans according to your recommended placement. If the placement diagnostic recommends that you review specific precalculus topics, we urge you to do this before enrolling.


Precalculus:

I need some guidance in how to solve the following precalc problem. I am not sure how to calculate it without being given speed. Any assistance or tips would be really appreciated. Thank you in advance!

Suppose your car speedometer is only giving accurate speeds when using a particular tire size. The diameter of the metal part of the wheel is 17 inches and the rubber portion of the tire is 6.5 inches. What is the angular speed?

Advice on getting ready for Calc 2?

How to get ready for Calc 2?

I start Calculus 2 in two months but am feeling uneasy and unprepared for it. I took Calc 1 in fall 2020 in an online, asynchronous format. I was able to scrape by with a C, but it was stressful especially with the circumstances. These next two months I really want to dive in and self-study and review Calc 1, trig, and algebra. What resources and tips do you have for really succeeding in Calc 2 and reviewing?

Finish precalculus 1

Soon I will start precalculus 2.

I did not really learn much. Other then I already knew. And those that I did learn. I forgot after the exam. I would end up with an A- or B+

I also used some notes and calculators as it was online class. My study would be that I would not focus on any of that. But about 3-4 days before the exams I would just go full mode and practice the sample exam too. That is how I would end up with good grade on exams.

Can I keep this same style of study until calculus 2? My original goal was all math classes of engineering, but I might go for computer science so only need until calculus 2 and then the computer math is given.

I did some calculus review exams they seems much easier than precalculus.


Math (MATH)

This is a draft edition of the 2021-2022 catalog. Information is subject to change. Go to MyTMCC for course descriptions and prerequisites in effect for the Fall registration cycle.

MATH 100 - Math for Allied Health Programs Units: 3

A review of basic mathematics with emphasis on those mathematical skills needed for the dental assisting program. This course will include a review of arithmetic, material on the metric system, apothecary system, dosages and solutions.

Transferability: May not transfer towards an NSHE bachelor's degree

MATH 105 - Applied Topics in Math Units: 3

A course including the following topics: review of arithmetic, algebra, geometry and graphical representation.

Transferability: May not transfer towards an NSHE bachelor's degree

MATH 106 - Geometry Units: 3

This course is designed to provide a basic working knowledge of practical geometry for students who have never taken a course in geometry or who need a refresher course. Theory is not emphasized. Some of the topics covered are: area of plane figures, similarity, volume of solids, angle measure, and properties of special triangles.

Transferability: May not transfer towards an NSHE bachelor's degree

Enrollment Requirements: Prerequisite: MATH 95 or equivalent or qualifying Accuplacer, ACT/SAT test results (taken within 2 years).

MATH 107 - Real Estate Math Units: 3

Review of basic arithmetic principles. A general mathematics course designed to assist the student who wishes to pass the state exam and the student who wants to be more proficient and knowledgeable in the real estate profession. Decimals, percentages, fractions, prorations, tax rate, interest, discount and depreciation are included.

Transferability: May not transfer towards an NSHE bachelor's degree

MATH 108 - Math for Technicians Units: 3

This applied mathematics course is designed to give the student math skills as they are applied to specific career choice areas. Topics for all individual applied areas (transportation, metalworking, construction, etc.) will include algebra and trigonometry, but the focus of the presentation and utilization will be specific to the industry area. The course will include demonstrations and hands-on exercises applying mathematics as it will be needed in the specific technical environment.

Transferability: May not transfer towards an NSHE bachelor's degree

Enrollment Requirements: Prerequisite: Qualifying ACCUPLACER, ACT/SAT test results. A graphing calculator may be required for this course.

MATH 119 - Fundamentals of College Mathematics II Units: 3

A continuation of Math 19 covering remaining topics of Math 120. Presentation is adapted to needs of students with learning or physical disabilities. (This course may be substituted for Math 120 in degrees and programs) Mathematical concepts particularly relevant to informed and aware citizenship in modern society. Topics covered include functions, graphs, problem solving, topics in finance, geometry, probability and statistics. Satisfies UNR core curriculum. Note: Computer use and graphing calculator may be required (TI-83/84 recommended). Note: Completing Math 119 is designed to be equivalent to completing Math 120. Therefore, Math 119 satisfies Math Gen Ed for AA and AAS but not AS.

Enrollment Requirements: Prerequisite: A grade of 'C' or better in MATH 19. A graphing calculator may be required.

MATH 120 - Fundamentals of College Mathematics Units: 3

This course covers the mathematical concepts particularly relevant to non-science majors. Topics covered include problem-solving, topics in finance, probability, statistics, and additional real-world applications. Satisfies UNR core curriculum. Note: Computer use and graphing calculator may be required (TI-83/84 recommended).

Enrollment Requirements: Prerequisite: A grade of 'A' or 'B' in MATH 95, or a grade of 'C' or better in MATH 96, or equivalent, or qualifying ACCUPLACER, ACT/SAT test results (within 2 years). A graphing calculator may be required.

Term Offered: All Semesters

MATH 122 - Number Concepts for Elementary School Teachers Units: 3

Mathematics needed by those teaching new-content mathematics courses at the elementary school level with emphasis on the structure of the real number system and its subsystems. Designed for students seeking a teaching certificate in elementary education. Open to others with approval of department chair. This course may be taken before, after or during the same semester as Math 123.

Enrollment Requirements: Prerequisite: A grade of 'C' or better in MATH 120 or MATH 126 or equivalent or qualifying ACCUPLACER, ACT/SAT test results (taken within 2 years). A graphing calculator may be required for this course.

Term Offered: Spring and Fall

MATH 123 - Statistical & Geometrical Concepts for Elementary School Teachers Units: 3

Mathematics needed by those teaching new-content mathematics courses at the elementary school level with emphasis on geometry, algebra, probability, and statistics. Designed for students seeking a teaching certification in elementary education. This course may be taken before, after or during the same semester as Math 122.

Enrollment Requirements: Prerequisite: A grade of 'C' or better in MATH 120 or MATH 126 or equivalent or qualifying ACCUPLACER, ACT/SAT test results (taken within 2 years). A graphing calculator may be required for this course.

Term Offered: Spring and Fall

MATH 124 - College Algebra Units: 3

The study of equations and inequalities, relations and functions, linear, quadratic, polynomial, rational, exponential, logarithmic and their applications.

Enrollment Requirements: Completion of Math 096, Math 120, ACT of 19, SAT 510, or Accuplacer Score of QAS 263

MATH 126 - Pre-Calculus I Units: 3

The study of functions, their properties, their graphs, and applications including polynomial, radical, rational, exponential and logarithmic functions. The course also covers the solving of equations, systems of equations, and inequalities.

Enrollment Requirements: A grade of 'C' or better in MATH 96 or equivalent or qualifying ACCUPLACER, ACT/SAT test results (within 2 years). Students may enroll concurrently with Math 26 without any prerequisite. A graphing calculator may be required (TI 83/84 recommended).

Term Offered: All Semesters

MATH 127 - Pre-Calculus II Units: 3

This course is a continuation of Math 126. It includes the study of circular functions, their graphs and applications, analytic trigonometry, the coordinate geometry of lines and conics and elementary vector algebra. Note: Computer use and graphing calculator may be required (TI-83/84 recommended).

Enrollment Requirements: Prerequisite: A grade of 'C' or better in MATH 126 or equivalent or qualifying ACCUPLACER, ACT/SAT test results (taken within 2 years). A graphing calculator may be required.

Term Offered: All Semesters

MATH 176 - Introductory Calculus for Business and Social Sciences Units: 3

Topics covered include graphing functions, derivatives, integrals, applications, the Fundamental Theorem of Calculus. This course is designed for business, social science or biological science majors.

Enrollment Requirements: Prerequisite: A grade of 'C' or better in MATH 126 or equivalent or qualifying ACCUPLACER, ACT/SAT test results (taken within 2 years). A graphing calculator may be required.

Term Offered: Spring and Fall

MATH 181 - Calculus I Units: 4

Topics covered include functions, the derivative, differentiation of functions, applications of the derivative, understanding the definite integral, finding integrals and applications of integrals. Throughout the course topics will be viewed geometrically, numerically and algebraically. This course is oriented toward students of mathematics, physical science and engineering. Satisfies UNR math core curriculum. Note: Computer use and graphing calculator may be required (TI-83/84 recommended).

Enrollment Requirements: Prerequisite: A grade of 'C' or better in MATH 127 or MATH 128 or equivalent or qualifying ACCUPLACER, ACT/SAT test results (within 2 years). A graphing calculator may be required.

Term Offered: Spring and Fall

MATH 182 - Calculus II Units: 4

A continuation of MATH 181. Topics covered include a continuation of the definite integral, finding integrals and applications of integrals, differential equations and approximations of functions with simpler functions. Throughout the course topics will be viewed geometrically, numerically and algebraically. This course is oriented toward students of mathematics, physical science and engineering. Note: Computer use and graphing calculator may be required (TI-83/84 recommended).

Enrollment Requirements: Prerequisite: A grade of 'C' or better in MATH 181 or equivalent or qualifying ACCUPLACER, ACT/SAT test results (taken within 2 years). A graphing calculator may be required.

Term Offered: Spring and Fall

MATH 19 - Fundamentals of College Mathematics I Units: 3

This is the first half of a 2-semester course covering Math 120 content. Presentation is adapted to needs of students with learning or physical disabilities. Enrollment by departmental permission. (Credit does not apply to any baccalaureate degree program.) Mathematical concepts particularly relevant to informed and aware citizenship in modern society. Topics covered include functions, graphs, problem solving, topics in finance, geometry, probability and statistics. Satisfies UNR core curriculum. Note: Computer use and graphing calculator may be required (TI-83/84 recommended).

Transferability: May not transfer towards an NSHE bachelor's degree

Enrollment Requirements: Prerequisite: A grade of 'C' or better in MATH 96 or equivalent or qualifying ACCUPLACER, ACT/SAT test results (within 2 years). A grade of 'B-' or better in MATH 95 in lieu of MATH 96.

MATH 20 - Learning Support for MATH 120/120E-2 credits Units: 2

Provides a review of algebra, corequisite mathematical support, and just in time material for MATH 120, Fundamentals of College Mathematics.

Transferability: May not transfer towards an NSHE bachelor's degree

MATH 24 - Learning Support for Math 124/124E - 3 credits Units: 3

This course provides the just-in-time algebraic support for Math 124. The course will refresh, review and introduce a variety of concepts to be successful in Math 124, College Algebra, including exponents and their properties, polynomials, rational and radical expressions, graphing, interval notation, proportions and variations.

Transferability: May not transfer towards an NSHE bachelor's degree

Enrollment Requirements: ACT of at least 17 or Accuplacer Score NGAR of at least 263.

MATH 26 - Learning support for Math 126 Units: 3

Provides a review of algebra, corequisite mathematical support, and just in time material for MATH 126, PreCalculus I.

Transferability: May not transfer towards an NSHE bachelor's degree

Enrollment Requirements: Must enroll concurrently with Math 126.

MATH 283 - Calculus III Units: 4

A continuation of Math 182. Topics covered include vectors, differentiating and integrating functions of many variables, optimization, parametric curves and surfaces, line integrals, flux integrals and vector fields. Throughout the course topics will be viewed geometrically, numerically and algebraically. This course is oriented toward students of mathematics, physical science and engineering. Note: Computer use and graphing calculator may be required (TI-83/84 recommended).

Enrollment Requirements: Prerequisite: A grade of 'C' or better in MATH 182 (taken within 2 years).

Term Offered: Spring and Fall

MATH 285 - Differential Equations Units: 3

Theory and solving techniques for constant and variable coefficient linear equations and a variety of non-linear equations. Emphasis on those differential equations arising from real world phenomena. Note: Computer use and graphing calculator may be required (TI-83/84 recommended).

Enrollment Requirements: Prerequisite: A grade of 'C' or better in MATH 182 or equivalent. A graphing calculator may be required for this course.

Term Offered: Spring and Fall

MATH 295 - Proof Writing for Math/Stat Major Units: 3

Foundations of mathematical proof writing for advanced courses in the Math/Stat majors. Proof methods will be applied to topics in logic mathematical induction elementary set theory functions properties of integers and real numbers.

Enrollment Requirements: Math 283 with a C or better

MATH 330 - Linear Algebra Units: 3

Vector analysis continued abstract vector spaces, bases, inner products, projections, orthogonal complements, least squares, linear maps, structure theorems, elementary spectral theory, applications.

Enrollment Requirements: Co-requisite: Math 283

MATH 90 - Continuing Studies in Math Units: 0.5-3

This developmental course is for assessment purposes. Developmental students may register for this course without taking Accuplacer.

Enrollment Requirements: Prerequisite: MATH 93 or equivalent or qualifying Accuplacer score, ACT/SAT test results.

Term Offered: Spring and Fall

MATH 95 - Elementary Algebra Units: 3

A first course in algebra. Topics covered include the fundamental operations on real numbers, first degree equations, inequalities in one variable, polynomials, integer exponents, solving quadratic equations by factoring. Note: Computer use and graphing calculator may be required (TI-83/84 recommended).

Enrollment Requirements: Prerequisite: SKC 80, CTM 85, CTM 86, CTM 87, qualifying ACCUPLACER, ACT/SAT test results or equivalent (taken within 2 years).

Term Offered: All Semesters

MATH 96 - Intermediate Algebra Units: 3

A second course in algebra. Topics covered include: solving quadratic, rational and radical equations, simplifying rational and radical expressions and complex numbers, and solving application problems. Note: Computer use and graphing calculator may be required (TI-83/84 recommended).

Enrollment Requirements: Prerequisite: A grade of 'C' or better in MATH 95, or Math 120, or equivalent or qualifying ACCUPLACER, ACT/SAT test results (taken within 2 years).

Term Offered: All Semesters

MATH 96A - Intermediate Algebra - Basic Properties Units: 1

This course is a co-requisite course for MATH 120E. Students need to be enrolled in both MATH 96A and MATH 120E. This course reviews the algebraic concepts needed to be successful in Math 120. Topics include, but are not limited to, exponents and radicals, solving linear, nonlinear, and absolute value equations and inequalities, and algebraic techniques involving exponents, radical, rational expressions and their applications.

Enrollment Requirements: You MUST enroll in MATH 120-2207, MATH 20-2207 meets just before the MATH 120-2207


Formulas from Trigonometry

Right-Angle Trigonometry

sin θ = opp hyp csc θ = hyp opp cos θ = adj hyp sec θ = hyp adj tan θ = opp adj cot θ = adj opp sin θ = opp hyp csc θ = hyp opp cos θ = adj hyp sec θ = hyp adj tan θ = opp adj cot θ = adj opp

Trigonometric Functions of Important Angles

Fundamental Identities

sin 2 θ + cos 2 θ = 1 sin ( − θ ) = − sin θ 1 + tan 2 θ = sec 2 θ cos ( − θ ) = cos θ 1 + cot 2 θ = csc 2 θ tan ( − θ ) = − tan θ sin ( π 2 − θ ) = cos θ sin ( θ + 2 π ) = sin θ cos ( π 2 − θ ) = sin θ cos ( θ + 2 π ) = cos θ tan ( π 2 − θ ) = cot θ tan ( θ + π ) = tan θ sin 2 θ + cos 2 θ = 1 sin ( − θ ) = − sin θ 1 + tan 2 θ = sec 2 θ cos ( − θ ) = cos θ 1 + cot 2 θ = csc 2 θ tan ( − θ ) = − tan θ sin ( π 2 − θ ) = cos θ sin ( θ + 2 π ) = sin θ cos ( π 2 − θ ) = sin θ cos ( θ + 2 π ) = cos θ tan ( π 2 − θ ) = cot θ tan ( θ + π ) = tan θ

Law of Sines

sin A a = sin B b = sin C c sin A a = sin B b = sin C c

Law of Cosines

a 2 = b 2 + c 2 − 2 b c cos A b 2 = a 2 + c 2 − 2 a c cos B c 2 = a 2 + b 2 − 2 a b cos C a 2 = b 2 + c 2 − 2 b c cos A b 2 = a 2 + c 2 − 2 a c cos B c 2 = a 2 + b 2 − 2 a b cos C

Addition and Subtraction Formulas

sin ( x + y ) = sin x cos y + cos x sin y sin ( x − y ) = sin x cos y − cos x sin y cos ( x + y ) = cos x cos y − sin x sin y cos ( x − y ) = cos x cos y + sin x sin y tan ( x + y ) = tan x + tan y 1 − tan x tan y tan ( x − y ) = tan x − tan y 1 + tan x tan y sin ( x + y ) = sin x cos y + cos x sin y sin ( x − y ) = sin x cos y − cos x sin y cos ( x + y ) = cos x cos y − sin x sin y cos ( x − y ) = cos x cos y + sin x sin y tan ( x + y ) = tan x + tan y 1 − tan x tan y tan ( x − y ) = tan x − tan y 1 + tan x tan y

Double-Angle Formulas

sin 2 x = 2 sin x cos x cos 2 x = cos 2 x − sin 2 x = 2 cos 2 x − 1 = 1 − 2 sin 2 x tan 2 x = 2 tan x 1 − tan 2 x sin 2 x = 2 sin x cos x cos 2 x = cos 2 x − sin 2 x = 2 cos 2 x − 1 = 1 − 2 sin 2 x tan 2 x = 2 tan x 1 − tan 2 x

Half-Angle Formulas

sin 2 x = 1 − cos 2 x 2 cos 2 x = 1 + cos 2 x 2 sin 2 x = 1 − cos 2 x 2 cos 2 x = 1 + cos 2 x 2

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