Mt/CS 401 Seminar

Spring 2004

PRELIMINARY Study Guide: Comprehensive Exam & Benchmark Test

January, 2004

Some topics to be covered on both the comprehensive exam and the benchmark test are listed here. Prior to the Exam, seminar participants will be invited to reshape this Study Guide to reflect those chapters of The Turing Omnibus which have been covered in class presentations.

The questions on the comprehensive exam are challenging problems which give the student an opportunity to demonstrate an ability to solve non-routine or ill-posed problems requiring creative thinking. Sample questions from previous exams are available.

The benchmark test gives the student an opportunity to demonstrate mastery of routine computations, and basic competence with tools of technology.

Basic Algebra and Analytic Geometry

Algebra and geometry are basic tools for solving problems that appear in many parts of everyday life. You should be able to:

evaluate a given function, f(x), at particular values (such as at x = 5, x = x+h, and x = g(x));

find the slope of a straight line from its graph or its algebraic formula;

find the product of a given pair of polynomial expressions;

simplify an algebraic expression;

solve linear, quadratic, and cubic equations (by factoring, using the quadratic formula, and/or using graphical methods);

sketch graphs of polynomials, trigonometric functions;

work with the equations for curves in the plane (including the conic sections);

interpret information which is presented in charts and graphs.

You should know and be able to use:

formulas for area of a square, rectangle, triangle, or circle;

formulas for volume of a rectangular box, or ordinary cylinder;

formulas for perimeter and circumference of simple geometric figures;

basic mathematics skills to estimate distanes and areas on ordinary maps.

You should be able to work with rectangular coordinates and polar coordinates in the plane.

Trigonometry

Trigonometry is useful for solving problems involving triangles. You should be familiar with basic right-triangle trigonometry:

The Trigonometric Relationships:

You should

• know and be able to use the definitions and relationships among thesix trigonometric functions in solving problems,
• be able to sketch graphs of the trigonometric functions.

The Unit Circle, the two Standard Triangles, and the Pythagorean Theorem:

You should be able to:

• construct (using diagrams and knowledge of the unit circle, the standard triangles, and symmetry) the values of the trigonometric relationships of certain angles;
• convert between degree and radian measure of angles;
• verify trigonometric relationships using the unit circle;
• use the Pythagorean theorem and the trigonometric versions of the Pythagorean theorem to solve problems involving triangles.

Logarithms & Exponents

Logarithms and exponents allow us to work more easily with very large and very small numbers. The internal computer representation of numbers uses logs and exponents. You should be able to:

do basic calculations involving logarithms and exponents;

work with problem situations involving logarithms (both natural logs and logs base 10), the number e, and the exponential function, exp(x);

do computations involving base 2 (binary), base 8 (octal), and base 16 (hexadecimal) numbers;

explain how binary, octal and hexademcimal representations are use by the computer;

sketch graphs of exponential and logarithmic and exponential functions;

solve application problems involving exponential or logarithmic growth or decay;

use exponential and logarithmic functions in problem situations.

Limits

You should be able to:

work with the concept of limit of a function;

evaluate expressions involving limits at infinity, and limits at a finite point;

work with real-world applications involving infinite limits, removable discontinuities, and jump discontinuities.

Boolean Algebra and Logical Reasoning

Boolean algebra provides the mathematical foundations for Computer Science. You should be able to:

set up a truth table or Venn diagram to represent a given statement;

use truth tables and Venn diagrams to determine whether two given statements are logically equivalent;

express Boolean functions in various ways using AND, OR, NOT, NAND, and NOR gates;

develop a Boolean function to represent a given task (such as lighting up a particular LED-segment given specified inputs;

draw the circuit diagram for a digital logic circuit to represent a given Boolean function;

use sets and Boolean algebra to reason about a problem situation;

use the rules of logic to interpret or simplify an expression or argument;

use the rules of logic to determine whether an argument or computer algorithm is valid (correct) or invalid (not correct);

negate an expression which has existential and/or universal quantifiers;

apply logical reasoning to structured programming and the development of computer-based algorithms.

Graph and Trees

You should be able to:

use appropriate terminology and representations for vertex-edge graphs and trees;

identify problem situations and data structures that can be represented using graphs and trees;

use graphs and/or trees to represent problem situations or computer algorithms;

use appropriate algorithms to solve counting, searching, and sorting applications problems;

use a recursive algorithm to solve certain data structures problems (such as traversing a tree), and/or us mathematical induction as a proof strategy for problems defined over as set like the set of positive integers.

Difference Quotients, Derivatives, and Applications of the Derivative

You should be able to

set up and simplify a difference quotient for a given function;

take the limit of a difference quotient as h (or delta-x) goes to 0, thus proving one of the basic derivative formulas;

calculate derivatives for polynomials, and for the six trigonometric functions;

use the product rule, the quotient rule, and the chain rule to calculate derivatives for simple combinations of functions;

use the concept of a difference quotient to solve position-velocity-acceleration problems;

use a difference quotient or derivative to solve problems involving rates of change and related rates;

use a difference quotient or derivative to solve maximum/minimum problems;

use the derivative to find slope and concavity of curves in the plane;

set up a difference quotient in an Excel spreadsheet, and use this to calculuate an approximate derivative.

Riemann Sums, Antiderivatives, and Definite Integrals

You should be able to:

set up a Riemann sum for a given function over a specified interval;

use a Riemann sum to estimate the total accumulation of a quantity -- such as total distance traveled or total profit;

use the power rule to integrate an expression of the form x^n, both for n /= 1 and for n = 1;

calculate antiderivatives and definite integrals for polynomials, and for the derivatives of the six trigonometric functions;

use simple substitutions to calculate an antiderivative (in other words, you should be able to find the antiderivative when the derivative was calculated using the chain rule);

use the method of partial fractions to simplify an expression so that it is easier to calculate an antiderivative;

use the method of integration by parts to calculate an antiderivative (that is, you should be able to find the antiderivative when the derivative was calculated using the product rule);

evaluate the definite integral of an expression of the form e^u du;

set up and evaluate a definite integral to solve certain application problems, such as to calculate of area, volume, arc length, surface area, total distance traveled, total profit or income, and so on;

apply the concept of an accumulated sum (i.e., a Riemann sum) to solve acceleration-velocity-position problems;

use definite integrals in applications involving business problems, problem situation in the natural sciences, and/or other real-life scenarios;

set up an Excel spreadsheet to evaluate an approximate value for a given definite integral;

use a definite integral to calculate the average of a continuous function;

sketch graphs showing the relationships among a given function, its derivative, and its antiderivative.

Interest Theory

Real life mathematics has applications to calculating the real cost of using a credit card, purchasing life insurance, and taking out a home mortgage loan. You should be able to use a spreadsheet and the concept of Riemann sums to calculate

the total interest you will be paying on a mortgage loan;

the cost of buying on credit (particularly if you do not pay off the full amount of the loan each month);

compounding interest on the outstanding balance of a loan.

Additional Topics selected from The Turing Omnibus

Seminar participants are expected to choose interesting chapters from The Turing Omnibus. These topics will also be represented on the comprehensive exam.

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Return to Sr. Barbara E. Reynolds Home Page.
Return to course list for 2003 -- 2004.
Return to Mt/CS 401: Assignments Assignments.
Go to Mt/CS 401: Seminar Syllabus.
Eventually the Revised Syllabus will be posted here, too.

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The easiest way to contact me is to send an email message to Sr. Barbara E. Reynolds.

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This page was updated on January 17, 2004.