Student Learning Outcomes MATH 100: BEGINNING ALGEBRA
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1 Student Learning Outcomes MATH 100: BEGINNING ALGEBRA I. Real Numbers 1. Perform the basic operations on fractions, reduce, and convert to decimals and percents. 2. Define, classify, and graph real numbers. 3. Understand and use inequality symbols. 4. Find the opposite and absolute value of a real number. 5. Perform the basic operations on signed numbers. 6. Evaluate powers and roots with a calculator. 7. Use the order of operations agreement to simplify expressions. 8. Evaluate algebraic expressions. 9. Identify and use the basic properties of real numbers. II. Linear Equations and Inequalities Student will be able to: 1. Understand and use the vocabulary of algebraic expressions 2. Simplify algebraic expressions by removing parentheses and combining like terms. 3. Use inverse operations to solve linear equations and inequalities. 4. Solve equations with variables on one and both sides of an equation. 5. Solve equations by first simplifying one or both sides of an equation. 6. Check whether a number is a solution to an equation. 7. Graph the solution of an inequality on a number line 8. Write ratios 9. Solve proportions 10. Solve fractional equations III. Formulas and Applications of Algebra 1. Evaluate formulas. 2. Solve for a variable in a formula (translate literal equations). 3. Translate English phrases into algebraic expressions (words to symbols). 4. Write expressions involving percent. 5. Solve word problems (applications) using algebraic models (five-step problem solving strategy). 6. Demonstrate knowledge of basic geometric concepts by evaluating formulas and solving world problems. IV. Exponents and Polynomials 1. Simplify expressions with integral exponents. 2. Convert numbers to and from scientific notation. 3. Identify the type and degree of polynomials. 4. Add, subtract, and multiply polynomials. 5. Divide a polynomial by a monomial. V. Factoring 1. Write the prime factorization of a number. 2. Determine the greatest common factor of two or more numbers. 3. Find the greatest common factor and factor out of a polynomial. 4. Factor the difference of two squares. 5. Factor simple and general trinomials using reverse of F.O.I.L. 6. Understand and use a general strategy for factoring polynomials (factor completely) 7. Recognize quadratic equations (standard form) and solve by factoring (use the zero product rule) 8. Solve word problems (applications) using factoring
2 MATH 106: INTERMEDIATE ALGEBRA 1. Perform all of the necessary operations connected with polynomials. 2. Differentiate the different types of factoring. 3. Simplify, add, subtract, multiply and divide rational expressions. 4. Solve both linear equations and inequalities. 5. Be aware of the exponential properties and be able to use each. 6. Graph linear equations. 7. Write linear equations given different information. 8. Perform all necessary operations involving radicals. 9. Solve systems of equations. 10. Solve quadratic equations. MATH 108: MATH OF FINANCE 1. The student will be able to perform conversions dealing with fractions, decimals and percents. 2. The student will be able to solve necessary word problems dealing with fractions, decimals, percents and proportions. 3. The student will be able to calculate discounts, markup, markdown, and other percents of changes. 4. The student will be able to differentiate between simple and compound interest and perform all necessary calculations. 5. The student will be able to identify all annuities and calculate any given value. 6. The student will be able to deal with consumer credit. 7. The student will be able to calculate depreciation using a variety of methods. MATH 111: SURVEY OF MATHEMATICS I. Sets and Set Operations 1. Describe sets verbally using appropriate mathematical terms (e.g., inclusive) and be able to write sets in roster form and set-builder notation. 2. Determine whether an object is an element of a set. 3. Determine whether a set is finite or infinite. 4. Determine whether two sets are equal, equivalent, or neither. 5. Determine the cardinal number of a set. 6. Determine whether sets are the empty set or the universal set. 7. Determine whether sets are subsets, proper subsets, or neither. 8. Develop a general formula for finding the number of distinct subsets of a given set and be able to use the formula. 9. Find the complement of a set and the intersection and union involving two or more sets. 10. Draw and use Venn diagrams to solve problems involving the intersection and union of sets. 11. Draw and use Venn Diagrams to verify the equality of sets. 12. Apply Venn diagrams to solve practical problems. II: Logic 1. Translate verbal sentences into symbolic form (and vice versa) using logical connectives,,,,. 2. Use the concept of necessary and sufficient to write the negation of statements containing universal quantifiers. 3. Determine the most dominant connective in a statement that contains more than one connective. 4. Construct truth tables involving compound statements with two or three variables. 5. Determine the truth value of a compound sentence. 6. Determine whether a compound statement is a tautology, logically false, or neither and to determine whether a tautology is an implication. 7. Use elementary logic to solve problems involving graphs (e.g., pie charts) and other practical problems (e.g., determining loan qualifications). 8. Write logically equivalent statements using DeMorgan s laws or the contrapositive.
3 9. Write the converse, inverse, and contrapositive of a condition statement. 10. Determine the validity of arguments by comparing the laws of inferences (Law of Detachment, Law of Contraposition, Law of Syllogism, Disjunctive Inference) to the statement. 11. Write short deductive proofs using logical symbolism. 12. Use Euler diagrams to determine the validity of syllogistic arguments. III. Probability 1. Find the empirical probability from contrived data and data gathered experimentally. 2. Find the theoretical probability of a single event occurring, including probabilities of 0 and Finding the odds for and against an event occurring and finding the probability of an event given the odds. 4. Draw a tree diagram to illustrate the possible outcomes of an experiment. 5. Use the basic counting principle to determine the number of possible outcomes. 6. Find the theoretical probability with and without replacement. 7. Determine whether given situations are mutually exclusive and find the theoretical probability. 8. Determine whether events are independent. 9. Find the conditional probability of an event, given that another event has occurred, by using a tree diagram or by reading a two-way table. 10. Find the permutation of an ordered arrangement of a set of objects, including duplication of objects, using a calculator. 11. Find the combination of a set of objects using a calculator. 12. Solve probability problems using combinations and binomial combinations. IV. Graphing, Linear Programming, and Matrices 1. Graph a straight line and determine the slope and intercepts using an equation or its graph. 2. Find the solution of two lines graphically, numerically, and algebraically in mathematical and applied situations (e.g., break-even point). 3. Perform elementary matrix operations (add, subtract, and multiply). 4. Use matrices to solve a system of equations. 5. Graph linear inequalities and determine the solution set of a system of linear inequalities. 6. Use linear programming to solve practical problems. MATH 115: MATH FOR ELEMENTARY TEACHERS I I. Numeration Systems 1. Represent numbers in various numeration systems 2. Represent a number in various bases (compose and decompose) 3. Convert a numeral from base ten to another base 4. Convert a numeral from one base to base ten 5. Perform arithmetic operations with numerals in bases other than ten. II. Operations with Natural Numbers, Whole Number, and Integers 1. Determine what properties hold for a set of numbers 2. Classify word problems by operation type 3. Perform arithmetic operations with integers 4. Determine why particular algorithms work (addition, subtraction, multiplication, and division) III. Number Theory 1. Find all factors of a number 2. Write the prime factorization of a number 3. Classify a number by the number of its factors 4. Test whether one number is divisible by another number
4 5. Find the greatest common factor of two or more numbers 6. Find the least common multiple of two or more numbers 7. Perform arithmetic operations in modulo m 8. Represent figurate numbers symbolically IV. The Real Number System 1. Recognize equivalent fractions 2. Model fractions with region, linear, and set models 3. Model operations with fractions 4. Simplify fractions 5. Classify word problems by operation category 6. Find a number between two other numbers 7. Represent quantities as ratios 8. Solve proportions 9. Convert fractions to decimals and decimals to fractions 10. Represent quantities as percents 11. Perform operations with decimals and percents 12. Find a number on the real number line MATH 116: MATHEMATICS FOR ELEMENTARY TEACHERS II I. Probability and Statistics 6. Determine the probability of an event occurring 7. Determine the sample space of an event occurring 8. Know when and how to use the counting principles 9. Find the mean, median, and mode (and determine its appropriateness) 10. Dispersion 11. Make graphical displays 12. Interpret information from a plot or a table 13. Determine intervals, percentiles, and quartiles from various distributions II. Patterns and Functions 5. Identify functional relationships from tables, graphs, and symbols 6. Generate output values when given input values 7. Identify a rule for a function from a table or a graph 8. Determine the domain and the range of a function 9. Use properties to solve equations for a variable 10. Find the rate of change of a function from a table, graph, or an equation III. Concepts of Geometry 9. Write definitions of terms with necessary and sufficient conditions 10. Recognize and use various geometric figures and shapes 11. Construct basic geometric shapes 12. Classify polygons according to their properties 13. Determine whether three given segment lengths could be used to form a triangle 14. Determine when two figures are congruent 15. Determine when two figures are similar 16. Use properties of figures to find angle measures and/or side lengths 17. Determine the measure of the angles in a polygon 18. Determine the measure of an angle in a regular polygon IV. Measurement 13. Find the length, area, perimeter/circumference, surface area, volume of various figures
5 14. Generate rectangles to meet specific criteria 15. Find the length of a side in a right triangle when given the other two sides 16. Prove the Pythagorean relationship 17. Identify various parts of two- and three-dimensional figures 18. Draw rectangular prisms from different views 19. Translate, rotate, reflect figures 20. Make tessellations 21. Identify vertex arrangements for tessellations MATH 121: COLLEGE ALGEBRA I: Algebra 1. Perform operations on polynomials: +, -, x, ), including synthetic ), and raising to a power 2. Factor expressions a. by grouping b. by synthetic division that are the sum or difference of cubes c. containing negative and fractional exponents 3. Perform operations on rational expressions a. Reduce, +, -, x, ) b. Simplify a complex fraction 4. Perform operations on radical expressions a. Simplify, +, -, x, ), raise to a power including complex numbers b. Rewrite radicals using fractional exponents 5. Perform operations on expressions with fractional and negative exponents a. Simplify, +, -, x, ), raise to a power b. Rewrite fractional exponents using radicals 6. Solve the following Equations algebraically and graphically on the TI-83 Plus calculator a. Linear b. Quadratic -- Algebraically by: Factoring, Quadratic Formula with Real and Complex Roots, Extracting the Roots with Real and Complex Roots, Completing the Square Method with Real and Complex Roots c. Higher Degree (Polynomials) d. Rational e. Radical f. Absolute Value 7. Solve the following Inequalities algebraically and graphically on the TI-83 Plus calculator and express the solution in Interval Notation a. Linear b. Quadratic c. Rational d. Absolute value 8. Perform the following in regard to Relations and Functions a. Find the value of a function using functional notation b. Determine whether a relation represents a function (from a set of ordered pairs and from a graph) c. Find the domain and range of a function d. Find the sum, difference, product and quotient of two functions e. Find the composite of a pair of functions (and its domain) f. Find the Difference Quotient of a Polynomial, Rational, and Radical Function 9. Hand graph the following functions a. Linear using Slope Y-Intercept (find the slope of a line and write an equation of a line). b. Quadratic (Find x and y-intercepts, find the axis of symmetry, and find the vertex c. Cubic d. Rational and determine asymptotes e. Radical f. Absolute Value g. Restricted domain, Split domain or Piecewise
6 h. Greatest Integer 10. Using the graphing calculator a. To graph a function and find/determine (x-and y-intercepts, zeros, intervals on which the function is increasing and decreasing, maximum and minimum and local minima and maxima, obtain information from or about the graph of a function b. Regression Program to find the Curve of Best Fit (linear, quadratic, power, polynomial cubic, exponential, logarithmic) 11. Solve systems of equations algebraically and graphically, by hand and graphing calculator a. That are linear and nonlinear, two variables b. *That are linear, three variables II. Trigonometry 1. Solve a right triangle using Right Triangle Trig and The Pythagorean Theorem 2. Convert between degrees, minutes, seconds and decimal form for angles 3. Convert from degrees to radians and from radians to degrees 4. Find the trigonometric functions of an angle of any size 5. Solve oblique triangles using the Law of Sines and Law of Cosines 6. Graph trigonometric functions 7. *Graph and solve vector problems III. Elementary Transcendental Functions 1. Use the Properties of logarithms to expand a log expression, and vice versa 2. Solve logarithm equations algebraically and graphically on the calculator 3. Graph logarithmic functions by hand and graphing calculator 4. Solve exponential equations algebraically and graphically on the calculator 5. Graph exponential functions by hand and the graphing calculator MATH 122: BASIC CALCULUS 1. Evaluate the limit of any function at any point. 2. Apply the concepts of limits to find the derivative of a function. 3. Evaluate both the average rated change and the instantaneous rate of change. 4. Differentiate any function using the appropriate rules. 5. Find both the velocity and acceleration for any given distance formula. 6. Find the marginal cost, marginal revenue and marginal profit for an economics application. 7. Solve related rate word problems. 8. Use the concepts of increasing functions, decreasing functions, concave up and concave down to graph function. 9. Solve optimization problems. 10. Integrate any function. 11. Find the area of a region between a function and the x-axis. 12. Find the area of a bounded region between two given functions. MATH 131: COLLEGE TRIGONOMETRY 1. Find the composition of two functions. 2. Find the inverse of a function. 3. Find using the graphing calculator the sine, cosine, tangent, cosecant, secant, and cotangent of any acute angle. 4. Find the trig values of special angles, 30 o and 60 o.
7 5. Solve right triangles using Right Triangle Trig and/or The Pythagorean Theorem. 6. Solve special triangles a. Right Triangle b. Isosceles Right Triangle 7. Solve applied problems involving right triangles. 8. Convert degrees to radians and vice versa. 9. Find the value of the 6 trig functions of any angle (degrees or radian measure). 10. Solve oblique triangles using the Law of Sines. 11. Solve oblique triangles using the Law of Cosines. 12. Solve applied problems involving oblique triangles. 13. Find the area of triangles using the formula A = ½ ab sin C and Heron s Formula 14. Find the domain and range of the trigonometric functions. 15. Graph the Sin Function. 16. Graph the Cos Function. 17. Graph the Shifts of Sin/Cos Functions. 18. Write equations of Sin and Cos Functions. 19. Graph the Tan and Cot Functions. 20. Graph the Sec and Csc Functions. 21. Graph the Inverse Sin Function. 22. Graph the Inverse Cos Function. 23. Simplify trigonometric expressions. 24. Prove trigonometric identities using fundamental identities and algebraic operations. 25. Verify trigonometric identities using the graphic calculator. 26. Solve trigonometric equations. 27. Find the Sum and Difference of Two Angles using the Formulas. 28. Use the Double Angle Formulas to find values. 29. Use the Half Angle Formulas to find values. 30. Write log b n = e in b e = n form and vice versa. 31. Evaluate e x and ln x functions using the calculator. 32. Graph e x 33. Graph ln x 34. Solve applied problems using log functions. 35. Expand a log expression using the Properties of Logs. 36. Solve log equations algebraically and graphically. 37. Solve exponential equations algebraically and graphically. MATH 141: STATISTICS I. Organizing and Presenting Sets of Data: 1. Describe methods of collecting data. 2. Describe methods of sampling. 3. Name different types of data. 4. Use the random number generator in the Appendix or on the calculator. 5. Construct a bar graph, pareto chart, circle graph, timeline graph, stem and leaf graph and a histogram. II. Analysis of Data - The Common Statistical Measure 1. Compute the mean, mode and median. 2. Explain the meaning of mean, mode and median. 3. Calculate the standard deviation for a sample, for a population. 4. Calculate the 5 number summary. 5. Identify any outliers. 6. Construct a box and whisker plot. III. Regression and Correlation of Paired Data 1. Draw a scatter plot of pair data. 2. Identify the type of regression.
8 3. Calculate the best fit line. 4. Calculate the linear correlation coefficient. 5. Interpret the value of the correlation coefficient. 6. Calculate the value of the coefficient of determination. 7. Interpret the value of the coefficient of determination. IV. Elementary Probability 1. Compute probabilities and their complements. 2. Apply the rules for compound probability. 3. Calculate conditional probabilities. 4. Identify permutations and combinations. 5. Compute permutations and combinations. V. Probability distributions For a discrete distribution: 1. Calculate probability distributions from given data. 2. Sketch the histogram of the distribution 3. Calculate the mean and standard deviation. For a binomial distribution: 1. Identify it as a binomial. 2. Evaluate exactly, at least, at most r successes. 3. Calculate the distribution. 4. Sketch the histogram of the probability distribution and the cumulative distribution. 5. Calculate the mean and standard deviation. For the normal distribution: 1. Identify the normal distribution. 2. Apply the Empirical Rule. 3. Calculate Z scores given x values. 4. Calculate x values given z scores. 5. Sketch the normal curve. 6. Compare Z scores. 7. Interpret Z scores. 8. Solve verbal problems. VI. Estimation of the Mean 1. Calculate the interval the mean falls in for large samples and small samples. 2. Interpret the interval the mean falls in for large samples, small samples and proportions. 3. Calculate the sample size for mean estimation. 4. Calculate the interval a proportion falls in. 5. Interpret the interval the proportion falls in. 6. Calculate the sample size for proportion estimation VII. Hypothesis Testing 1. Describe the hypothesis and the null hypothesis. 2. Test the hypothesis for the mean of large and small samples. 3. Test the hypothesis for a proportion. MATH 161: CALCULUS I I: Functions and Graphs 1. Find the slope between two points and of a line 2. Sketch the graphs of algebraic functions (polynomials, rational, radical, piecewise-defined, and absolute value). 3. Find the value of a function using the equation or the graph.
9 4. Evaluate composite functions. 5. Sketch the graphs of conics (circles, ellipses and hyperbolas). II: Limits 1. Estimate the limit of algebraic and trigonometric functions using tables and graphs. 2. Find the limit of algebraic or trigonometric functions using algebra (cancellation and rationalization). 3. Determine when the limit of a function does not exist. 4. Finding one-sided limits. 5. Using the definition of continuity, determine whether an algebraic function is continuous. 6. For functions that are discontinuous, determine whether the function has removable or non-removable discontinuity. III: Differentiation 1. Find the derivative of an algebraic function by finding the limit of the secant line. 2. Determine where an algebraic function is differentiable. 3. Find the derivative of algebraic and trigonometric functions using basic rules, the product rule, and the quotient rule. 4. Find the second and third (higher) derivative of a function. 5. Find the derivative of composite algebraic and trigonometric function using the chain rule. 6. Determine whether a function is written in implicit or explicit form. 7. Find the derivative of a function using implicit differentiation. 8. Find the derivatives of natural logarithmic and exponential functions. IV: Applications of Differentiation 1. Find a related rate 2. Use related rates to solve real-world problems 3. Find a critical number of a function. 4. Find the extrema on a closed interval using the first derivative. 5. Apply Rolle s Theorem in order to find all values of c in an open interval such that f (c) = Apply the Mean Value theorem to find the all values of c in an open interval such that (c, f(c)) is on the tangent line at x = c. 7. Determine the intervals on which a function is increasing or decreasing. 8. Apply the First Derivative Test to find relative extrema of a function. 9. Determine intervals on which a function is concave upward or concave downward. 10. Find any points of inflection of the graph of a function. 11. Apply the Second Derivative Test to find relative extrema of a function. 12. Determine limits (finite and infinite) at infinity. 13. Determine horizontal asymptotes, if any, of the graph of a function. 14. Analyze and sketch the graph of a function. 15. Solve applied minimum and maximum problems. 16. Approximate a zero of a function using Newton s Method. 17. Find the tangent line approximation of a function. 18. Compare the value of the differential, dy, with the actual change in y. 19. Find the differential of a function using differentiation formulas. V: (OPTIONAL) Antidifferentiation Students will be able: 1. Write the general solution of a differential equation. 2. Find indefinite integral for antiderivatives. 3. Use basic integration rules to find antiderivatives. 4. Find a particular solution of a differential equation. 5. Draw a slope field. 6. Use Sigma notation to write and evaluate a sum. 7. Approximate the area of a plane region. 8. Find the area of a plane region using limits. 9. Evaluate a definite integral using limits. 10. Evaluate a definite integral using properties of definite integrals.
10 MATH 162: CALCULUS II I: The Definite Integral 1. Use sigma notation to write and evaluate a sum. 2. Find the area under a curve using Rieimann Sums. 3. Evaluate a definite integral using properties of definite integrals. 4. Find the area under a curve using the Fundamental Theorem of Calculus. 5. Find the average value of function using the Mean Value Theorem. II: Applications of Integrals 1. Find the area of a region formed by two curves using the Fundamental Theorem of Calculus. 2. Find the volume of revolution using the Disk or Washer Method. 3. Find the volume of revolution using the Shell Method. 4. Find the length of an arc using integrals. 5. Solve problems with variable force (liquid pressure and moments at the discretion of the instructor). 6. Find the surface area of any figure. III: Transcendental Functions 1. Find integrals involving trigonometric, logarithmic, and exponential functions. 2. Differentiate inverse trigonometric functions. IV: Techniques of Integration 1. Find indefinite and definite integrals using substitution. 2. Find indefinite and definite integrals using integration by parts. 3. Find indefinite and definite integrals using trigonometric substitution. 4. Find indefinite and definite integrals of partial fractions. V: Numerical Methods 1. Find the area under a curve using Simpson s Rule. 2. Find the area under a curve using the Trapezoidal Rule. VI. OPTIONAL TOPICS 1. Find the derivative and integrals of hyperbolic functions. 2. Approximate a function using a Taylor Polynomial. 3. Approximate a function using Cubic Splines. MATH 262: CALCULUS III I. Conics 1. Write equations for a given parabola, ellipse or hyperbola. 2. Recognize equations for parabolas, ellipses and hyperbolas and determine important details such as vertices and foci. II. Parametric Equations
11 1. Graph parametric equations and determine curve orientations. 2. Find the slope of the tangent line to a point on a curve given by a set of parametric equations. 3. Find the arc length of a segment of a curve given by a set of parametric equations. III. Polar Coordinates 1. Graph polar equations. 2. Convert rectangular equations to polar form and vice versa. 3. Find the slope of the tangent line to a point on a curve given by a polar equation. 4. Find the area of a polar region. 5. Find the arc length of a segment of a curve given by a polar equation. IV. Space Coordinates 1. Locate points in P 3 given in rectangular, cylindrical and spherical coordinates. 2. Recognize and find equations of lines and planes in space. 3. Recognize and find equations of cylindrical surfaces, quadric surfaces and surfaces of revolution in space. V. Vectors 1. Perform vector operations and manipulate vectors using vector space properties. 2. Find the dot product and norm of vectors in P 2 and P Find vector projections and angles between vectors. 4. Solve basic force and work problems using vectors. 5. Find the cross product of vectors in space. 6. Determine if a set of vectors forms a basis. VI. Vector-valued functions and Elementary Differential Geometry 1. Perform basic calculus on vector-valued functions. 2. Solve projectile motion problems using vector-valued functions. 3. Calculate the unit tangent vector, the unit normal vector and the unit binormal vector at a point on a space curve described by a vector-valued position function. 4. Calculate the curvature and torsion of a space curve. 5. Find the tangential and centripetal components of acceleration. 6. Reparameterize a curve as a unit speed curve with the arc length parameter. VII. Functions of Several Variables 1. Find limits of functions of several variables. 2. Determine continuity of a function of two variables at a point. 3. Compute partial derivatives of multivariable functions. 4. Determine the differentiability of a function of two variables by examining the total differential. 5. Use the total differential to approximate measurement errors and function value changes. 6. Find derivatives using the chain rule for functions of several variables. 7. Find partial derivatives of an implicitly defined function of two or three variables. 8. Find the directional derivative and gradient of a function of two or three variables. 9. Write the equation of the tangent plane and normal line to a point on a surface. 10. Locate extrema of functions of two variables on open and closed domains using the second partial derivative test. 11. Solve applied optimization problems involving functions of two or three variables. 12. Use the method of Lagrange multipliers to solve constrained optimization problems. VIII. Multiple Integration 1. Evaluate iterated integrals and switch the order of integration. 2. Find volumes of solids by calculating appropriate double integrals in rectangular and polar coordinates. 3. Find the center of mass and moments of mass of a plane lamina of variable density. 4. Find surface area using a double integral. 5. Evaluate triple integrals and use them to find volumes in rectangular, cylindrical and spherical coordinates.
12 6. Find the center of mass, moments of mass and moments of inertia of a solid region of variable density. 7. Use a Jacobian to make a change of variables in a double integral. (optional) IX. Vector Analysis (optional) 1. Identify conservative vector fields. 2. Find the divergence and curl of a vector field. 3. Evaluate line integrals of curves and vector fields. 4. Use Green s theorem to evaluate line integrals. MATH 264: DIFFERENTIAL EQUATIONS I. First Order Differential Equations 1. Distinguish between linear, nonlinear, partial and ordinary differential equations. 2. State the basic existence theorem for 1st order ODE s and use the theorem to determine a solution interval. 3. Recognize and solve a variable separable differential equation. 4. Recognize and solve a homogeneous differential equation. 5. Recognize and solve an exact differential equation. 6. Recognize and solve a linear differential equation by use of an integrating factor. 7. Recognize and solve equations of Bernoulli, Ricatti and Clairaut. 8. Make a change of variables to reduce a differential equation to a known form. 9. Find particular solutions to initial value problems. 10. Solve basic application problems described by first order differential equations. II. Linear Differential Equations of Higher Order 1. Use the existence theorem for boundary value problems to determine uniqueness of solutions. 2. Use the Wronskian to determine if a set of functions is linearly independent. 3. Build solutions to differential equations by superposition of known solutions. 4. Find the complete solution of a nonhomogeneous differential equation as a linear combination of the complementary function and a particular solution. 5. Construct a second solution to a second order differential equation by reduction of order. 6. Find the complete solution of a homogeneous differential equation with constant coefficients by examining the characteristic equation and its roots. 7. Find the complete solution of a nonhomogeneous differential equation with constant coefficients by the method of undetermined coefficients. 8. Write a differential equation with constant coefficients in operator form and find the complete solution by using an annihilator operator. 9. Find the complete solution of a differential equation with constant coefficients by variation of parameters. 10. Solve basic application problems described by second order linear differential equations with constant coefficients. III. Differential Equations with Variable Coefficients 1. Solve a Cauchy-Euler Equation. 2. Identify ordinary and singular points. 3. Find power series solutions about ordinary points. 4. Find power series solutions about singular points. IV. The Laplace Transform 1. Find the Laplace transform of a function by definition and by use of a table. 2. Find the inverse Laplace transform of a function. 3. Write piecewise functions using the unit step function. 4. Find transforms using the first and second translation theorems. 5. Find the convolution of two functions and the transform of a convolution. 6. Find the transforms of derivatives and integrals. 7. Find the transform of a periodic function. 8. Solve a basic integrodifferential equation using the Laplace transform.
13 9. Solve linear differential equations with constant coefficients and unit step input functions using the Laplace transform. V. Systems of Differential Equations 1. Write a system in operator notation and solve by elimination. 2. Solve a system by the Laplace transform method (optional). 3. Solve a homogeneous linear system by the eigenvalue method. 4. Solve a nonhomogeneous linear system by variation of parameters. VI. Numerical Methods for Ordinary Differential Equations (Optional) 1. Obtain an approximate solution function value to a initial value problem using the Runge-Kutta method. 2. Obtain an approximate set of solution function values to a second order boundary value problem using a finite difference equation.
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