# Precalculus

### MATH 2312

• State Approval Code: 2701015819
• Semester Credit Hours: 3
• Lecture Hours per Week: 3
• Contact Hours per Semester: 48

### Catalog Description

Analytic and numerical treatment of functions, their graphs, inverse functions, polynomial functions, rational and irrational functions, exponential and logarithmic functions, trigonometric and inverse trigonometric functions, and graphical analysis of slope related to derivatives and area related to integration. Lecture hours = 3, Lab hours = 0.

### Prerequisites

TSIP math completed and high school precalculus or MATH 1316

### Basic Intellectual Compentencies in the Core Curriculum

• Listening
• Critical thinking
• Computer literacy

### Perspectives in the Core Curriculum

• Use logical reasoning in problem solving.

### Mathematics

• To apply arithmetic, algebraic, geometric, higher-order thinking, and statistical methods to modeling and solving real-world situations.
• To represent and evaluate basic mathematical information verbally, numerically, graphically, and symbolically.
• To expand mathematical reasoning skills and formal logic to develop convincing mathematical arguments.
• To use appropriate technology to enhance mathematical thinking and understanding and to solve mathematical problems and judge the reasonableness of the results.
• To interpret mathematical models such as formulas, graphs, tables and schematics, and draw inferences from them.
• To recognize the limitations of mathematical and statistical models.
• To develop the view that mathematics is an evolving discipline, interrelated with human culture, and understand its connections to other disciplines.

### Instructional Goals and Purposes

Panola College's instructional goals include 1) creating an academic atmosphere in which students may develop their intellects and skills and 2) providing courses so students may receive a certificate/an associate degree or transfer to a senior institution that offers baccalaureate degrees.

### General Course Objectives

Successful completion of this course will promote the general student learning outcomes listed below. The student will be able to:
1.Improve problem-solving skills through solving application problems.
2.Demonstrate at the completion of the course the algebraic manipulation and problem solving skills necessary to be successful in future coursework.
3.Develop the vocabulary and methodology necessary for reading and understanding scientific and mathematical literature.
4.Construct appropriate mathematical models to solve applications.
5.Interpret and apply mathematical concepts.
6.Use multiple approaches - physical, symbolic, and verbal - to solve application problems.

### Specific Course Objectives

Upon successful completion of the course, the student will be able to:
1.Apply coordinate geometry formulas.
2.Model application problems using functions.
3.Model and solve variation problems.
4.Perform transformations on functions.
5.Find real and complex zeros of polynomial functions.
6.Graph rational, exponential, and logarithmic functions.
7.Model and solve problems involving exponential and logarithmic functions.
8.Model and solve trigonometry problems involving both right triangles and oblique triangles.
9.Graph trigonometric functions and their inverses.
10.Solve trigonometric identities and trigonometric equations.
11.Write a sum or difference of fractional expressions as a single fraction.
12.Model and solve problems involving conic section formulas for a circle, parabola, ellipse, and hyperbola.
13.Graph polar and parametric equations.
14.Model and solve problems involving vectors.
15.Find limits of functions numerically, algebraically, and graphically.
16.Apply the definition of a derivative.

### General Description of Each Lecture or Discussion

Students will be required to do the following:
REVIEW OF ALGEBRA SKILLS
•Simplify algebraic expressions, exponents and radicals.
•Solve fractional equations and inequalities.
•Model and solve problems involving linear equations.
•Apply the formulas of coordinate geometry (distance, midpoint).
•Write the equation of a circle given its center and radius.
•Graph circles, linear functions, and inequalities.
•Write equations of linear functions given slope, points, parallel and perpendicular lines.
•Define a function, domain, and range.
•Model and solve problems involving variation functions.
•Find the average rate of change, increasing and decreasing intervals, and extreme values of functions.
•Apply transformations to functions.
•Model problems using functions.
•Combine functions by adding, subtracting, multiplying, dividing, and composition.
•Define one-to-one functions and find their inverses.
•Define a polynomial function and sketch its graph using end behavior, zeros, local extrema, and test points.
•Divide polynomials by using synthetic division.
•Define and apply the remainder and the factor theorems.
•Find the rational zeros of a polynomial by use of the rational zeros theorem, Descartes' rule of signs, and the upper/lower bound theorem.
•Add, subtract, multiply, and divide complex numbers.
•Find the complex roots of quadratic equations.
•Apply the Fundamental Theorem of Algebra and the Complete Factorization Theorem to find the complex roots of a polynomial function.
•Write a polynomial function give its complex and real roots.
•Find the vertical and horizontal asymptotes of a rational function.
•Sketch the graph of a rational function.
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
•Sketch the graph of an exponential and logarithmic function and find the domain and range.
•Model and solve problems involving compound interest.
•Define the natural logarithm and its properties.
•Apply the Laws of Logarithms to simplify and solve logarithmic equations.
•Solve exponential equations.
•Model and solve problems involving exponential and logarithmic functions.
TRIGONOMETRIC FUNCTIONS
•Define the unit circle and use reference numbers to find terminal points.
•Define the trigonometric functions, their even-odd properties, their signs in different quadrants, and the domains and ranges of each.
•Sketch the graphs of the trigonometric functions.
•Find the period and amplitude of trigonometric functions.
•Graph transformations of the trigonometric functions.
•Model and solve problems involving the formulas for length of a circular arc, area of a sector of a circle, and linear and angular speed.
•Model and solve problems involving trigonometry of right triangles.
•Use reference numbers on the unit circle or reference triangles to evaluate trigonometric functions.
•Apply the Law of Sines and the Law of Cosines to solve oblique triangle problems.
•Prove Trigonometric Identities.
•Apply the Addition and Subtraction Formulas to problems and identities.
•Apply the Sum of Sines and Cosines Formulas.
•Apply the Double-Angle and Half-Angle Formulas to problems and identities.
•Apply the Product-to-Sum and Sum-to-Product Formulas.
•Define and sketch the inverse trigonometric functions.
•Solve trigonometric equations involving single and multiple angles.
VECTORS
•Understand the geometric and the analytic description of vectors.
•Find the magnitude, horizontal and vertical components of a vector.
•Find the resultant force of two or more vectors.
•Apply vectors to model and solve problems involving velocity and force.
•Find the dot product of two vectors and use it to find the angle between the two vectors.
•Find the component vector of u along v.
•Find the projection of u onto v.
•Apply vectors to solve problems involving work.
LIMITS
•Find the limit of a function numerically, algebraically, and graphically.
•Define a one-sided limit.
•Write the definition of a limit.
•Apply the Limit Laws.
•Find and prove limits by applying right- and left-hand limits.
•Write the definition of a derivative.
•Find the equation of a line tangent to a curve at a given point.
•Find the instantaneous velocity of a falling object.
•Find the limit of a function at infinity.
•Find the horizontal asymptote of a rational function by applying limits at infinity.
•Approximate the area under a curve.
•Find the exact area under a curve using limits.

### Methods of Instruction/Course Format/Delivery

Students will be required to do the following:
REVIEW OF ALGEBRA SKILLS
•Simplify algebraic expressions, exponents and radicals.
•Solve fractional equations and inequalities.
•Model and solve problems involving linear equations.
•Apply the formulas of coordinate geometry (distance, midpoint).
•Write the equation of a circle given its center and radius.
•Graph circles, linear functions, and inequalities.
•Write equations of linear functions given slope, points, parallel and perpendicular lines.
•Define a function, domain, and range.
•Model and solve problems involving variation functions.
4
•Find the average rate of change, increasing and decreasing intervals, and extreme values of functions.
•Apply transformations to functions.
•Model problems using functions.
•Combine functions by adding, subtracting, multiplying, dividing, and composition.
•Define one-to-one functions and find their inverses.
•Define a polynomial function and sketch its graph using end behavior, zeros, local extrema, and test points.
•Divide polynomials by using synthetic division.
•Define and apply the remainder and the factor theorems.
•Find the rational zeros of a polynomial by use of the rational zeros theorem, Descartes' rule of signs, and the upper/lower bound theorem.
•Add, subtract, multiply, and divide complex numbers.
•Find the complex roots of quadratic equations.
•Apply the Fundamental Theorem of Algebra and the Complete Factorization Theorem to find the complex roots of a polynomial function.
•Write a polynomial function give its complex and real roots.
•Find the vertical and horizontal asymptotes of a rational function.
•Sketch the graph of a rational function.
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
•Sketch the graph of an exponential and logarithmic function and find the domain and range.
•Model and solve problems involving compound interest.
•Define the natural logarithm and its properties.
•Apply the Laws of Logarithms to simplify and solve logarithmic equations.
•Solve exponential equations.
•Model and solve problems involving exponential and logarithmic functions.
TRIGONOMETRIC FUNCTIONS
•Define the unit circle and use reference numbers to find terminal points.
•Define the trigonometric functions, their even-odd properties, their signs in different quadrants, and the domains and ranges of each.
•Sketch the graphs of the trigonometric functions.
•Find the period and amplitude of trigonometric functions.
•Graph transformations of the trigonometric functions.
•Model and solve problems involving the formulas for length of a circular arc, area of a sector of a circle, and linear and angular speed.
•Model and solve problems involving trigonometry of right triangles.
•Use reference numbers on the unit circle or reference triangles to evaluate trigonometric functions.
•Apply the Law of Sines and the Law of Cosines to solve oblique triangle problems.
•Prove Trigonometric Identities.
•Apply the Addition and Subtraction Formulas to problems and identities.
•Apply the Sum of Sines and Cosines Formulas.
•Apply the Double-Angle and Half-Angle Formulas to problems and identities.
•Apply the Product-to-Sum and Sum-to-Product Formulas.
•Define and sketch the inverse trigonometric functions.
•Solve trigonometric equations involving single and multiple angles.
VECTORS
5
•Understand the geometric and the analytic description of vectors.
•Find the magnitude, horizontal and vertical components of a vector.
•Find the resultant force of two or more vectors.
•Apply vectors to model and solve problems involving velocity and force.
•Find the dot product of two vectors and use it to find the angle between the two vectors.
•Find the component vector of u along v.
•Find the projection of u onto v.
•Apply vectors to solve problems involving work.
LIMITS
•Find the limit of a function numerically, algebraically, and graphically.
•Define a one-sided limit.
•Write the definition of a limit.
•Apply the Limit Laws.
•Find and prove limits by applying right- and left-hand limits.
•Write the definition of a derivative.
•Find the equation of a line tangent to a curve at a given point.
•Find the instantaneous velocity of a falling object.
•Find the limit of a function at infinity.
•Find the horizontal asymptote of a rational function by applying limits at infinity.
•Approximate the area under a curve.
•Find the exact area under a curve using limits.

### Assessment

Faculty may assign both in- and out-of-class activities to evaluate students' knowledge and abilities. Faculty may choose from the following methods:
•Attendance
•Book reviews
•Class preparedness and participation
•Collaborative learning projects
•Compositions
•Exams/tests/quizzes
•Homework
•Internet
•Journals
•Library assignments
•Research papers
•Scientific observations
•Student-teacher conferences
•Written assignments
Students' final grades are determined by:
Exams           30% to 50%
Homework/Quizzes      20% to 30%
Other      0% to 10%
Final Exam      20% to 30%

### Text, Required Readings, Materials, and Supplies

•Precalculus, 8th Edition. Larson and Falvo. Cengage Learning, 2011.
•Scientific calculator (graphing preferred).