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by: Zechariah Hilpert
Zechariah Hilpert
GPA 3.8


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Class Notes
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This 7 page Class Notes was uploaded by Zechariah Hilpert on Thursday September 17, 2015. The Class Notes belongs to MATH 112 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/205299/math-112-university-of-wisconsin-madison in Mathematics (M) at University of Wisconsin - Madison.


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Date Created: 09/17/15
Review for Final The purpose of this review sheet is just to remind you of the topics covered in Math 112 since the third midterm You may wish to refer to the other review sheets for topics to review from other sections The homework teptbooh and midterm eccams are good places to nd practice problems This list includes a sample of topics from the sections that have been covered however it does not include everything you need to know You will not be given any formulas and you may not bring any notes to the eccam You are not allowed to use a graphing or programmable calculator on the exam Review material from previous chapters 61 Systems of linear equations 0 Substitution o AdditionSubtraction 0 Word problems 0 Possibilites for two linear equations with two unknowns o Graphing linear systems in two variables 62 Gaussian Elimination 0 Elementary Operations 0 Triangular form back substitiution o ln nitely many solutions no solutions 66 Nonlinear Systems of Equations 0 Substitution o Graphing nonlinear equations in two variables intersections correspond to real solutions 0 No solutions 0 Convert to linear system 0 Exponentials and Logarithms 71 Complex Numbers 0 i2 71 computations with i 0 real part imaginary part 0 equality operations complex conjugate 2 o roots of quadratics 72 Polynomial Division 0 What is a polynomial 0 Division Algorithm pd Rz quotient remainder 0 Synthetic division 73 Remainder and Factor Theorems 0 zero root of a polynomial o Remainder theorem applications 0 Factor theorem applications 0 Find all roots of given polynomial 0 Make a polynomial satisfying given conditions 74 Fundamental Theorem of Algebra 0 The Fundamental Theorem of Algebra only applies to polynomials 0 Linear Factors Theorem 0 Write a polynomial as the product of linear factors 0 polynomial of degree n 2 1 has n roots counting multiplicity 0 Make a polynomial sats ying given conditions if possible 0 Relationship between roots and factors 92 Binomial Theorem 0 What is a binomial o Pascal7s triangle 0 Factorials and the binomial coef cent n o The binomial theorem 1 b n an kbk l o rth term SJM THUA 0 Find a term coef cent in an expansion 93 Sequences and Series numerical sequence nth term is can compute terms from formula or recursive de nition series sums sigma notation collapsing sum 95 Geometric Sequences and Series 0 geometric sequence common ratio 0 nth term can arn l a1 7 r 1 7 r i i i i i a 0 Sum of in nite geometric series for M lt 1 is S 1 0 Sum of rst 71 terms Sn A Partial List of Topics Math 112 Spring 2009 o This is a partial compilation of a majority of the topics covered this semester and may not include everything which might appear on the exam 0 The purpose of this review sheet to remind you of the different types of material we have covered throughout the semester 0 You should review your homework and previous exams 0 You will not be given any formulas and you may not bring any notes to the exam 0 No calculators will be permitted on this exam Review amp Appendix B Sections 15 amp 17 o exponent rules7 rational and negative 39 CharaCteriStiCS 0f graphs 0 expressions with radicals 39 I and y Intercepts 0 Operations with fractions Symmetry o Re ecting a line segment 0 compound fractions 0 test an equation for symmetry 0 Simpli cation 0 Equation for a circle I 7 h2 y 7 k2 7 2 o Factoring o completing 12 121 to be a square Sections 11 14 Section 16 o absolute value as a distance yg 7 yl Ay o slope formula m 7 7 12 7 11 AI 0 pointslope form y 7 yl mz 7 11 o if pq 0 then either p 0 q 0 or both 0 distance formula d 12 7 zl2 7 yg 7 y12 o slopeintercept form y mm b o midpoint formula M lt 0 standard form A1 By C 0 where A7 B C are integers 11 I2 91 y2 2 7 2 0 two nonvertical lines are parallel if their slopes satisfy m1 m2 0 two nonvertical lines are perpendicular if their slopes satisfy 1 m1 77172 Section 21 7b i V122 7 4 o 0412 121 c 0 has solutions I 2 w which come from a completing the square 0 write a quadratic equation with given roots and coef cients 0 discriminant 0 sum and product of roots 0 factor to solve Section 22 0 taking nth roots of equations even vsi odd 0 equations of quadratic type 0 equations with fractions 0 equations with roots 0 extraneous roots always check your answer 0 equations with absolute value Section 24 0 Practice these Section 25 and 26 o interval notation 0 compound inequalities strings of inequalities o inequalities with absolute value 0 rational and polynomial inequalities Section 31 amp 3 2 0 De nition of a function 0 DomainRange from a function de nition or its graph 0 Vertical line test 0 Graph the 6 basic functions 0 Piecewise functions Section 33 Average rate of change of 161 on Mn is w related to slope o Simplifying a difference quotient 0 De nitions turning point maximum value minimum value increas ing decreasing 0 Velocity Section 34 o Translations horizontal and vertical 0 Re ections over the z and y axes 0 Determine how two graphs are related 0 Getting order correct Exi f7z c Section 35 De nitions of f ggtltzgt f 7 ggtltzgt7 faxes and my 0 De nition of f o g 0 Getting a value for a composition from a graph table or algebra 0 Domain and range 0 Express a function as a composition of simpler functions Section 36 0 De nition 0 Conditions for a function to have an inverse 0 Horizontal line test 0 Finding an inverse function 0 Graphing inverse functions 0 Domain and range of inverse functions Section 41 0 Uses of linear functions cost depreciation position 0 What does slope represent 0 What does the constant term represent 0 Find a linear function from initial data 0 Which linear functions have inverses Section 42 0 Standard form for a quadratic function y az 7 h2 k 0 Completing the square 0 Graphing Vertex Axis of Symmetry intercepts 0 Maximum or Mininum outputs and their corresponding inputs 0 Maximum or minimum outputs of a function related to a quadratic Section 44 0 Geometry Formulas for rectangles circles triangles boxes 0 Assigning variables 0 Setting up equations 0 Substitute to get one equation With one variable Section 45 0 Similar to 44 Where the corresponding function is always either quadratic or related to a quadratic 0 Use ideas from 42 to nd the maximum or minimum Section 46 amp 47 0 Finding intercepts of polynomialrational functions 0 nding vertical and horizontal asymptotes for rational functions 0 Establishing When a graph is positive negative 0 Finding Where the graph of a rational function crosses its horizontal asymptote if any 0 Graphing Section 51 amp 52 0 Simpli cation Exponent rules 0 Solve basic exponential equations 0 Characteristics of y by for b gt 0 and I f 1 Domain Range intercepts Asymptotes o shape for 12 gt1 as opposed to 0 lt I lt1 0 Graphing as per 34 o lnverses properties of inverses o e 2 2718 note 6 gt1 Section 53 Section 55 o logbz is the inverse function of by 0 Checking solutions 0 Evaluating basic logarithmic expressions 0 Use of log and exponent rules 0 Characteristics ofy 0951 for b gt 1 Domain Range lntercepts 0 Apply logarithms or exponentials to both sides to an equation or Asymptotes inequality 0 Estimating logarithms o Quadratic equation types 0 Graphing logarithms per 34 o If I gt 1 then by and logbz are both increasing in their domains Which means applying either to both sides of an inequality does not 0 Finding the domain of logarithmic functions Change the direction of the inequality SOlVing equations With logarithms o If 0 lt b lt 1 then by and log are both decreasing in their domains Section 54 Which means applying either to both sides of an inequality DOES change the direction of the inequality 0 logbPQiH 10gbPQ 7 o Solve inequalities With variables in exponents or in logarithms 10gbPn 7 0 Checking domain blogb p Section 56 10gb Eb v o Formulas for compound interest and continual compound interest log 1 i i i 0 Effective interest rate 0 Use properties to shorten or expand expressions o Doubling time 0 Solving equations Section 57 0 Finding intercepts 0 Population models 0 Changing base 0 Finding growth constants 0 Relative growth rates 0 Doubling time tripling time o Radioactive decay 0 Half Life Section 61 Section 73 0 Solving by substitution 0 zero root of a polynomial 0 Solving by Addquot39 0 t 39 Linear 39 39 o Remainder theorem and its applications 0 Word problems 0 Factor theorem and its applications 0 Possible solution sets for two linear equations and two unknowns 0 Finding all roots of a given polynomial o Graphing linear systems in two variables 0 Construct a polynomial satisfying a list of criteria Section 66 Section 74 o Substitution o The Fundamental Theorem of Algebra only applies to polynomials o Graphing nonlinear equations in two variables intersections corre 0 Linear Factors theorem spond to real solutions 0 Write a polynomial as the product of linear factors 0 No solutions how to tell o polynomials of degree n 2 l have n roots counting multiplicity 0 Converting to a linear system if possible 0 Make a polynomial satisfying given criteria if possible 0 Exponentials and logarithms o The relationship between roots and factors Section 71 o i2 71 and other computations with i 0 real and imaginary parts of a complex number 0 equality operations and complex conjugates 2 o roots of quadratics being complex Section 72 o What is a polynomial o The division algorithm 101 dzqz Rz quotient and re mainder 0 Synthetic division when you can use it


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