CS442 - Additional Assignments

Q&A

		MT360A - Calculus

Class Outline (taken from Regis outline for class)

General Notes

Calculus is dependent on all of the mathematics you have been able to perform before this class.  Calculus requires a slightly different mind-set that the mathematics you have had before, but if you use the logic you have gained from the other mathematics classes, you should be able to do just fine.  Do not fight thinking in a different way, just enjoy the class and "go with the flow."  ;-)  You will do well.

Questions and Answers

As I receive questions, I will create a file that will contain answers to these questions.

Homework Status

You can check your homework here (eventually).

 Learning Topic #1:  Review: Functions and Trigonometry

 This topic includes a review of functions and concepts from college trigonometry.

 Topic Outcomes

 After completing this topic, students will be able to:

• Find the slope of a line, given two points.

• Find the slope of a line, given a linear function.

• Simplify rational functions.

• Given two functions, use function composition to create new functions.

• Find trigonometric ratios for given right triangles.

• Evaluate trigonometric ratios for angles that are not in the first quadrant.

• Manipulate trigonometric functions.

• Apply basic trigonometric identities.

• Use a calculator to evaluate and graph the following types of functions:

• The facilitator will show examples of evaluating limits graphically, algebraically, and intuitively.

• Polynomial functions

• Rational functions

• Exponential functions

• Trigonometric functions - Solve trigonometric equations.

 Discussion Questions

• What are several types of change that are important to understand?

• What are the positive and negative aspects of representing functions graphically?

• What is the difference between discrete and continuous models and when do you think this makes a difference?

•  Describe a procedure that could be used to solve a right triangle.

Suggested Classroom Activities

• The facilitator and the students will discuss the idea of a function and how it relates to its inverse (graphically and analytically).

• The class will evaluate functions.

• The class will combine functions by addition, subtraction, multiplication, division, and composition.

• The facilitator will demonstrate several useful features of a TI 83 Plus in graphing functions.

• The class will use calculators to analyze functions numerically and graphically.

• The class will review the six trigonometric functions.

• The class will work on applications of right triangle.

• The facilitator will derive basic trigonometric identities and their applications.

• The class will prove some basic trigonometric identities and apply the same in solving trigonometric equations.

Suggested Enrichment Activities

•  Investigate rational functions using a graphing calculator, paying particular attention to local extreme values, asymptotes, and other discontinuities.

Learning Topic #2:  Limits and Continuity

This topic includes the historical rationale for the concept of a limit, and explores limits numerically, graphically and formally.  This definition is then applied to explain the concept of continuity.

 Topic Outcomes

 Upon completion of this topic, students will be able to:

• Provide a definition of a limit.

• Interpret the existence of a limit graphically.

• State algebraic and graphical examples of the existence or non-existence of a limit.

• Find limits numerically, graphically, and algebraically.

• Use the definition of continuity to determine the intervals where a function is continuous.

Discussion Questions

• Graphically, how would you recognize the points or intervals where limits do not exist?

• What algebraic techniques help us locate points of discontinuity?

• Which functions are continuous everywhere?

Suggested Classroom Activities

• The facilitator will lead a discussion of the concept of a limit and some applications.

• The facilitator will show examples of evaluating limits graphically, algebraically, and intuitively

• The facilitator will graphically illustrate the difference between continuous and discontinuous functions

• The class will analyze the definition of continuity at a point and at an interval. The class will then use this definition to determine continuity of some algebraic and trigonometric functions.

Suggested Enrichment Activities

•  Explore how graphing calculators display points of discontinuity.

Learning Topic #3:    Derivatives

This topic includes introduction of the derivative as the limit of a difference quotient, statement of the formal definition of the derivative.

Topic Outcomes

Upon completion of this topic, students will be able to:

•  Find the slope of the secant line for two points from the graph of a function.

• Express the slope of a secant line as a difference quotient.

• Find the equation of the secant line for two points from the graph of a function.

• Find the slope of the tangent line at one point of the graph of a function.

•  Find the equation of the tangent line for one point of the graph of a function.

• Find the instantaneous rate of change of a function for a given x-value.

•  Determine graphically whether a function is locally linear.

• State the definition of instantaneous rate of change of a function for a given x-value.

• State the limit definition for the derivative function.

• Use the limit definition to find the derivative function for polynomials and radical functions.

• Estimate the graph of a derivative function from a given function.

 Discussion Questions

• Why is the concept of a limit necessary to understand derivatives?

• What is the difference between instantaneous and average velocity?

• What is a secant line?

• Explain the relationship between the slope of a tangent line and derivative.

Suggested Classroom Activities

• The facilitator and the class will develop the connection between limits, slopes of lines, velocity, and derivative.

• The class will evaluate limit of the difference quotient and relate this to the value of the derivative.

• The class will analyze graphically analyze points where the derivative is not defined.

• The facilitator will demonstrate methods of determining points of non-differentiability.

Suggested Enrichment Activities

• Experiment with the derivative and tangent functions on the graphing calculator.

Learning Topic #4: Techniques of Differentiation

This topic focuses on the algebraic methods of differentiation

Topic Outcomes:

Upon completion of this topic, students will be able to:

• Apply the appropriate derivative formula.

• Identify problems that require the use of the Chain Rule.

• Apply the Chain Rule correctly.

• Find the derivative of trigonometric functions.

• Find the derivative of logarithmic functions.

• Apply the method of logarithmic differentiation.

• Find the derivative of exponential functions.

• Find higher order derivatives.

Discussion Questions

• When is the power rule used?

• Can quotients be expressed as products?

• What is the product rule?

• What is the quotient rule?

• How is the derivative of the sum/difference of functions evaluated?

• What is logarithmic differentiation?

• How do you separate a composition of functions into its component functions?

• How is the product rule different (in application and form) from the chain rule?

Suggested Classroom Activities

• The facilitator will demonstrate the application of the product rule, quotient rule, and sum/difference rule to evaluate the derivative of a function.

• The facilitator will demonstrate the application of the derivative formulas for logarithmic and exponential functions.

• The facilitator will present an example of the application of the chain rule.

• The class will discuss how the values of the derivative will apply to velocity, acceleration, and slope of tangent lines.

• The class will work on a set of problems involving finding derivatives.

• The class will discuss several derivative problems and determine which method is appropriate in solving the problems.

Suggested Enrichment Activities

• The class will use a graphing calculator to find values of derivatives at a point.

Learning Topic #5: More Derivatives & Applications

This topic includes evaluating the derivative of functions expressed implicitly, evaluating derivatives of inverse trigonometric functions, and solving problems on applications of derivatives.

Topic Outcomes

Upon completion of this topic, students will be able to:

• Derive the formula for the derivative of inverse trigonometric functions.

• Apply the appropriate derivative formula for inverse trigonometric functions together with previously established rules and formulas to evaluate the derivative of a function.

• Apply the method of implicit differentiation to evaluate the derivative of a function.

• Solve related rate problems.

• Determine if a function is increasing or decreasing over an interval.

• Determine if the graph of a function is concave up or concave down over an interval.

• Determine relative extrema.

Discussion Questions

• How is the derivative of the inverse of a trigonometric function derived?

• What information about a function is derived from the first derivative?

• What information about a function is derived from the second derivative?

• When is implicit differentiation applied?

• What are related rate problems?

• What is a relative extremum?

• Describe a procedure that will determine the relative minimum and relative maximum of a function.

Suggested Classroom Activities

• The facilitator will explain the interpretation of rates of change as a derivative with respect to time.

• The class will solve related rate problems.

• The facilitator will demonstrate how derivatives can be used to approximate nonlinear functions by simpler linear functions.

• The facilitator will illustrate the process of implicit differentiation.

• The class will discuss several derivative problems and determine which method is appropriate in solving the problems.

• The class will practice using the method of implicit differentiation.

• The facilitator will illustrate the process of analyzing a function by using its first and second derivative.

• The class will work on problems that will determine the relative extrema of a function.

Suggested Enrichment Activities

·      Use CAS to verify solutions to chain rule and implicit differentiation problems.

Learning Topic #6: Applications of Differentiation

This topic includes absolute extreme values of functions on a closed interval and its applications, rectilinear motion, differential, and Newton’s method.

Topic Outcomes:

After completing this topic, the student should be able to: 

• Find the absolute extremes for a function on a closed interval graphically.

• Find the absolute extremes for a function on a closed interval algebraically.

• Apply a four-step process to set up and solve applied optimization problems.

• Given a position function, produce a time vs. position graph.

• Given a position function, determine a velocity function and an acceleration function.

• Given a position function, find the extreme positions for a specified time interval.

•  Define the differential.

• Use differentials to estimate the value of a function near a known value.

• Iterate a function.

• Use a formula to determine the intersection of a tangent line to a function with the x-axis.

• Use Newton’s method to estimate the root of a function.

• Show graphically a situation where Newton’s method would fail.

Discussion Questions

• What is a differential?

• What is error propagation?

• What are some applications of differentials?

• Explain the concept behind Newton’s Method.

• How is the velocity function determined from the position function?

• Explain how the absolute extrema is determined.

Suggested Classroom Activities

• The facilitator will illustrate the process of finding extreme values of a function.

• The class will solve a problem on finding the maximum and minimum value of a function.

• The facilitator will demonstrate the use of derivatives to solve optimization problems.

• The facilitator will introduce the concepts of related rates and differentials.

• The class will apply the concepts above to related rate problems.

• The class will apply differentials to solving problems involving error propagation.

• The class will apply Newton’s method to solve an equation.

• The class will use derivatives to solve rectilinear motion problems.

Learning Topic #7:  Antidifferentiation

This topic includes antiderivatives and the idea of area as a limit of a summation.

Topic Outcomes:

 After completing this topic, the student should be able to: 

• Sketch a graphical illustration of the mean value theorem.

• Find a family of antiderivatives for a power.

• Find a family of antiderivatives for sin (x), cos (x), sec ² (x), and sec(x) tan (x).

• Find a family of antiderivatives for exponential (base e) function.

• Solve initial value problems involving functions with simple antiderivatives.

• Given a linear, quadratic, or sinusoidal velocity function and an initial position, find a position function.

• Find antiderivatives of functions that are constant multiples, sums, or differences of other functions with known antiderivatives.

• Represent sums using sigma notation.

• Manipulate sums expressed in sigma notation algebraically.

• Evaluate sums using a calculator.

• Estimate area between the graph of a function and the x-axis using rectangles.

• State the definition of a Riemann sum.

• Define areas as limits of Riemann sums.

 Discussion Questions

• What is an antiderivative?

• Explain how the velocity function relate to the velocity function.

• What is a sigma notation?

• What is the syntax for using the calculator to evaluate the sum of finite number of terms?

Suggested Classroom Activities

• The facilitator explains the Mean Value Theorem.

• Students work on problems involving the Mean Value Theorem.

• The facilitator leads a review of the application of the derivative formulas and use the results to find an antiderivative for each function.

• The facilitator leads a discussion of the sigma notation and the symbols used.

• The students work on evaluating sums manually and with the use of calculators.

• The facilitator leads activities on approximating areas using the left-endpoint, right-endpoint, and midpoint methods.

• The facilitator explains the relationship between the approximate area under a curve and the Riemann Sum.

Learning Topic #8:  Definite Integrals

This topic includes evaluating definite integrals using the Fundamental Theorem of Calculus, the Second Fundamental Theorem of Calculus, and the Mean Value Theorem.

Topic Outcomes:

After completing this topic, the student should be able to:

• Relate Riemann Sum to Definite Integral

• Evaluate definite integrals using formulas from geometry.

• Evaluate definite integrals using the Fundamental Theorem of Calculus Part I.

• Use the TI-83 calculator to find an approximate value of a definite integral.

• Evaluate the derivative of an integral function by using the Fundamental Theorem of Calculus Part II.

• Evaluate definite integrals by u-substitution.

• Demonstrate the relationship between the area under a curve and the definite integral.

Discussion Questions

• How does the Riemann Sum relate to the definite integral?

• How is a definite integral evaluated?

• How does the definite integral relate to the area under a curve?

Suggested Classroom Activities

• The facilitator shows an example on how the definite integral can be evaluated using an area formula from geometry.

• The facilitator explains the First Fundamental Theorem of Calculus.

• The class works on problems applying the First Fundamental Theorem of Calculus.

• The facilitator demonstrate evaluating the application of the Second Fundamental Theorem of Calculus.

• The class works on evaluating definite integrals by u-substitution.

Resources

 Textbooks and Study Guides

• Ayres, Frank and Elliott Mendleson; Schaum’s Outline of Theory and Problems of Differential and Integral Calculus, McGraw-Hill 1990; ISBN: 0-07-002662-9

• Passow, Eli; Schaum’s Outline of Theory and Problems of Understanding Calculus Concepts, McGraw-Hill 1996; ISBN: 0-07-048738-3

• Thompson, Silvanus P., Calculus Made Easy, 3^rd Edition, St. Martins Press, 1988, ISBN: 0-312-11410-9

 Software

Student versions of each of the following packages are available through the bookstore.  The academic price may apply to students.  One of these packages may be useful for learning and experimenting, but check with your instructor first to determine if one may be preferable over the others.  In each case, you will have to learn some programming syntax, but it will facilitate the learning process and will help with the application of the calculus to later classes and work situations.

·        Mathcad

·        Matlab

·        Mathematica

Internet

The following sites contain tutoring information and may be searched as an aid to learning.

• http://www.hofstra.edu/~matscw/realworld.html - Online supplementary information.

• http://www.ccwf.cc.utexas.edu/~egumtow/calculus/index.html - “Help With Calculus”

• http://www.netsrq.com/~hahn/notes.html - “Calculus Tutor Notes and Notation”

• http://www-cm.math.uiuc.edu// - “Calculus and Mathematica”

This site was last updated 07/03/06

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