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# PRECALCULUS MATH 120

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This 8 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 120 at University of Washington taught by Andrew Loveless in Fall. Since its upload, it has received 182 views. For similar materials see /class/192076/math-120-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15

Math 120 Final Exam Review Checklist The nal exam in Math 120 is comprehensive You need to understand all the major concepts from the course In order to be ready for the nal exam7 you are expected to 1 Work through several of the old nal exams and come to me or the math study center when you 2 encounter dif culties Several means 5 but the more the better Look back through the homework and lectures to get an idea of the key concepts To help you with the second point above7 l have gone through the homework and listed the classic problems from each chapter that illustrate the main concepts involved Read through these problems and try to remember how you did them If you nd a few problems that were especially hard for you7 then go back and review more homework problems from that section and visit the math study center for help H E0 00 4 U 03 5 00 H H H D Chapters 1 and 2 Speed7 rates7 distance formula 187 247 26 Chapters 3 and 4 Circle7 linear models7 intersections 347 377 4 67 48 Chapters 5 and 6 Functional notation7 graphs7 multipart functions 527 637 66 Chapter 7 Quadratic modeling 767 7127 714 Chapters 87 97 107 11 Composition7 Moving Graphs7 lnverses 847 927 1047 118 Chapter 12 Linear to linear modeling 1287 1210 Chapters 13 and 14 Arc Length7 area of wedge7 angular speed 1337 1437 147 Chapter 15 and 22 Circular Motion7 trig functions 1547 15137 15187 227 Chapter 16 Know the graphs of cos7 sinz and tan along with the standard identities from class Chapter 17 and 18 Sinusoidal modeling 18107 1812 Chapters 197 20 and 21 Exponential modeling 2027 21117 2116 Chapter 23 Linear parametric Equations Know the assigned problems well Math 120 Exam I Review Autumn 2005 Coverage Chapters 1 8 In class and on this review sheet I will review some key points of the course so far However you are expected to know ALL material that we have covered up to this point The Quick and Dirty Review 1 Unit Conversion and Rates 0 Multiply by 1 and Aim for units 0 Distance Speed x Time Speed QiTS Time W N Imposing Coordinates 0 Choose an origin label all points and find all equations 0 The Distance Formula d 2 7 1 yg 7 y12 03 Circles o The circle with center h k and radius r is given by the equation x 7 h2 y 7 k2 r2 5 Linear Modeling 0 Finding the equation for a linear model 0 Slopes for parallel and perpendicular lines U I Quadratic Modeling 0 Completing the square and constructing a quadratic model 0 Finding the maximum or minimum 9 Functions 0 Know what a function is Domain Range equations for semicircles etc o Multipart Functions 0 Compositions 1 Miscellaneous Important Things to Study 0 Finding the intersection of curves 0 Organized and systematic problems solving VETS method The More Elaborate Review 1 Chapters 1 and 2 Warming Up and Imposing Coordinates 5280 ft 1 min 0 We viewed basic conversion as multiplying by 1 in a convenient way 1 mi 60 sec etc rate m ATime 0 Using the distance formula Often you are given a speed and asked for time to do so you need to use the distance formula and them time 2 distance speed 0 Use the VETS method of solve problems a Visualize Choose an origin impose a coordinate simple and label all points b Equations Write an equation for every curve in the problem c Translate Translate the question into mathematical terms involving your equations d Solve Solve for what you can 2 Chapter 3 Three Simple Curves 0 Every horizontal line is of the form y k and every vertical line is of the form x h o A circle with center h k and radius T has the equation x 7 h2 y 7 k2 r2 0 To solve when a line crosses a circle Solve for z or y in the equation for the line and replace z or y in the equation for the circle 3 Chapter 4 Linear Modeling 0 Given two points we find the equation for the line by 1 finding the slope and 2 plugging into the linear model o Slope m 2 yr 2711 0 Linear Model y mz 7 zl yl where zhyl is any points on the line 0 Parallel lines have the same slope If two lines are perpendicular there slopes are nega tive reciprocals of one another Note that we often find perpendicular lines when we are trying to find the location when an object is closest to a line 0 In this section we first used the quadratic formula when where intersection circles and lines The quadratic formula says that the equation of bx c 0 has solutions 7 7bixb274ac 7 2a 4 Chapter 5 Functions and Graphs o A function is a procedure that has a unique output for every input The set of all allowed inputs is called the domain The set of all corresponding outputs is called the range 0 We discussed the vertical line test for seeing if a graph corresponds to a function 5 Chapter 6 Graphical Analysis 0 We discussed what it means for a function to be positive and negative We also discussed what it means for a function to be increasing and decreasing o A circle is not a function but we can split it up into two function which we called the upper and lower semicircles If h k is the center of the circle and r is the radius then these functions are given by The upper semicircle y k 4 M 7 z 7 h The lower semicircle y k 7 4 r2 7 z 7 h2 0 Understand how multipart functions work Specifically given a graph you should be able to write the corresponding multipart function You should also be able to solve equations involving multipart functions Recall we solved each part separately and checked if the solution was in the appropriate domain 6 Chapter 7 Quadratic Modeling 0 Understand how to work with functions of the form f azz bx c In particular you should be able to rewrite them in the form f az 7 h2 k by completing the square Recall h k is the vertex and a will indicate if the graph opens upward or downward corresponding to a gt 0 or a lt 0 respectively 0 Be able to construct quadratic models from 3 points 0 Recognize when a question is asking for a maximum or minimum of a quadratic model and find it my finding the vertex 7 Chapter 8 Composition 0 Be able to comfortably work with composition 215 you should understand what is meant by fgz and 0 Be able to work with domains and ranges of compositions Especially when multipart functions are involved Math 120 Final Exam Review Autumn 2005 The final exam is cumulative It covers Chapters 1 23 That is it could include ANYTHING from this quarter The questions will force you to apply material from lecture the text and or the homework A good idea is to first review those topics which you struggled with this quarter It is also a good idea to look at old exam questions from Taggart and Conroy which can be found at http wwwmathwashingtonedu m120testindexhtml The website includes review sheets for the material from Chapters 1 16 This review sheet discusses some of the key points from Chapters 17 23 1 Sinusoidal Modeling and Inverse Circular Functions Ch 17 and 18 0 There are three main components to answering questions in these sections a Constructing and sketching the model b Solving equations involving the model using inverse circular functions c Interpreting your solutions ie finding other solutions 0 The general sinusoidal model is y Asz n 7 0 D A amplitude MAX Y VALUE MIN Y VALUE MAX Y VALUE MIN Y VALUE 2 D mean B period how long it takes the wave to repeat distance from peak to peak O phase shift x coordjnate of a peak 7 an x coordinate where the graph crosses the mean line and is increasing 0 Recall that when you use sz n 1z you only get the principal solution That is the solutions which is closest to the phase shift 0 0 Be able to find the symmetric solution finding the distance to the nearest peak or valley and using this to get the solution on the other side of the peak or valley 2 Exponential Modeling and Logarithms Ch 19 20 and 21 o The general exponential model is of the form y Aob which can also be written in the form y A05 Recall that the relationship is a lnb 0 Understand how to find the general exponential model when you are given two pieces of information 0 Be able to use the basic natural logarithm rules to solve equations with variables in the exponent lnab lna lnb lna 7 lnb lnbm xlnb 3 Parametric Equations and Linear Motion Ch 22 and 23 0 Understand the basic idea behind parametric equations In particular how do you sketch a graph when given parametric equations 1 0 Recall how we give parametric equations for circular motion 0 In general given the following A circle with radius r and center zc yo angular speed of w in the counterclockwise direction in units RADIANS per time A start angle of 60 The parametric equation for the circular motion is zt zc rcos00 wt yt ye 719271090 wt 0 Understand how to model motion at a constant speed on a straight line 0 In general given the following s the speed along the straight line Um the speed in the z direction 1 the speed in the y direction 1111 is the starting point and 2112 is the point reached after T units of time The parametric equations for the linear motion is zt zl vmt yt 11 vyt and we have the relationships v21 U9291 m T y T 52 1 1 i the slope of the line NE 0 Usually in linear motion problems you are given only a couple of pieces of informa tion and you have solve for the other parts Math 120 Exam II Review Autumn 2005 Coverage Chapters 9 17 In class and on this review sheet I will review some key points of the course so far However you are expected to know ALL material that we have covered in these chapters As another study tool please look at the practice exams online at http wwwmathwashingtonedu m120 testindexhtml 1 Constructing Graphs o Shift horizontally by h Replace x by z 7 h 1 rr 0 Re ect over y axis Replace x by ix 1 rr 0 Horizontal Dilation Replace x by 11 H c If c gt 1 then the graph is horizontally stretched If 0 lt c lt 1 then the graph is horizontally compressed 0 Vertical information is identical to the info above except involving y 0 Recall how to identify the order of operations 2 Arithmetic and Functions 0 You should know how to work with a function that involves more than one multipart function Le be able to find the multipart rule See the homework in Ch 10 3 Inverse Functions 0 Given a function you should be able to know the steps to find the inverse function 0 Also understand how to working with non invertible functions such as y 2 4 Rational Functions 0 Be able to find zeros vertical asymptotes and horizontal asymptotes of rational func tions 0 Be able to find a linear to linear model from a story problem 5 Angles Arc Length and Areas of Wedges 0 You should be comfortable working with degree and radian measures of angles 0 Understand and be able to use the formulas for arc length and area of a wedge 0 Recall 180 degrees 2 7r radians we use this to convert If 6 is in degrees then Arc Length s amp6r and Area of Wedge 3 26 If 6 is in radians then Arc Length s r and Area of Wedge 219 0 Review the basic angles You should be able to identify the radian measures 7r 6 7r4 7r 3 7r 2 7r 37139 2 and 27139 and what angles they represent on a circle 6 Circular Motion i change in angle 0 Angular Speed w 7 W 0 Linear Speed 2 2 W change in time 0 You should be able to work with RPM39s and you should know how to convert between angular speed and linear speed Remember you need to know the radius 0 3 key formulas which only apply when 6 is in RADIAN S and w is in RADIANS per unit time 5 r6 Arc Length radiusangle 6 wt Angle angular speed time 2 7w Linear Speed 2 radiusangular speed 0 You should be comfortable with Belt and Wheel problems 7 Circular Functions 827109 hopposite 6086 adlacent tame opposite ypotenuse hypotenuse adjacent 0 Know how to use these functions to answer questions about right triangles 0 Given a circle of radius r with the center as the origin consider the point that is obtained by rotating by an angle of 6 in standard position Then the 221 coordinates of this point are given by z 00056 and y r52716 0 Be able to use the coordinate formula to answer questions about locations on a circle even when the information given does not start in standard position 1 tan6 0 Understand how to use 20716 and 7 to find the slopes of lines with given angles 8 Trigonometric Functions 0 Be aware of the basic graphs of 52716 005 6 and 20716 0 Know the following key identities 527126 00526 1 please review how to use this notation 527176 752716 and 00576 0056 52716 52716 22m and 0056 0056 22m for 71 07 i1 i2 0056 52716 and 52716 0056 7 9 Sinusoidal Modeling 0 The general sinusoidal model is y A5271 7 0 D A amplitude MAX Y VALUE 7 MIN Y VALUE 2 MAX Y VALUE MIN Y VALUE 2 Dmean B period how long it takes the wave to repeat distance from peak to peak O phase shift x coordinate of a peak 7 an x coordinate where the graph crosses the mean line and is increasing

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