cKathryn Bollinger, February 7, 2011 1Concepts to Know #1Math 1411.1-1.5, 2.1-2.7• 1.1 - The Cartesian Coordinate Systemx and y axis (label your scales on graphs)Ordered pairs (points): (x, y)Origin: (0,0)• 1.2 - Straight LinesSlope = m =y2−y1x2−x1Positive slope: line rises from lft to rtNegative slope: line falls from lft to rtZero slope: Horizontal lineUndefined Slope: Vertical lineEquations of LinesPt-Slope Form: y − y1= m(x − x1)Slope-Int Form: y = mx + bGeneral Form: Ax + By + C = 0Horizontal line: y = aVertical line: x = bInterceptsx-intercept = point where line crosses x-axis:(#,0)y-intercept = point where line crosses y-axis:(0,#)Parallel Lines =⇒ m1= m2(same slopes/diff. y-int)Perpendicular Lines =⇒ m1m2= −1(neg. recipr ocal slopes)• 1.3 - Linear Functions and Math. ModelsFunctionsDomainRangeIndependent VariableDependent VariableLinear FunctionsLinear Depreciation: V (t) = mt + bm = rate of depreciationb = value of asset at time = 0Scrap Value = lowest value asset attainsLinear Cost, Revenue and ProfitCost: C(x) = cx + F where c is the costto pr oduce each unit and F is th e fixedcostsRevenue: R(x) = sx where s is theselling price of each unitProfit: P (x) = R(x) − C(x)Linear Supply and DemandS(x) = p = mx + bD(x) = p = mx + b**All points on supply and demandcurves are of the form (x, p) =(quantity, price)!!**• 1.4 - Intersection of Straight LinesBreak-Even PointR(x) = C(x) (or P (x) = 0 to find quantity)x = break-even quantityy = break-even revenue/costEquilibrium PointSupply = Demandx = equilibrium q uantityy = equilibrium price• 1.5 - The Method of Least Squares/LinearRegressionBe able to use LinReg on your calculator to findthe least-sq. lineCorrelation coefficient (r) - determines the amountof the data explained by the line(Want |r| close to 1)Be able to predict values, using the least-sq. linecKathryn Bollinger, February 7, 2011 2• 2.1 - Systems of Linear EquationsTwo linear eqns =⇒ three casesUnique Soln (intersecting lines =⇒ m16= m2)No Soln (parallel lines =⇒m1= m2and b16= b2)Infinitely Many S olns (same line =⇒m1= m2and b1= b2)General soln (parametric soln)Specific solnsSetting up systems of equationsDEFINE YOUR VARIABLES!• 2.4 - MatricesSize (dimension): mxn (m = # rows, n = #columns)Matrix elements: aij(element in row i and col j)Equality =⇒ all corresponding entries equalAddition/Subtraction (matrices must be the samesize)Matrix Trans pose (AT): switch rows and colsScalar Multiplication: multiply every entry by thescalar• 2.2/2.3 - Solving System of EquationsGauss-Jordan (GJ) Elimination (Row Operations)Interchange any two equationsMultiply an eqn by a non-zero constantAdd a multiple of one eqn to anotherRow-Reduced Echelon FormAll zero rows must be below all non-zero rowsThe first non-zero entry in each row is 1 (lead-ing 1)In any two success ive (non-zero) r ows, theleading 1 in the lower row lies to the rightof the leading 1 in the upper rowIf a column contain s a leading 1, then theother entries in that column are zerosRREF on your calculatorSolving a s y stemPut system in “nice” formPlace “nice” system in an augmented matrixUse RREF on you r calculator to r educe yoursystemRead system (eqns) from reduced systemFind soln• 2.5 - Multiplication of MatricesThe # of cols in the left matrix must equal the #of rows in the right m atrix. (Inner dimensionsequal.)If A is (mxn) and B is (nxp), then AB is (mxp).(Outer dimensions give ans wer size.)ORDER IS IMPORTANT!Know how to mu ltiply matrices by hand.Identity Matrix: Square matrix (# rows = # cols)with 1’s along the diagonal (from upper lft tolower right) and 0’s elsewhereBe able to rep resent a system of eqns as a matrixeqn: AX = BPut system in “nice”formA = coefficient m atrixX = variable m atrixB = constant matrix• 2.6 - Inverse of a Square MatrixMatrix must be squareNot all matrices have an inverse (“singular” meansno inverse)Inverse of A = A−1AA−1= A−1A = IA system in form AX = B =⇒ X = A−1B ifthe inverse exists (know how to solve a matrixequation)• 2.7 - Leontief Input-Output ModelBe able to set-up the input-output matrix, AUnderstand how to read each element in the input-output matrixKnow how to find how much of each prod-uct/service should be produced to satisfy de-mand (Be able to solve X = AX + D for X,knowing w hat each of these matrices means)Know how to find how much of each prod-uct/service is consumed internally by an econ-omy wh ile meeting
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