Math 4C Fall 2008 Final Exam Study Guide Format 12 questions, some multi-part. Questions will be similar to sample problems in this study guide, homework problems, lecture examples or examples from the text. The final exam is comprehensive. Protocol Wednesday, December 10, 3-6 pm in YORK 2622 (tentative). No notes nor calculators allowed. Bring photo ID. No credit for unsupported work. No makeup test; however lowest midterm percentage will be automatically replaced by the final exam percentage if it is higher. Do work on final exam (blue book not required) or attach scratch paper. Study Guide Review homework, lecture notes, and notes from reading the textbook. Work, review sample problems below. Study with one or two other students. Seek help at office hours; ask specific questions. Topics, Sample Problems Supp Expressions and functions 1.3.57, 69, 71, 81; 1.R.3, 5, 9 distinguish between expressions and functions define polynomial evaluate polynomials perform operations on polynomials, e.g., multiplication factor polynomials over the integers, the rationals, the reals, the complex numbers Supp Rational expressions, solving equations evaluate and simplify rational expressions add, subtract, multiply, and divide rational expressions determine if a value is a solution to an equation solve equations by inverse operations 12.7, Partial fractions 12.R.89; 12.T.19 12.8 determine form of partial fraction decomposition (non-repeated linear factors, repeated linear factors, non-repeated quadratic factors, repeated quadratic factors) solve partial fraction decompositions by equating coefficients or choosing convenient values 12.2 Polynomial division 12.R.1, 5; 12.T.5, 9 divide polynomials using long and synthetic division to find quotients and remainders 12.3- Roots of polynomial equations 12.R.13, 17, 19, 31, 37, 43, 47, 63, 67; 12.T.1, 2, 12.6 3, 4, 8, 10, 12, 14, 15, 17 use various theorems (Remainder, Factor, Linear Factors, Fundamental Theorem of Algebra, Location or Intermediate Value, relationship between coefficients and roots or Vieta's, Rational Roots, Upper and Lower Bound, Descartes Rule of Signs, Conjugate Roots, Irrational Conjugate Roots) to solve polynomial equations 5.4 Properties of logarithms 5.R.11, 14, 65 use properties of logarithms to find values of logarithms use properties of logarithms to simplify or expand logarithmic expressions 5.5 Equations involving exponents and logarithms 5.R.15, 35; 5.T.13 solve exponential and logarithmic equations 5.6 Applications 5.R.103; 5.T.6
solve compound interest problems 5.7 Exponential growth and decay 5.R.3, 78, 83 solve exponential growth problems solve exponential decay problems 10.1, Systems of linear equations, Gaussian elimination 10.2 recognize when a system of linear equations has one solution, no solutions, or an infinite number of solutions 3.1 Functions determine domain, range, independent and dependent variables 3.R.1, 43, 49, 53, 102, 125; 3.T.1, 2 compute values of functions using numbers, using literals 3.R.11, 69, 73, 77, 79, 105, 114, 118; 3.T.10 compute difference quotients 3.R.5, 89, 129; 3.T.5 find slope and average rate of change, use!y!x notation 3.R.8, 13 solve application problems 3.T.17 3.2 Graphs of functions use Vertical Line Test to determine if a graph represents a function find x- and y-intercepts; positive and negative parts of graph; increasing, decreasing, and level parts of graph; turning points (local or relative extrema); global (absolute) extrema; slope, secant lines, and average rate of change; concavity and (estimate) inflection points; symmetry, odd and even functions; vertical and horizontal asymptotes; sketch graphs 3.R.10, 101, 103, 104, 107, 108, 109, 110, 111, 116, 117, 123, 124, 127, 128; 3.T.9 3.3 Graphing transformations relate vertical and horizontal translations (shifts), reflections, and dilations of a graph to its function 3.R.6, 15, 126 recognize transformations from the function sketch graphs by applying transformations of a basic graph 3.R.9, 21, 29; 3.T.7, 8, 15 3.4 Combining functions find sum, difference, product, and quotient of two functions 3.R.4ab, 81, 119; 3.T.3a find composition of two functions 3.R.2bc, 3, 4c, 85, 86, 87, 120, 121, 122; 3.T.3bc decompose a function 3.R.63 3.5 Inverse functions know properties or inverse functions and their graphs (monotonicity, one-to-one, Horizontal Line Test, compositions of function and its inverse) find inverse function, if it exists, algebraically 3.R.39, 41, 91, 97, 98; 3.T.6 sketch inverse function from graph of function by reflection through y = x 3.R.7 restrict domain to find inverse function 3.R.112, 113 solve application problems 3.R.16; 3.T.16 4.1 Linear functions find equations of lines in slope-intercept, point-slope, standard, and double-intercept forms 4.R.1, 15, 19; 4.T.1
graph linear functions find equations of lines with graphs that are parallel or perpendicular to a given linear graph 4.R.17 4.2 Quadratic functions transform equations of quadratic functions among general, vertex, and factored forms (use completing the square where appropriate) find vertex and axis of symmetry, x- and y-intercepts, vertex of graph of quadratic function (parabola) 4.R.2, 6, 21, 4.T.2, 5 sketch graphs using transformations of y = x 2 sketch graphs using vertex, x- and y-intercepts find discriminant and use discriminant to find number of zeros (roots) find function from its graph solve application problems 4.R.20, 27, 31; 4.T.6 4.5 Maximum and minimum values find maximum and minimum values by analyzing functions containing quadratic expressions 4.R.33, 35, 37; 4.T.12 Supp Power functions graphs of power functions given values, find a power function 4.6 Polynomial functions find degree of polynomial function relate linear factors, zeros, roots, and x-intercepts 4.R.5 use properties of graphs (continuity, smoothness, maximum number of turning points, end behavior, behavior near x-intercepts) as aids in graphing 4.R.53, 55; 4.T.3, 10, 15 4.7 Rational functions graph rational functions (domain, range, intercepts, asymptotes) translations, reflections, and dilations 4.R.9, 11, 63; 4.T.13 5.1 Exponential functions graph exponential functions (domain, range, intercepts, asymptotes) translations, reflections, and dilations 5.T.1 5.2 Natural exponential function graph natural exponential function (domain, range, intercepts, asymptotes) translations, reflections, and dilations 5.R.29 5.3 Logarithmic functions graph logarithmic functions (domain, range, intercepts, asymptotes) translations, reflections, and dilations 5.R.77 6.1-4 Solving triangles 6.R.25, 33, 63, 67 solve right triangle applications find area of SAS triangles 9.1 Laws of Sines and Cosines 9.R.1, 5, 7, 21 use Law of Sines to solve ASA, SAA, and SSA triangles use Law of Cosines to solve SAS and SSS triangles use Heron's formula to find area of SSS triangles 6.4-5, Trigonometric functions of angles 7.R.19 7.3 find values of trig functions of any angle defined by a point on the plane
find values of trig functions of any angle defined by a point on unit circle find values of trig functions of real numbers (radians) use properties of sine function (odd) and cosine function (even) to find values of trig functions 7.1, 7.2 Radian measure and geometry 7.T.7 convert between radians and degrees solve problems involving arc length and sector area solve problems involving angular speed and linear speed 7.4, 7.5 Graphs of trigonometric functions 7.R.7, 45, 51 graph sine and cosine functions (domain, range, intercepts, asymptotes, period) translations, reflections, and dilations (amplitude, phase shift, period) determine amplitude, period, phase shift, vertical shift find equation from graph of sine or cosine function 7.7 Graphs of other trigonometric functions 7.R.53a graph tangent, cotangent, cosecant, and secant functions (domain, range, intercepts, asymptotes, period) translations, reflections, dilations 12.1 Complex numbers (rectangular form) conjugates of complex numbers complex number arithmetic (adding, subtracting, multiplying, dividing, powers, roots) in rectangular form 12.R.71, 73, 75, 77, 79, 81c, 85; 12.T.16 plotting complex numbers in the complex plane 13.6 Complex numbers (trigonometric or polar form) modulus and argument of complex numbers converting between complex numbers in rectangular and trig forms 13.R.79 multiplying and dividing complex numbers in trig form 13.R.81, 83 DeMovire's theorem (finding powers and roots of complex numbers in trig form) 13.R.87, 91 6.2, Trig identities 6.R.39, 45; 7.R.37 6.5, use basic trigonometric identities to find missing values of trig functions 7.3 simplify trigonometric expressions using identities prove trigonometric identities 8.1 Addition formulas 8.R.5, 11; 8.T.1 apply sine, cosine, and tangent addition/subtraction formulas 8.2 Double-angle and half-angle formulas 8.R.19, 23, 27, 103a; 8.T.3 apply double-angle formulas apply half-angle formulas 8.4 Trigonometric equations 8.R.51, 53; 8.T.5, 7 solve trigonometric equations 8.5 Inverse trig functions 8.R.83, 89; 8.T.11, 15 evaluate expressions involving inverse trigonometric functions 13.3, Sequences, Limits of Sequences 13.R.35, 46, 52, 55; 13.T.8 13.4, given a general term, write the first four terms of the sequence 13.5 given the first four terms of a sequence, find a possible general term determine if a sequence converges or diverges (and find its limit if it converges) find the sum of an infinite geometric sequence that converges
Supp Limits of Functions determine if a function has a limit as x approaches infinity (and find the limit if it exists) determine if a function has a limit as x approaches some finite value c (and find the limit if it exists) determine one-sided limits, if they exist use limits to determine if a function is continuous at a point determine limits (if they exist) for undefined expressions (indeterminate forms) use laws of limits to determine limits of functions Supp Rates of Change and Graphical Interpretations calculating average velocity of a position function interpreting average velocity as slope of secant line of graph of a position function finding equation of secant line through two points of a graph estimating instantaneous velocity of a position function interpret instantaneous velocity as slope of tangent line of graph of a position function finding equation of tangent line at a point of a graph Supp Instantaneous Rate of Change and Derivative of a Function at a Point calculate instantaneous rate of change of a function at a point using limit definition calculating derivative of a function at a point using limit definition interpreting instantaneous rate of change of a function at a point as slope of tangent line at the point finding equation of tangent line at a point of a graph understanding when a derivative of a function may fail to exist at a point