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1、Jim Jack (J)MATH 1314 - College AlgebraAdminClassGradingAttendance1.1 Numbers, Data, and Problem SolvingNatural NumbersIntegersRationalIrrationalPEMDAS1. Parenthesis2. Exponents3. Multiplication and Division, left to right4. Addition and Subtraction, left to rightScientific NotationA real number r i
2、s in scientific notation when r is written as , where and n is an integer.Dist to sun = 93,000,000Pop of world in 2050 = 9,000,000,000Light travels in 1 mile = Mass of the earth kgCalculator ExerciseFind the speed the earth travels in space relative to the sun in miles per hour.Find the volume of a
3、soda can with radius 1.4” and height 5”. Can it hold 16oz?1.2 Visualizing and Graphing DataLow temperatures in Minneapolis for six nights-12-4-821189maximum, minimum, mean, medianA relation is a set of ordered pairs.Portland rainfall1234567891011126.23.93.62.32.01.50.51.11.63.13.26.4Domain=Range=202
4、040401.21.11.51.6Domain=Range=Portland rainfall1234567891011126.23.93.62.32.01.50.51.11.63.13.26.4Distance formulaDistance between two moving cars100 50 KC50100Linear ApproximationMidpoint formulaThe midpoint of a line segment with endpoints and in the xy plane isPopulation of the USIn 1990, the pop
5、ulation of the US was 249 million, and in 2010, it was 308 million.Find the population in 2000.A circle is the set of points equidistant from a center point.Graph Finding the equation of a circlea) Radius = 4, Center is b) center with a point on the circle A line is drawn from the point to the point
6、 . State the equation of a circle for which this line is the circles diameter.Completing the squareskipFind the center and the radius of the circle:1.3 Functions and their representationsThunder and lightning speed of sound=1050 ft/sec51015202512345Rule: Compute y by dividing x by 5meaningful input
7、domaincorresponding output rangeVerbal: Divide x by 5 to obtain y milesNumerical:12345670.20.40.60.81.01.21.4Diagrammatic:51awAx102bxBy153cyCz204zDsymbolic: Leonhard Euler(1707-1783)graphical:Graph A function is a relation in which each element in the domain corresponds to exactly one element in the
8、 range. exactly one y function (rule for finding y)Ordered pairs DomainRangeLet DomainDomainRangeFind domain and range graphicallyAir cools at a rate of 3.6 F per 1,000 ft altitudeFigure for , where x is in thousands ft.Verbal:Symbolic:Graphical:Numerical:Crutch length trial and error or (65” height
9、) skip exSymbolic:Graphical:Numerical:Identifying functionsVertical line test1.4 Types of Functions and Their Rates of ChangeDescribe datamake predictionsLinearNon-linearConstantWindspeed at Hilo, HIMayJuneJulyAugSeptOctNovDec77777777f=A function represented by , where b is a fixed number, is a cons
10、tant function.Discrete functionContinuous functionOther constant fcns Thermostat, speed controlLinear functionsA car is initially located 30 miles N of the Red River on I-35 traveling northbound at 60mph.0123453090150210270330f=A function represented by , where a and b are fixed, is a linear functio
11、n.Wages earned at $8.25/hrTuition at $75/credit hr + $55 student feeDistance traveled by light at 186,000 miles/sec.Slope as a rate of changeThe slope of the line passing through the points and isPositive slopeNegative slopeNo slopeFind the slope of a line through and Songs that can be stored on x G
12、B of Ipod memory20,000songs 80GbPass thru origin?Slope?Rate of change?Non-linear functionsx01234f(x)014916average temperature in each monthheight of a child at age 2 through 18Linear and non-linear dataskip05101520-4-2024-30369571014190123411111111110136103691215Recognizing linear functionsIncreasin
13、g and Decreasing FunctionsSuppose f is a function defined over interval I.For :a) f increases on I if b) f decreases on I if Interval NotationAverage rate of change, non-linear functionrate of change from x = 1 to x = 3rate of change of US pop, 1800 1840, 1900 1940YearPopulation180051840171900761940
14、132The difference quotientFind first, then difference quotient.Ch 2 Linear Functions and Equations2.1 Equations of linesPoint-slope formx intercept , y intercept Slope passing through Equation of a line passing through , Alternate method slope-intercept formEquation of a line passing through , Apple
15、 sold 55m iPods in 2008, and 43m in 2011 Determine x and y interceptsEquations of horizontal and vertical linesAn equation of a horizontal line with y intercept is . An equation of the vertical line with x intercept is .Vertical and horizontal lines through Finding parallel linesTwo lines with slope
16、s and , neither of which are vertical, are parallel if and only if .Find the equation of a line parallel to the line , passing through .Finding perpendicular linesTwo lines with slopes and , neither of which are vertical, are perpendicular if and only if their slopes have product , i.e.Find the equa
17、tion of a line perpendicular to the line , passing through .Determining a rectangleInterpolation and ExtrapolationIn 2005, about $350M was spent on US musical downloads. This amount reached $1350M in 2010. Model, find value in 2008, 2003.20032005200820103501350Investments in cloud computingYear20052
18、006200720082009Investments26113195299374Model the dataEstimate Investments in 20142.2 Linear EquationsEquation statement that two expressions are equalSolve find variable value(s) that make stmt trueSolution set set of the values that make it trueEquivalent two stmts with same solution setContradict
19、ion equation with no solution setIdentity equation with infinite solution setA linear equation in one variable is an equation that can be written in the form(an equation that is not linear is called non-linear)Addition propertyMultiplication property Solving linear equations symbolicallySolving line
20、ar equations symbolicallyEliminating fractions and decimalsThe linear function estimates iPod sales (million) during year x. When will sales reach 27 million?Contradictions, Identities, Conditional EquationsAn equation with one expression on each side of the = sign can be solved by the intersection
21、of graphs method.Using a graphing calculator; 1. Set equal to the left side, and equal to the right side of the equation.2. Graph both on one set of coordinates3. Points of intersection are ordered pairs that make both sides true at the same time. These ordered pairs solve the equation.The market sh
22、are of music on CDs held from 1987 to 1998 could be modeled by . During the same period, cassette tape sales could be modeled by . (x=0 indicates year 1987)Percentages: Suppose 76% of all bicycle riders do not wear helmets. There are 38.7 million riders that do not wear helmets, Find total ridership
23、.Solving for a variableProblem Solving1. Read problem, understand it. Assign variable to what you must find. Write other quantities in terms of this variable.2. Model it, write the equation. Diagram it if necessary.3. Solve the equation, state solution4. Check your work.Work problemsA pump can empty
24、 a gasoline tank in 5 hours. A smaller pump can empty the same tank in 9 hours. How long will it take both pumps working together to empty the tank?Motion problemsIn one hour, an athlete runs 10.1 miles by running some at 8mph, the rest at 11 mph. How long did the athlete run at each speed?Similar t
25、rianglesA person 6 feet tall stands 17 feet from a streetlight, and casts an 8 ft shadow. Estimate the height of the streetlight.MixturePure water is being added to 153 ml of a solution of 30% hydrochloric acid. How much pure water should be added to dilute to 13%?2.3 Linear InequalitiesSizing appro
26、ximateMax takeoff weight of aircraftInterstate speedA linear inequality in one variable is an inequality that can be written in the form:, .Properties of inequalitiesLet a, b, and c represent real numbers1. and are equivalent.2. If , then and are equivalent.3. If , then and are equivalent.Solve symb
27、olicallySolve symbolicallySolve graphicallyThe daily payment processing for the company Square grew from $1M in March 2011 to $11M in March 2012. Model this growth and find when daily volume was $8.5M or less.x intercept methodNumerical SolutionsLet cost of manufacturing be and revenue be expressed
28、by . Profit is expressed by revenue minus cost.Boundary number (break even point)x171819202122230102030Solve numerically.x1234567y181383-2-7-12x4.34.44.54.64.74.84.9y1.51.50-.5-1-1.5Compound inequalitiesSpeed limit on Interstate 35Sunset in BostonIn Boston, the sun set at 7PM on the 82nd day (22 Mar
29、), and at 8PM on the 136th day (15 May).Find the days when the sun set between 7:15PM and 7:45PM. (do not include fall dates).SymbolicGraphical 2.4 More Modeling with FunctionsTo model a quantity that is changing at a constant rate, with, f(x)=(const rate of chg)x + (initial value)Model each situati
30、on and state the domain:(a) In 2011, the average cost of attending a public college was $8200, and it is projected to increase, on average, by $600 per year until 2014.(b) A cars initial speed is 50 mph, then it begins to slow down at a constant rate of 10 mph each second.A 100 gallon tank is initia
31、lly full of water and being drained at 5 gallons per minute.formula?How much water is in the tank after 4 minutes? emissionsx miles240360680800y pounds150230435510Slopes? Linear? Piecewise defined functionsFirst-Class Mail Rates First ounce $0.46Each additional ounce $0.20First-Class Mail Rates Weig
32、ht notover(ounces) Rate 1 0.46 2 0.66 Domain?3 0.86 4 1.06 Range?TornadoFujita Scale Housing startsYear2000200520112012Homes1.31.70.40.5Graph, interpret, continuous? modelEvaluating a piecewise-defined functionPiecewise defined functionDomain?Continuous?Greatest Integer Function, the greatest intege
33、r xDirect variationy is directly proportional to x, or y varies directly with x, if there is a non-zero number such thatThe number k is called the constant of proportionality or the constant of variation.Wages $57.75 for 7 hours workSuppose T varies directly with x, and that when . Find T when .A 12
34、-pound weight is hung on a spring and it stretches 2 inches.Find the spring constant (const of var)How far will it stretch with a 19lb weight on it?Megabytes needed for y sec of musicx (MB)0.230.491.161.275.00y (sec)10.722.855.260.246.546.547.647.4Computing for each entry 3rd row, approximate consta
35、nt of proportionality.Using constant of proportionality, how much music would a FD hold?2.5 Absolute Value Equations and InequalitiesAbsolute value functionPrimary square root If , then .Absolute Value EquationsFor any positive number k, is equivalent to GraphicalNumericalTblSet SymbolicInterstate s
36、peed limits: Two absolute valuesAbsolute Value InequalitiesLet the solutions to be and , with , and .1. is equivalent to .2. is equivalent to .Absolute Value Inequalities (Alternative Method)Let k be a positive number.1. is equivalent to 2. is equivalent to Error tolerances on iPhone 4sThe iPhone 4s
37、 is 4.5” high. Suppose actual height A of any particular phone has an error tolerance of 0.005”.Model and quantifyCh 3 Quadratic Functions and Equations3.1 Quadratic Functions and ModelsLinearLet a, b, and c be real numbers with . A function represented by is a quadratic function.Domain is , leading
38、 coefficient is a.Graph parabolaVertex, Axis of Symmetry Standard formVertex formThe parabolic graph of with has vertex . Its graph opens upward when , and opens downward when .Vertex form from graphConverting to standard formConverting to vertex form The vertex of the graph with is the point .Find
39、the vertex of .Symbolic, Graphical, NumericalNote: for axis of symmetry ,Converting to vertex form:Graph A farmer is fencing a rectangular area for cattle using the straight portion of a river as one side of the rectangle. If the farmer has 2400 feet of fence, find the dimensions of the rectangle that will give the maximum area for the cattle.Models and Applications A