Created:   Fri, 12 /21 / 2007 — Revised:   Sun, 01 / 06 / 2007

Spring 2008 Courses Taught by Doug Jones

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Hello, Fall Students!

Main Themes for the Spring 2008 Semester

As you can see from the panel of courses on the left, I am teaching 3 classes this Spring Semester — Ordinary Differential Equations (ODE), Honors Precalc/Trig, and Calc 3. But above and beyond the actual subject material of these courses, I want to talk to you about a "tent," so to speak, under which this thing called education takes place. ( "Learning takes place in the pavilion of the mind." ) This tent is held up by a framework of ideas, common to all, which, if properly investigated and developed, will help you to not only tie together and make sense of the subject material within your particular course, but also will provide you with a repertoire of general methods of learning and understanding which will be of life-long benefit to you in all your intellectual pursuits.

So, in addition to the actual subject matter covered in your course, I would like you to keep always in mind the following considerations:

  • Your learning style
    • What IS my LEARNING STYLE?  You must know, or at least have some idea of how you learn. How can you hope to improve your learning process if you don't even know what it is?
    • As a matter of fact, just WHAT IS "LEARNING," anyway?
    • Ask yourself the question: "Is this material being presented to me in 'my' strongest learning style?"
    • Ponder these questions: "After the class is over, after the lecture is over, what do I have to do to really learn the stuff we covered in class? That is, in what manner will I need to further study and review this material in order to gain an adequate understanding of it? How can I wrap my brain around this stuff?"
    • And finally: "How do I make the information 'stick' in my mind, so that I won't forget it?"
    • Here are some links that may help you: — PLEASE NOTE: In none of the REFERENCES below am I advocating, promoting, or advising you to purchase or sign-up for anything advertised on those pages. I have, however, found that each link does provide some relevant, free information.

  • The intellectual relationship between Teacher and Learner.
    • Consider the paradigm of the "Big T, Little l" transitioning into the "Little t, big L" —   where
      • T and t stand for the teacher's playing either a greater or lesser role in your education (I am the teacher), and
      • L and l stand for the learner's (that's you) playing a greater or lesser role in your education.
    • At any stage of your education, there is always the question to be asked and answered — "Which of us is the lead-agent in your education?"
    • We all start at the   T, l level (Big T, little l) with the teacher providing most of the "driving force" in your education,
    • And we should progress to the t, L level (Little t, big L) with you, the student-learner, providing most of the "driving force" attendant to your education.
    • The question you must constantly ask is "Where am I now on the road from  T, l  to  t, L?

  • The notion of   CRITICAL THINKING
    • Who is telling me this?
    • What is she or he trying to "sell" me?
    • Should I simply accept what's being said?
    • Are we examining all the possibilities?
    • What other data or facts are relevant to decision-making with respect to this issue?
    • Is the reasoning involved in this issue sound?
    • Are the supposed "facts" in this discussion true?
    • And, even if I accept the argument, is there a better way to state it?

  • On what level of thought is my current thinking taking place, vis-a-vis Bloom's Taxonomy?
    • Just as there are many levels on the learning path discussed above, there are many levels on the thinking path or understanding path, as it might be called.
    • These levels are described in BLOOM'S TAXONOMY. I invite you to click on this link, visit the site and study the "triangle."


The Courses

SCNS Number: MAP 2302

Prepared by: D. Jones
Date: February, 2003

Prerequisite: MAC 2312 with a grade of 'C' or better. Calculus II.
   Methods of solutions of ordinary differential equations, linear and non- linear, systems of linear differential equations, boundary value problems. methods include operators undetermined coefficients, variation of parameters. LaPlace transforms, and series solutions. There is also utilization of the CAS (Computer Algebra System) MAPLE. A graphing calculator is required. Lecture 3 hours.

    Since this is an introductory course, its primary goal is to teach the student how to solve a fairly representative assortment of types of differential equations which will be useful in her/his future course-work and in his/her future profession. A secondary goal, and one which takes longer to achieve, is that of helping the student to see how to construct mathematical models which are fairly faithful translations of specific physical problems, and which, naturally, result in differential equations, or system of differential equations. This secondary goal is really the more important of the two, but it cannot be accomplished within the framework of MAP 2302; it can at most be begun here with its development continued in subsequent courses.

At the end of this course the student should be able to:
  • Solve differential equations employing the techniques of
    • Separation of Variables
    • Homogeneous Equations
    • Exact Equations
    • Linear First-Order Equations
  • Solve certain Initial Value Problems and Boundary Value Problems
  • Identify Linearly Independent Solutions
  • Construct a Second Solution from a known Solution
  • Utilize the Method of Undetermined Coefficients
  • Manipulate Differential Operators
  • Utilize the Method of Variation of Parameters
  • Set Up and Solve certain Harmonic Motion problems
  • Solve Systems of Linear Differential Equations
  • Utilize the Laplace Transform in the solution of certain Differential Equations

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Honors Precalculus Algebra and Trig // MAC 2147 - 55091

Course Description: (5 Semester Hours) OD
Prerequisites: Honors Authorization. This course serves as a prerequisite for MAC2311, Calculus with Analytic Geometry I.
Topics include: properties and graphs of polynomial, rational, exponential, and logarithmic functions; solutions of higher degree polynomial equations; solutions of systems of equations using matrices and determinants; sequences and series; the binomial theorem; an introduction to conic sections; trigonometric functions of angles and real numbers, along with their graphs and inverses; solutions of triangles and other applications; trigonometric identities; conditional trigonometric equations; complex numbers in trigonometric form and DeMoivre’s Theorem; vectors and polar coordinates; and piecewise defined functions. A graphing calculator is required. Please check with your instructor for the most appropriate one for the course. This course may not be taken for credit by any student who already has a grade in MAC2140 or in MAC 2114. Lecture, 5 hours college credit.

Objective of the Course:
To combine Precalculus Algebra and Trigonometry into one 5 semester-hour course in order to allow students with adequate mathematics background to complete the prerequisites for Calculus I in one course instead of two. This would possibly reduce the number of excess hours a student who is majoring in science, engineering, or mathematics may need to take.

Performance Goals of the Course: Upon the completion of this course, the student should be able to:
  1. Use and define trigonometric terminology concerning standard position of any angle, positive or negative.
  2. Know the radian measure for angles that are multiples of 1/4 and 1/6 of the circumference of a unit circle.
  3. Write all angles coterminal with a given angle; find the measure of the least positive angle coterminal with any given angle.
  4. Convert any angle given in degrees to radians, and in radians to degrees.
  5. Find the length of an arc intercepted by a given central angle on a circle of given radius.
  6. Find the linear and angular speed of an object in circular motion.
  7. Know the coordinates of the points associated with arc lengths of multiples of π/6 and π/4.
  8. Give the trigonometric function values of multiples of π/4 and π/6 without using tables or a calculator.
  9. Define the six trigonometric functions by use of the unit circle.
  10. Find the function values of any angle measure or any real number using a hand calculator.
  11. Define the trigonometric functions in terms of the lengths of the sides of a right triangle.
  12. State the reciprocal trigonometric functions.
  13. State and prove the Pythagorean identities.
  14. Given one function value of an acute angle, find the other five function values of the angle.
  15. Know what is meant by "solving" a right triangle.
  16. Solve right triangles and applied problems involving them, and gain facility in the use of graphing calculators in solving trigonometry problems.
  17. Define the trigonometric functions in terms of angles in standard position using x, y, and r.
  18. Give the domain and range of the six trigonometric functions and tell in which quadrants the functions values are positive or negative.
  19. Given one function value of an angle and some additional information ( e.g., the quadrant in which it terminates or that some function value is negative ), find the other five function values of the angle.
  20. Sketch the graphs of y = a sin (bx - c) and y = a cos (bx - c), and determine amplitude, period, and phase shift; sketch the graph of y = a tan (bx - c) and determine period and phase shift.
  21. Sketch the graphs of the six trigonometric functions and give their period, domain, and range.
  22. Use addition of ordinates or the graphing calculator to graph the sum of two functions. (optional)
  23. Define and use the inverse functions of the trigonometric functions.
  24. Give the domain and range of the arc sin, arc cos, and arc tan functions; graph these three inverse functions; give the domain and range of the arc sec, arc csc, and arc cot functions; find function values.
  25. Simplify expressions involving compositions of functions and inverses [ e.g., cos (arcsin (a/b))].
  26. Use the basic identities to verify other identities.
  27. Solve trigonometric equations, including those involving half-angles and multiple angles.
  28. State all and prove at least some of the sum and difference formulas for sin, cos, and tan.
  29. Use the formulas in #28 to find function values and to simplify selected trigonometric expressions.
  30. State and use the cofunction identities for sin, cos, and tan.
  31. State all and prove at least some of the half-angle and double angle formulas for sin, cos, and tan.
  32. Use the half-angle and double-angle formulas for sin, cos, and tan.
  33. Use the double-angle (half-angle) identities to find function values of twice (half) an angle when the value of one of its functions is given.
  34. State the identities for trigonometric functions of (-?).
  35. State the Law of Sines and the Law of Cosines.
  36. Use the Law of Sines and the Law of Cosines in applied problems.
  37. Find the area of a triangle using the formula Area = (1/2)ab sine C.
  38. Find the area of a triangle using Heron's formula. (optional)
  39. Graph and use vectors given their magnitudes and direction angles.
  40. Work with two dimentional vectors as ordered pairs of real numbers.
  41. Find the resultant of two vectors both algebraically and geometrically.
  42. Identify the scalar multiple of a vector.
  43. State all and prove some of the properties of vectors under the operations of vector addition and multiplication by a scalar.
  44. Solve applied problems involving vector sums.
  45. Resolve a vector into a linear combination of the unit vectors i and j.
  46. Find the dot product of two vectors.
  47. Find the angle between two vectors given in component form.
  48. Solve applications such as work problems or orthogonal projections. (optional)
  49. Define and graph complex numbers.
  50. Change a complex number given in trigonometric form to standard form ( a + bi ), and vice-versa.
  51. State and use the theorems for product and quotients of complex numbers in trigonometric form.
  52. State DeMoivre's Theorm and use it to raise a complex number to a positive integer power.
  53. Find the n nth roots of a complex number.
  54. Find all solutions to polynomial equations of the form xn + c = 0.
  55. Change a point written in rectangular coordinates to polar coordinates, and vice-versa.
  56. Transform an equation in polar form to rectangular form and vice-versa.
  57. Know the definition of polynomial functions and the characteristics of their graphs, with emphasis on local behavior and end behavior.
  58. Use a graphing calculator to approximate the x-intercepts of the graph of a polynomial function.
  59. Use long division on polynomials to find the quotient and the remainder, and correctly write the polynomial in the form P(x) = Q(x) D(x) + R(x).
  60. Use, and interpret the results of, synthetic division when dividing a polynomial by x c, c is a real number.
  61. State, prove, and use the Remainder Theorem and the Factor Theorem.
  62. List all possible rational zeros of a polynomial with integer coefficients. Identify multiplicity of roots in factored polynomials.
  63. Use Descartes’ Rule of Signs to determine the possible number of positive real zeros and negative real zeros of a polynomial.
  64. Find the smallest positive integer upper bound and the largest negative integer lower bound for the real roots of a polynomial equation, using synthetic division.
  65. Perform arithmetic operations on complex numbers.
  66. Graph a complex number and find its modulus and its conjugate.
  67. State the Fundamental Theorem of Algebra.
  68. Use the Conjugate Roots Theorem and the Complete Factorization Theorem to find a polynomial equation having a given rational number and a given nonreal number as two of its roots.
  69. Explain why the Conjugate Roots Theorem guarantees that a polynomial of odd degree and real coefficients has at least one real zero.
  70. Find all the exact roots of a polynomial equation by using a combination of the theorems and techniques presented in the text.
  71. Define a rational function and use algebraic methods to find its x- and y-intercepts, and any vertical, horizontal, and slant asymptotes of its graph.
  72. Sketch the graph of a rational function either by algebraic analysis or by using a graphing calculator.
  73. Identify the properties, domain, and range of exponential functions and sketch their graphs.
  74. Identify the natural exponential function and its connection with compound interest.
  75. Solve applied problems involving population growth, radioactive decay, and compound interest.
  76. Recognize logarithmic functions as inverses of exponential functions.
  77. State the domain and range of a logarithmic function and draw its graph.
  78. Prove the properties of logarithms and use them correctly.
  79. Solve exponential and logarithmic equations without calculators (when possible) and with calculators (when necessary).
  80. Know and use the change-of-base formula for logarithms.
  81. Using algebraic methods, solve exponential and logarithmic equations relating to applied problems, and approximate exact solutions by using a calculator.
  82. Solve systems of linear and nonlinear equations by the substitution and elimination methods and by the use of a graphing calculator.
  83. Compute the determinants of 2x2 and 3x3 matrices by expansion by cofactors and by row and column transformations.
  84. Use a graphing calculator to compute the determinant of a small-dimensional square matrix (2x2 through 5x5).
  85. Know and use Cramer’s Rule to solve “square” systems of linear equations, with and without a graphing calculator.
  86. By algebraic methods, find the partial fraction decomposition of a rational function, whether the denominator is a product of distinct or repeated linear or irreducible quadratic factors.
  87. State the locus definitions of a parabola, an ellipse, and a hyperbola, and derive equations for these conic sections when the axes of symmetry are horizontal or vertical.
  88. Given the equation of a parabola, find its vertex, its focus, its directrix, and its axis of symmetry, and sketch the graph.
  89. Find the equation of a parabola, given a minimum amount of information about its vertex, focus, directrix, focal width, and axis of symmetry.
  90. Given the equation of an ellipse, find its center, foci, major and minor axes, intercepts, and eccentricity, and sketch the graph; conversely, given minimal information about these features of the graph of an ellipse, find the equation.
  91. Given the equation of a hyperbola, find the asymptotes and intercepts of the graph and draw the graph.
  92. Recognize and sketch the graphs of conic sections whose centers have been shifted horizontally or vertically.
  93. Know the notation and terminology associated with sequences and series.
  94. List the terms of a sequence defined either explicitly or recursively.
  95. Expand a sum given in sigma notation.
  96. Write the nth term of a sequence, given the first several terms of the sequence.
  97. Identify arithmetic sequences by finding the common difference.
  98. Determine any of the following for an arithmetic sequence, given any of the others: two specific terms, one term and the common difference, the sum of a finite number of terms.
  99. Identify geometric sequences by finding the common ratio.
  100. Determine any of the following for a geometric sequence, given any of the others: two specific terms, one term and the common ratio, the sum of a finite number of terms.
  101. Find the sum of an infinite geometric sequence with a common ratio of absolute value less than one.
  102. Calculate expressions involving factorial notation.
  103. State the Binomial Theorem, and use it to expand small (2nd through 6th) powers of binomials.
  104. Find the rth term of the expansion of a power of a binomial.
  105. Understand the definition of piecewise defined functions and be able to graph such functions by hand.
We are also going to "do" some logic!

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Calculus 3 with Analytic Geometry //
MAC 2313 - 55105
MAC2313 Calculus with Analytic Geometry III (4) FA SP
  • Prerequisite: MAC 2312 with a grade of C or better.
  • Topics include
    • vectors,
    • equations of planes and lines in space,
    • vector-valued functions   —   including
      • the unit tangent and unit normal vectors,
      • velocity and acceleration of objects in space,
      • curvature.
    • multivariable functions,
    • the differential and integral calculus of multivariable functions, and
    • line and surface integrals  —   including
      • Green’s Theorem,
      • the Divergence Theorem,
      • Stokes’s Theorem.
There is also use of the CAS (Computer Algebra System) MAPLE.
This is a WEB assisted course. A graphing calculator is required. Check with your instructor for the most appropriate one. Lecture 5 hours. Special fee.

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Class Times and Places  —  Where to Find Me.
Ref # Course or Office Time Days Room
55107 ODE // MAP 2302 9:05-9:55MWF AC 112
SM 243
55091 PRECALC/TRIG // MAC 2147 11:15-12:40 MWF FPA 121
55105 CALC 3 // MAC 23131:25-2:15MTWRF AC 112
 OFFICE 2:15-5:30 MTWR SM 243
 OFFICE OBA ( Or By Appointment )

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