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MOHAWK VALLEY COMMUNITY COLLEGE

UTICA, NEW YORK

COURSE OUTLINE

FUNDAMENTALS OF COLLEGE MATHEMATICS 2

MA122

Reviewed and Found Acceptable by Robert Bernstein – 5/02

Reviewed and Found Acceptable by Robert Bernstein – 5/03

Revised by Robert Bernstein – 1/04

Reviewed and Found Acceptable by Robert Bernstein – 5/04

Reviewed and Revised by Robert Bernstein – 1/05

Reviewed and Revised by Mark Miller – 5/05

Reviewed and Revised by Mark Miller – 5/06

Reviewed and Revised by Robert Bernstein – 5/07

Reviewed and Found Acceptable by Robert Bernstein – 5/08

Reviewed and Revised by Robert Bernstein – 1/09

Reviewed and Revised by Robert Bernstein – 5/09

Reviewed and Found Acceptable by Russell Penner – 5/10

Reviewed and Found Acceptable by Russell Penner – 5/11

Reviewed and Found Acceptable by Robert Mineo – 5/12

Reviewed and Found Acceptable by Robert Mineo – 5/13

Reviewed and Revised by Robert Mineo – 1/14

Reviewed and Found Acceptable by Robert Mineo – 5/14

Reviewed and Found Acceptable by Robert Mineo – 5/15

Reviewed and Found Acceptable by Robert Mineo – 5/16

Reviewed and Revised by Robert Mineo – 5/17

Reviewed and Found Acceptable by Robert Mineo – 5/18

Reviewed and Found Acceptable by Robert Mineo – 5/19

Reviewed and Found Acceptable by Robert Mineo – 5/20

Reviewed and Found Acceptable by Robert Mineo – 5/21

Course Outline

TITLE: Fundamentals of College Mathematics 2

CATALOG NO.: MA122

CREDIT HOURS: 4

LAB. HOURS: 0

PREREQUISITES: MA121 Fundamentals of College Mathematics 1

CATALOG

DESCRIPTION: This is the second of a two-course

sequence for students in programs that require mathematics through polynomial calculus. Topics include complex numbers, exponential and logarithmic functions, analytic geometry, limits, derivatives and integrals of polynomial functions, applications of the derivative, and area under a curve.

COURSE TEACHING GOALS FOR ALL TOPICS:

GOAL A: Use mathematical processes to acquire and convey

knowledge.

GOAL B: Systemically solve problems.

SUNY Learning Outcomes

1. The student will develop well reasoned arguments.

2. The student will identify, analyze, and evaluate arguments as they occur in their own and other’s work.

3. The student will demonstrate the ability to interpret and draw inferences from mathematical models such as formulas, graphs, tables, and schematics.

4. The student will demonstrate the ability to represent mathematical information symbolically, visually, numerically, and verbally.

5. The student will demonstrate the ability to employ quantitative methods such as arithmetic, algebra, geometry, or statistics to solve problems.

6. The student will demonstrate the ability to estimate and check mathematical results for reasonableness.

7. The student will demonstrate the ability to recognize the limits of mathematical and statistical methods.

TOPIC 1. COMPLEX NUMBERS

The rectangular, polar, and exponential forms of complex numbers are studied. Methods of adding, subtracting, multiplying, dividing, raising to powers, and finding roots of complex numbers are covered.

Topic Goal: To help students develop an understanding of complex numbers, and how to perform operations

on complex numbers.

Student Outcomes:

The student will:

1.1 Express complex numbers in the form a + bj

1.2 Simplify integral powers of j

1.3 Convert complex numbers from one form (rectangular, polar, or exponential) to another

1.4 Add, subtract, multiply, and divide complex numbers in rectangular form

5. Represent complex numbers graphically (as vectors in the complex plane)

1.6 Add and subtract complex numbers graphically

1.7 Determine the modulus and argument of a complex number

in polar form

1.8 Multiply and divide complex numbers in polar or exponential form

1.9 Raise complex numbers to powers and find roots of complex numbers in exponential form and in

polar form (De Moivre’s Theorem)

TOPIC 2. EXPONENTIAL AND LOGARITHMIC FUNCTIONS

The exponential and logarithmic functions are studied. The relationship between exp. and log. functions is discussed. Basic exponential and log. functions are graphed. Certain types of equations are graphed on linear, log., and semi-log paper. Computations are performed using the properties of logarithms.

Topic Goal: To help students develop an understanding of exponential functions, log. functions, and

their graphs.

Student Outcomes:

The student will:

2.1 Convert from exponential form to logarithmic form, and

vice-versa

2.2 Demonstrate an understanding of the properties of exponents and logarithms

2.3 Discuss the properties of basic exponential and

logarithmic functions and their graphs

4. Sketch the graphs of basic exponential and logarithmic functions

2.5 Demonstrate how to use a calculator to approximate natural and common logarithms

2.6 Solve exponential and logarithmic equations

2.7 Solve applied problems involving exponential or

logarithmic equations

8. Demonstrate an understanding of how to use the “change- of-base formula for logarithms”

2.9 Graph certain types of equations (involving powers or exponential expressions) on log-log or semilog graph paper

TOPIC 3. TOPICS IN PLANE ANALYTIC GEOMETRY

Topics include the distance formula, slopes, lines, circles, parabolas, ellipses, and hyperbolas. Emphasis should be placed on the definitions of these curves and their graphs. The student should be able to recognize or write the equations of the above curves in their “standard forms”.

Topic Goal: To help students develop an understanding of the equations and graphs of the conic sections.

Student Outcomes:

The student will:

3.1 Use the distance formula (for two points in the plane)

3.2 Demonstrate an understanding of the slope of a line, the inclination of a line, and the relationship between them

3.3 Determine whether two lines are parallel,

perpendicular, or neither

3.4 Graph any line

3.5 Find an equation of a line, given two pieces of information about the line

3.6 Demonstrate an understanding of the various forms of the equation of a line, and be able to convert to the slope-intercept form or the general form from another form

7. Define circle, parabola, ellipse, and hyperbola

8. Convert an equation of any of these four curves to standard form or general form (if necessary, by completing the square(s))

9. Given an equation of one of these four curves, identify key features and draw its graph

10. Given key features write an equation of any of the four curves

3.11 Demonstrate an understanding of translation of axes and its effect on equations

TOPIC 4. TOPICS IN DIFFERENTIAL CALCULUS

Included are the computation of limits, the “delta-process”, the definition of the derivative of a function, interpretations of the derivative, “rules” (properties) of differentiation, differentials, and applications of the derivative.

Topic Goal: To help students develop an understanding of limits, derivatives, how to differentiate functions, and how derivatives can be applied in various types of problems.

Student Outcomes:

The student will:

4.1 Demonstrate an understanding of limits and continuity

4.2 Calculate limits or show that a limit does not exist

4.3 Distinguish between average rate of change and instantaneous rate of change

4.4 Define the derivative of a function

5. Use the definition of the derivative to find derivatives of functions

6. Demonstrate how to use the rules of

differentiation to find derivatives of various functions

4.7 Use the chain rule for derivatives

4.8 Use implicit differentiation to find derivatives

4.9 Demonstrate an understanding of the slope of a tangent line, and its relation to the derivative

4.10 Find equations for the tangent line and the normal line to the graph of a given function at a given point

4.11 Provide complete solutions to related-rates problems

4.12 Use the 1st and 2nd derivatives to get information (such as where the function is increasing, where it’s

decreasing, concavity, relative extrema, inflection points) and then use this information to sketch the

graph of a function

4.13 Determine higher-order derivatives of select functions

4.14 Solve applied maximum/minimum problems

4.15 Demonstrate an understanding of differentials, and how they can be used in estimating “errors”

TOPIC 5. TOPICS IN INTEGRAL CALCULUS

Included are antiderivatives, the indefinite integral, the definite integral, and the area under a curve.

Topic Goal: To help students develop an understanding of indefinite and definite integrals, how to integrate certain types of functions, and how to evaluate definite integrals for those types of functions.

Student Outcomes:

The student will:

1. Discuss similarities and differences of the

antiderivative, indefinite integral, and definite

integral

2. Demonstrate an understanding of certain properties of

integrals

5.3 Evaluate the constant of integration, given sufficient information

5.4 Approximate (by partitioning an interval into subintervals of equal width) the area under a curve

5. Use integration to find the exact area under a curve

6. Determine numerical approximation to definite integrals using The Trapezoidal Rule or Simpson’s Rule

5.7 Evaluate definite integrals

Teaching Guide

Title: Fundamentals of College Mathematics 2

Catalog No.: MA122

Credit hours: 4

Lab Hours: 0

Prerequisite: MA121 Fundamentals of College Mathematics 1

Catalog

Description: This is the second of a two-course

sequence for students in programs that require mathematics through polynomial calculus. Topics include complex numbers, exponential and logarithmic functions, analytic geometry, limits, derivatives and integrals of polynomial functions, applications of the derivative, and area under a curve.

NOTE: Due to the widespread use of the calculator, at least a few test problems should be designed to insure that the student has learned the fundamental concepts involved and does not depend on the calculator to do the thinking as well as computing. For example, when finding sin(220o), the student should be required to show a sketch and indicate the reference angle. The student should show equations involved and all required back-up work.

Text: Basic Technical Mathematics with Calculus, 11th edition by Allyn J. Washington, Richard Evans, 2018; Pearson.

Chapter 12 Complex Numbers 5 hours

Section

12-l Basic Definitions

12-2 Basic Operations with Complex Numbers

12-3 Graphical Representation of Complex Numbers

12-4 Polar Form of a Complex Number

12-5 Exponential Form of a Complex Number

12-6 Products, Quotients, Powers and Roots of Complex Numbers (see note)

12-7 Omit

NOTE: The instructor should provide additional applications of complex numbers by evaluating determinants in which the entries are complex numbers. In addition, solving systems of equations with complex coefficients is appropriate.

Chapter 13 Exponential and Logarithmic Functions 6 hours

Section

13-1 Exponential Functions

13-2 Logarithmic Functions

13-3 Properties of Logarithms

13-4 Logarithms to the Base 10 (see note)

13-5 Natural Logarithms

13-6 Exponential and Logarithmic Equations

13-7 Graphs on Logarithmic and Semilogarithmic Paper

NOTE: The logarithmic function should be treated as a function, rather than a computational tool.

Chapter 21 Plane Analytic Geometry 9 hours

Section

21-1 Basic Definitions

21-2 The Straight Line

21-3 The Circle

21-4 The Parabola

21-5 The Ellipse

21-6 The Hyperbola

21-7 Translation of Axes

21-8 Omit

21-9 Omit

21-10 Omit

NOTE: The emphasis in Chapter 21 is to be placed on the use of and familiarization with the equations of the conics, rather than their derivations.

Chapter 23 The Derivative 12 hours

Section

23-1 Limits

23-2 The Slope of a Tangent to a Curve

23-3 The Derivative

23-4 The Derivative as an Instantaneous Rate of Change

23-5 Derivatives of Polynomials

23-6 Derivatives of Products and Quotients of Functions

23-7 The Derivative of a Power of a Function

23-8 Differentiation of Implicit Functions

23-9 Higher Derivatives

Chapter 24 Applications of the Derivative 10 hours

Section

24-1 Tangents and Normals

24-2 Optional (see below)

24-3 Omit

24-4 Related Rates

24-5 Using Derivatives in Curve Sketching

24-6 Omit

24-7 Applied Maximum and Minimum Problems

24-8 Differentials and Linear Approximations

Chapter 25 Integration 12 hours

Section

25-1 Antiderivatives

25-2 The Indefinite Integral

25-3 The Area Under a Curve

25-4 The Definite Integral

25-5 Optional (see below)

25-6 Optional (see below)

OPTIONAL TOPICS: Since the technology student has facility with the calculator, the following topics provide appropriate applications for the technology student. The instructor should strive to cover at least two of the following three topics.

24-2 Newton's Method for Solving Equations

25-5 Numerical Integration: The Trapezoidal Rule

25-6 Simpson's Rule

Examinations: The teaching guide allows 4 additional hours for the in-class assessment of student learning. A two hour comprehensive final examination will also be given.

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