The Department of Mathematics, Applied Mathematics and Statistics offers proficiency exams for Math 121, 122, 125, 126, 223 and 224. These exams are given twice each year, shortly before classes start in the fall and spring semesters, on a schedule determined by the Office of Undergraduate Studies (*not by the Department of **Mathematics, Applied Mathematics and Statistics)*. In recent years, the exams have been offered on the Friday before classes begin. The next date the exams will be given is January 11, 2019. **Proficiency exams are NOT given on request at other times! Students are expected to research these scheduling issues BEFORE making travel plans.**

It is not possible to complete two proficiency exams in a single three hour period. Students who wish to test out of two math courses will have to do this by taking separate exams in August and in January. Students are allowed only one attempt at a proficiency exam. If you do not pass, you will need to take the course.

To be certain that an exam is available for you, you** must **register for a proficiency exam in advance (*preferably at least 24 hours in advance*) by contacting Prof. Butler. (*If no one requests an exam for a particular course, an exam for that course may not be available. If the number of students who show up for an exam exceeds the number of available exams, those who registered in advance will be given priority*.) Email registration to chris@case.edu Your message should include the following information:

- The email address to which your results should be sent, preferably an official CWRU email address;
- Which of the six possible exams you plan to take. If you aren’t certain about this, consult either Prof. Butler or your academic advisor.
- Your class (
*first year, second year, junior or senior*) and your major(*s*) or intended major(*s*).

A description of the material covered by each exam is provided below. There are not sample exams available. The point of these proficiency exams is to provide students who believe they know the material an opportunity to demonstrate that they should be excused from the corresponding course. The Math proficiency exams are similar to the final exams given in each course; problems are modeled on those of past course exams and on homework problems from texts used in these courses. To receive proficiency credit, your performance on the exam must be equivalent to or better than a C grade in the course (*in the neighborhood of 70%*). Proficiency credit is indicated on your transcript by the symbols PR; no letter grade is given and proficiency credit does not count as part of your GPA. There is no penalty, and no permanent university record, if you do not pass a proficiency exam.

You are not allowed to use your own formula “cheat” sheet. You need to supply your own calculator, pencils and erasers. There are no restrictions on the kind of calculator you use for the exam, but computers, smartphones and other devices with communication capabilities may not be used during the test. You may use the calculator to do routine arithmetic computations only. All details of differentiation and integration must be shown. You may not use your calculator to store notes or formulas.

You might consider the following factors when deciding whether or not to take a proficiency exam:

- You may not learn very much in a class if you already know the material well enough to pass the proficiency exam. Passing the course by proficiency will free up your schedule so that you may take another course in its place.
- You only get one chance at a proficiency exam, if you do not pass the exam, you will need to take the class.
- Many students may start at the second or third course of a sequence after skipping earlier courses by proficiency. Some of these students have difficulty adjusting to the demands of a college course. While most students make this adjustment in a few weeks, some never do and struggle through the entire semester. In a worst case scenario, you might drop the higher-level course and decide to take the lower-level course for a grade the following semester. This would actually put you behind a semester since you could have just taken the lower-level course initially.

**EXAM COVERAGE**

Course descriptions, along with a more detailed listing of topics which might be tested on a proficiency exam.

**MATH 121. Calculus for Science and Engineering I (4)**

Functions, analytic geometry of lines and polynomials, limits, derivatives of algebraic and trigonometric functions. Definite integral, antiderivatives, fundamental theorem of calculus, change of variables. Prereq: Three and one half years of high school mathematics. Book: *Calculus Early Transcendentals*Third Edition by Rogawski and Adams, Chapters: 1-6

Topics: Real Numbers and Graphs, Functions, Inverse and Exponential Functions, Limits, Continuity, Computing Limits, Trig Limits, Limits at Infinity, I.V.T. , Formal Definition of Limit, Definition of Derivative, Derivative, Product and Quotient Rules, Rates of Change, Higher Derivatives, Trig Derivatives, The Chain Rule, Implicit Differentiation, Derivatives of Inverse Trig Functions, Derivatives of Exponentials, Derivatives of Hyperbolic Trig, Related Rates, Linear Approximations, Extreme Values, M.V.T. , Graphing, l’Hospital’s Rule, Graphing, Applied Max/Min, Newton’s method, Area, The Definite Integral, The Indefinite Integral, FTC Part I/II, Substitution, Transcendental Functions, Exponential Growth, Area, Average value, Volume by Slicing, Volume by Cylindrical Shells, Work

**MATH 122. Calculus for Science and Engineering II (4)**

Continuation of MATH 121. Exponentials and logarithms, growth and decay, inverse trigonometric functions, related rates, basic techniques of integration, area and volume, polar coordinates, parametric equations. Taylor polynomials and Taylor’s theorem. Prereq: MATH 121. Book: *Calculus Early Transcendentals*Third Edition by Rogawski and Adams, Chapters: 7-12

Topics: Parts, Trig. Integrals, Trig. Substitution, Hyperbolic Functions, Partial Fractions, Improper Integrals, Numerical Integration, Arc Length, Fluid Pressure, Center of Mass, Taylor Polynomials, Differential Equations, y’= k(y – b) , Euler’s Method, The Logistic Equation, First Order Linear, Applications of First Order, Sequences, Series, Integral Test, Comparison Test, Ratio and Root Test, Absolute and Conditional Convergence, Power Series, Taylor Series, Parametric Equations, Arc Length, Polar Coordinates, Area in Polar, Conic Sections, Vectors, Lines, Dot Product, Cross Product, Planes

**MATH 125. Mathematics I (4)**

The first semester of a two semester introduction to college mathematics. The course covers some finite mathematics (including discrete probability) and differential calculus with applications (including exponential, logarithmic, and trigonometric functions and difference equations). Prereq: Three and one-half years of high school mathematics.

Books: *College Mathematics for Business, Economics, Life Sciences,…*Chapters 7-8, 10-12; and *Additional Calculus Topics*, Chapter 2.1; bothby Barnett, Ziegler, Byleen (13^{th}ed).

Topics: Sets, counting, permutations, combinations, probability, conditional probability, independence, Bayes’ theorem, limits, continuity, limits at infinity, definition of derivative, rates of change, product and quotient rules, derivatives of exponential and log functions, derivatives of trig functions, trig models, trig derivatives, differentials, marginal analysis, chain rule, implicit differentiation, related rates, elasticity of demand, graphing, l’Hôpital’s rule, extreme values, optimization, difference equations, fixed points, stability, linear approximations, Taylor polynomials.

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**MATH 126. Mathematics II (4)**

Continuation of MATH 125 covering integral calculus (including trigonometric functions), continuous probability, matrix algebra, differential equations, calculus of several variables, and applications. Prereq: MATH 125.

Books: *College Mathematics for Business, Economics, Life Sciences,…*Chapters 4, 8.5, 13-15; and *Additional Calculus Topics*, Chapters 1, 3; bothby Barnett, Ziegler, Byleen (13^{th}ed).

Topics: antiderivatives, u-substitution, trig integrals, definite integrals, FTC, integration by parts, area between curves, future value, consumer surplus, gini index, improper integrals, approximations, discrete random variables, expected value, binomial distributions, continuous random variables, cumulative distribution functions, mean, variance, standard deviation, exponential and normal distributions, Chebyshev’s inequality, Gauss-Jordan elimination, matrix operations, inverse matrices, slope fields, separable and first order linear DEs and their applications, second order DEs, functions of several variables, partial derivatives, local extrema and saddle points, Lagrange multipliers, least squares regression, double integrals, triple integrals.

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**MATH 223. Calculus for Science and Engineering III (3)**

Introduction to vector algebra; lines and planes. Functions of several variables: partial derivatives, gradients, chain rule, directional derivative, maxima/minima. Multiple integrals, cylindrical and spherical coordinates. Derivatives of vector valued functions, velocity and acceleration. Vector fields, line integrals, Green’s theorem. Prereq: MATH 122. Book: *Calculus Early Transcendentals*Third Edition by Rogawki and Adams, Chapters: 12-17

Topics: Dot and Cross Product, Planes, Quadratic Surfaces, Cylindrical and Spherical Coordinates, Vector Valued Functions, Arc Length and Speed, Unit Tangent – Curvature, Motion Along a Curve, Functions of Several Variables, Limits, Partial Derivatives, Tangent Planes,

Directional Derivative, Gradient Vector, Chain Rule, Max and Min, Lagrange Multipliers, Double Integrals, Triple Integrals, Double Integrals in Polar, Triple Integrals in Cylindrical and Spherical Coordinates, Change of Variables, Vector Fields, Line Integrals, Conservative Vector Fields, Surface Integrals, Green’s Theorem

**MATH 224. Elementary Differential Equations (3)**

A first course in ordinary differential equations. First order equations and applications, linear equations with constant coefficients, linear systems, Laplace transforms, numerical methods of solution. Prereq: MATH 223.

Book: *Differential Equations*, by Blanchard, Devaney, and Hall (4^{th}ed), Chapters 1-7.

Topics: DEs as models, separable DEs, slope fields, uniqueness and existence, phase lines, bifurcations, linearity, first order linear DEs, systems of DEs, system geometry, damping, decoupling, SIR models, Lorenz equations, linearity for systems, straight-line solutions, solutions to systems with real eigenvalues, complex eigenvalues, repeated eigenvalues, zero eigenvalues, second-order linear DEs as systems, the trace-determinant plane, systems with forcing, sinusoidal forcing, resonance, steady state solutions, linearization, nullclines, Hamiltonian systems, dissipative systems, Laplace transformations, step functions, Dirac delta functions, convolutions, numerical methods (Euler, Euler for systems, Improved Euler, Runge Kutta).