Review Sheet for MATH 124 at UA
Review Sheet for MATH 124 at UA
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WileyPLUS MATH 124 129 223 5th ed WileyPLUS Home Hel Contact us Loout HughesHallett Calculus Single and Multivariable 5e Chapter 12 Functions of Several Van39ables 39 l sun we HI WU v Reading content D121 Functions of Two Variables D 122 Graphs of Functions of Two Variables D 123 Contour Diagrams D 124 Linear Functions D 125 Functions of Three Variables D 126 Limits and Continuity DChapter Summary Review Exercises and Problems for Chapter Twelve D Check Your Understanding D Projects for Chapter Twelve El Student Solutions Manual p Graphing Calculator Manual p Focus on Theory p Web Quizzes i i i CHAPTER Functions of Several Variables 12 121 Functions of Two Variables Function Notation Suppose you want to calculate your monthly payment on a veyear car loan this depends on both the amount of money you borrow and the interest rate These quantities can vary separately the loan amount can change while the interest rate remains the same or the interest rate can change while the loan amount remains the same To calculate your monthly payment you need to know both If the monthly payment is m the loan amount is ESL and the interest rate is r then we eXpress the fact that m is a function of L and r by writing m f L if This is just like the function notation of onevariable calculus The variable m is called the dependent variable and the variables L and r are called the independent variables The letter f stands for the function or rule that gives the value of m corresponding to given values of L and r A function of two variables can be represented graphically numerically by a table of values or algebraically by a formula In this section we give examples of each leClDocuments20and20ettingsmthDesktopindex uni him 1 of 198262009 82002 AM WileyPLUS Graphical Example A Weather Map Figure shows a weather map from a newspaper What information does it convey It displays the predicted high temperature T in degrees Fahrenheit OF throughout the US on that day The curves on the map called isotherms separate the country into zones according to whether T is in the 60s 70s 80s 90s or 100s so means same and therm means heat Notice that the isotherm separating the 80s and 90s zones connects all the points where the temperature is exactly 900F 603 male 05 50 Bols ms 705 S I 805 cm 05 Figure 121 Weather map showing predicted high temperatures T on a summer day leClDocuments20and20ettingsmthDesktopindex uni htm 2 of 198262009 82002 AM WileyPLUS Example 1 Estimate the predicted value of T in Boise Idaho Topeka Kansas and Buffalo New York Solution Boise and Buffalo are in the 70s region and Topeka is in the 80s region Thus the predicted temperature in Boise and Buffalo is between 70 and 80 while the predicted temperature in Topeka is between 80 and 90 In fact we can say more Although both Boise and Buffalo are in the 70s Boise is quite close to the T 70 isotherm whereas Buffalo is quite close to the T 80 isotherm So we estimate the temperature to be in the low 70s in Boise and in the high 70s in Buffalo Topeka is about halfway between the T 80 isotherm and the T 90 isotherm Thus we guess the temperature in Topeka to be in the mid 80s In fact the actual high temperatures for that day were 710F for Boise 79oF for Buffalo and 86oF for Topeka The predicted high temperature T illustrated by the weather map is a function of that is depends on two variables often longitude and latitude or miles eastwest and miles northsouth of a xed point say Topeka The weather map in Figure is called a contour map or contour diagram of that function Section m shows another way of visualizing functions of two variables using surfaces Section E looks at contour maps in detail Numerical Example Beef Consumption Suppose you are a beef producer and you want to know how much beef people will buy This depends on how much money people have and on the price of beef The consumption of beef C in pounds per week per household is a function of household income I in thousands of dollars per year and the price of beef p in dollars per pound In function notation we write if I f U 1 I Table 121 contains values of this function Values of p are shown across the top values of I are down the left side and corresponding values of f I p are given in the tablel For example to nd the value leClD0cuments20and20ettingsmthDesktopindex uni hlm 3 of 198262009 82002 AM WileyPLUS of f 40 350 we look in the row corresponding to I 40 under p 350 where we nd the number 405 Thus 40 EU 4 13 This means that on average if a household s income is 40000 a year and the price of beef is 350 lb the family will buy 405 lbs of beef per week Table 121 Quantity ofBeefBought PoundsH ousehold Week Price of beef lb 300 350 400 450 20 265 259 251 243 Household 40 414 405 394 388 Income 60 511 500 497 484 per year I 1000 80 535 529 519 507 100 579 577 560 553 Notice how this differs from the table of values of a onevariable function where one row or one column is enough to list the values of the function Here many rows and columns are needed because the function has a value for every pair of values of the independent variables Algebraic Examples Formulas In both the weather map and beef consumption examples there is no formula for the underlying function That is usually the case for functions representing reallife data On the other hand for many idealized models in physics engineering or economics there are exact formulas leClDocuments20and20ettingsmthDesktopindex uni htm 4 of 198262009 82002 AM WileyPLUS Example 2 Give a formula for the functionM f B t where M is the amount of money in a bank account I years after an initial investment of B dollars if interest is accrued at a rate of 5 per year compounded annually Solution Annual compounding means thatM increases by a factor of 105 every year so 7 a r M j alllJ Example 3 A cylinder with closed ends has radius r and height k If its volume is Vand its surface area is A find formulas for the functions V f r h and A gr h Solution Since the area of the circular base is TEr2 we have 39 r 2 Area of39basa Height 2 The surface area of the side is the circumference of the bottom 27V times the height h giving 275 Thus A 50quot 32 Area of39base fl of39side L A Tour of 3Space In Section 122 we see how to visualize a function of two variables as a surface in space Now we see how to locate points in threedimensional space 3space leClD0cuments20and20ettingsmthDesktopindex uni htm 5 of 198262009 82002 AM WileyPLUS Imagine three coordinate axes meeting at the origin a vertical axis and two horizontal axes at right angles to each other See Figure Think of the xyplane as being horizontal while the zaxis extends vertically above and below the plane The labels x y and 2 show which part of each axis is positive the other side is negative We generally use right handed axes in which looking down the positive zaxis gives the usual view of the xyplane We specify a point in 3space by giving its coordinates x y z with respect to these axes Think of the coordinates as instructions telling you how to get to the point start at the origin go x units along the xaxis then y units in the direction parallel to the yaxis and nally 2 units in the direction parallel to the zaxis The coordinates can be positive zero or negative a zero coordinate means don t move in this direction and a negative coordinate means go in the negative direction parallel to this axis For example the origin has coordinates 0 0 0 since we get there from the origin by doing nothing at all Figure 122 Coordinate axes in threedimensional space Example 4 Describe the position of the points with coordinates l 2 3 and 0 0 l Solution We get to the point 1 2 3 by starting at the origin going 1 unit along the xaxis 2 units in the direction parallel to the yaxis and 3 units up in the direction parallel to the z axis See Figure 123 leClDocuments20and20ettingsmthDesktopindex uni htm 6 of 198262009 82002 AM WileyPLUS Figure 123 The pointl 2 3 in 3space To get to 0 0 l we don t move at all in the x and y directions but move 1 unit in the negative 2 direction So the point is on the negative z aXis See Figure 124 You can check that the position of the point is independent of the order of the x y and z displacements I 39 I39J mu 1 7 Figure 124 The point 0 0 l in 3space leClDocuments20and20ettingsmthDesktopindex uni htm 7 of 198262009 82002 AM WileyPLUS Example 5 You start at the origin go along the yaXis a distance of 2 units in the positive direction and then move vertically upward a distance of 1 unit What are the coordinates of your nal position Solution You started at the point 0 0 0 When you went along the yaXis your ycoordinate increased to 2 Moving vertically increased your z coordinate to 1 your xcoordinate did not change because you did not move in the x direction So your nal coordinates are 0 2 1 See Figure Q Figure 125 The point 0 2 1 is reached by moving 2 along the yaXis and 1 upward It is often helpful to picture a three dimensional coordinate system in terms of a room The origin is a corner at oor level where two walls meet the oor The z aXis is the vertical intersection of the two walls the x and yaxes are the intersections of each wall with the oor Points with negative coordinates lie behind a wall in the next room or below the oor Graphing Equations in 3Space We can graph an equation involving the variables x y and z in 3space such a graph is a picture of all points x y 2 that satisfy the equation leClD0cuments20and20ettingsmthDesktopindex uni him 8 of 198262009 82002 AM WileyPLUS Example 6 What do the graphs ofthe equations 2 0 z 3 and z 1 look like Solution To graph 2 0 we visualize the set of points whose zcoordinate is zero If the z coordinate is 0 then we must be at the same vertical level as the origin that is we are in the horizontal plane containing the origin So the graph of z 0 is the middle plane in Figure The graph of z 3 is a plane parallel to the graph of z 0 but three units above it The graph of z 1 is a plane parallel to the graph of z 0 but one unit below it Figure 126 The planesz 1 z 0 andz 3 The plane 2 0 contains the x and ycoordinate axes and is called the xyplane There are two other coordinate planes The yzplane contains both the y and the zaxes and the xzplane contains the x and zaxes See Figure 127 leClDocuments20and20ettingsmthDesktopindex uni htm 9 of 198262009 82002 AM WileyPLUS l ru plane Jlrpla e Luzplane Figure 127 The three coordinate planes Example 7 Which ofthe pointsA l l 0 B 0 3 4 C 2 2 l andD 0 4 0 lies closest to the xzplane Which point lies on the yaXis Solution The magnitude of the ycoordinate gives the distance to the xzplane The pointA lies closest to that plane because it has the smallest ycoordinate in magnitude To get to a point on the yaXis we move along the yaXis but we don t move at all in the x or 2 directions Thus a point on the yaXis has both its x and z coordinates equal to zero The only point of the four that satis es this is D See Figure leClDocuments20and20ettingsmthDesktopindex unihtm 10 0f198262009 82002 AM WileyPLUS Figure 128 Which point lies closest to the xzplane Which point lies on the yaXis In general if a point has one of its coordinates equal to zero it lies in one of the coordinate planes If a point has two of its coordinates equal to zero it lies on one of the coordinate axes Example 8 You are 2 units below the xyplane and in the yzplane What are your coordinates Solution Since you are 2 units below the xyplane your zcoordinate is 2 Since you are in the yz planeyour xcoordinate is 0 your ycoordinate can be anything Thus you are at the point 0 y 2 The set of all such points forms a line parallel to the yaXis 2 units below the xyplane and in the yzplane See Figure 129 leClDocuments20and20ettingsmthDesktopindex uni htm 11 of 198262009 82002 AM WileyPLUS Figure 129 The line x 02 2 Example 9 You are standing at the point 4 5 2 looking at the point 05 0 3 Are you looking up or down Solution The point you are standing at has z coordinate 2 whereas the point you are looking at has z coordinate 3 hence you are looking up leClDocuments20and20ettingsma1hDesktopindex uni htm 12 of 198262009 82002 AM WileyPLUS Example 10 Imagine that the yzplane in Figure 127 is a page of this book Describe the region behind the page algebraically Solution The positive part of the xaXis pokes out of the page moving in the positive x direction brings you out in front of the page The region behind the page corresponds to negative values of x so it is the set of all points in 3space satisfying the inequality x lt 0 Distance between Two Points In 2space the formula for the distance between two points x y and a b is given by Distance Iquot x a l y 331 The distance between two points x y z and a b c in 3space is represented by PG in Figure 1210 The side PE is parallel to the xaXis EF is parallel to the yaXis and F G is parallel to the z aXis leClDocuments20and20ettingsmthDesktopindex uni htm 13 of 198262009 82002 AM WileyPLUS Figure 1210 The diagonal PG gives the distance between the points x y z and a b C Using Pythagoras theorem twice gives GSquot FF FE 293313 Eisj I gFGja39d39 x az it y t 2 quot39 Thus a formula for the distance between the points x y z and a b c in 3space is r 39 s 1 s a Distance 2 I39I r vx a 1 as C Example 11 Find the distance between 1 2 1 and 3 1 2 Solution Distance p39llj311 277 1 232 7 2 1quot 2 139 1E 424 Exam ple 1 2 Find an expression for the distance from the origin to the point x y 2 Solution The origin has coordinates 0 0 0 so the distance from the origin to x y z is given by Distance 2 I39lll 032 4 quot3 1312 112 2 Illquot2 l 33 l a leClD0cuments20and20ettingsmthDesktopindex uni htm 14 of 198262009 82002 AM WileyPLUS Exam pie 1 3 Find an equation for a sphere of radius 1 with center at the origin Solution The sphere consists of all points x y 2 whose distance from the origin is 1 that is which satisfy the equation I39llxz ya Isa This is an equation for the sphere If we square both sides we get the equation in the form x2 l39 Jig 7 3922 Note that this equation represents the surface of the sphere The solid ball enclosed by the sphere is represented by the inequality x2 y2 22 S l Exercises and Problems for Section 121 Exercises 1 Which ofthe pointsA 23 92 48 B 60 0 0 C 60 l 92 is closest to the yzplane Which lies on the xzplane Which is farthest from the xyplane 2 Which ofthe pointsA 13 27 0 B 09 0 32 C 25 01 03 is closest to the yz plane Which one lies on the xzplane Which one is farthest from the xyplane 3 Which ofthe pointsP l 2 l and Q 2 0 0 is closest to the origin 4 Which two ofthe three points P1 l 2 3 P2 3 2 l and P3 l l 0 are closest to each other 5 You are at the point 3 l l standing upright and facing the yzplane You walk 2 units forward turn left and walk another 2 units What is your nal position From the point of View of an observer looking at the coordinate system in Figure m are you in front of or behind the yzplane To the left or to the right of the xzplane Above or below the xyplane leClD0cuments20and20ettingsmthDesktopindex uni him 15 of 198262009 82002 AM WileyPLUS 6 You are at the point 1 3 3 standing upright and facing the yzplane You walk 2 units forward turn left and walk for another 2 units What is your nal position From the point of view of an observer looking at the coordinate system in Figure m are you in front of or behind the yzplane To the left or to the right of the xzplane Above or below the xyplane Sketch graphs of the equations in Exercises 1 and 2 in 3space 7 x 3 8 y l 9 z 2 and y 4 10 Find the equation of the sphere of radius 5 centered at the origin 11 Find the equation of the sphere of radius 5 centered at l 2 3 12 Find the equation of the vertical plane perpendicular to the yaxis and through the point 2 3 4 Exercises Q E and Q refer to the map in Figure 121 13 Give the range of daily high temperatures for a Pennsylvania b North Dakota c California 14 Sketch a possible graph of the predicted high temperature T on a line northsouth through Topeka 15 Sketch possible graphs of the predicted high temperature on a northsouth line and an eastwest line through Boise For Exercises E u and refer to Table 12 l wherep is the price ofbeefand is annual household income 16 Give tables for beef consumption as a function of p with I fixed at 20 and I 100 Give tables for beef consumption as a function of I with p xed at p 300 and p 400 Comment on what you see in the tables 17 Make a table of the proportion P of household income spent on beef per week as a function of price and income Note thatP is the fraction of income spent on beef 18 How does beef consumption vary as a function of household income if the price of beef is held constant leClDocuments20and20ettingsmthDesktopindex uni htm 16 of 198262009 82002 AM WileyPLUS Problems 19 The temperature adjusted for windchill is a temperature which tells you how cold it feels as a result of the combination of wind and temperature See Table 122 Table 122 Temperature Adjustedfor Wind Chill 0 F as a Function osz39rzdSpeed and Temperature Temperature OF 35 30 25 20 15 10 5 0 5 31 25 19 13 7 1 5 11 Wind 10 27 21 15 9 3 4 10 16 Speed 15 25 19 13 6 0 7 13 19 mph 20 24 17 11 4 2 9 15 22 25 23 16 9 3 4 11 17 24 a If the temperature is 00F and the wind speed is 15 mph how cold does it feel b If the temperature is 350F what wind speed makes it feel like 24oF c If the temperature is 250F what wind speed makes it feel like 12oF d If the wind is blowing at 20 mph what temperature feels like 00F 20 Using Table 122 make tables of the temperature adjusted for windchill as a function of wind speed for temperatures of 200F and 00F 21 Using Table 122 make tables of the temperature adjusted for windchill as a function of temperature for wind speeds of 5 mph and 20 mph 22 The balance B in dollars in a bank account depends on the amount deposited A dollars the annual interest rate r and the time t in months since the deposit so B f A r t a Is fan increasing or decreasing function of A Of r Of t b Interpret the statement f 1250 1 25 z 1276 Give units leClDocuments20and20ettingsmthDesktopindex uni htm 17 of 198262009 82002 AM WileyPLUS 23 24 25 26 27 28 29 30 31 32 33 The monthly payments P dollars on a mortgage in which A dollars were borrowed at an annual interest rate of r for t years is given by P f A r I Is fan increasing or decreasing function of A Of r Of t A car rental company charges 40 a day and 15 cents a mile for its cars a Write a formula for the cost C of renting a car as a function f of the number of days d and the number of miles driven m b IfC fd m ndf5 300 and interpret it The gravitational force F nevvtons exerted on an object by the earth depends on its mass m kilograms and its distance r meters from the center of the earth so F f m r Interpret the following statement in terms of gravitation f 100 7000000 z 820 Consider the acceleration due to gravity g at a height h above the surface of a planet of mass m a If m is held constant is g an increasing or decreasing function of h Why b If h is held constant is g an increasing or decreasing function of m Why A cube is located such that its top four comers have the coordinates l 2 2 l 3 2 4 2 2 and 4 3 2 Give the coordinates of the center of the cube Describe the set of points whose distance from the xaXis is 2 Describe the set of points whose distance from the xaXis equals the distance from the yzplane Find a formula for the shortest distance between a point a b c and the yaXis Find the equations of planes that just touch the sphere x 22 y 32 z 32 l6 and are parallel to a The xyplane b The yzplane c The xzplane Find an equation of the largest sphere contained in the cube determined by the planes x 2 x 6 y5y9andzlz3 Which ofthe points P1 3 2 15 P2 0 10 0 P3 6 5 3 and P4 4 2 7 is closest to P 6 0 4 leClDocuments20and20ettingsmthDesktopindex uni htm 18 of 198262009 82002 AM WileyPLUS 34 3 Find the equations of the circles if any where the sphere x l2 y 32 z 22 4 intersects each coordinate plane b Find the points if any where this sphere intersects each coordinate aXis 35 A rectangular solid lies with its length parallel to the yaXis and its top and bottom faces parallel to the plane 2 0 Ifthe center ofthe object is at l l 2 and it has a length of 13 a height of5 and a width of 6 give the coordinates of all eight corners and draw the gure labeling the eight corners leClDocuments20and20ettingsmthDesktopindex uni htm 19 of 198262009 82002 AM
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