Basic Matrix Operations Seion 1 - Video A G2 G. C. 7 [2 Gz m = 34 n 6 15 Notation rows columns = m + n letters Capital 2x3 Practice Problem A3xz = are 1 X -X X 3x5 5 > - A = 2 x zc = 3x5 = 1x5D = 1x1 to in given give a name A = a +2 method B to I an an na ↑ an Matrices 2x2 used a ij matrices i = now jecolumn a us describe convenient information table ↓ ↓ ↓ e Matrix transportation 2e + ·= 17 f A = ( Rows columns-goods Let is customers - A a the matrix of transpose order of (1xn) Vector column rector its Al(OA) is 13 > - , 2 , 1 , 6) January (mrb) ( · · A + B = A-A = to add - = sales February A + B B + A 0 (zero to 15551 (ii) B= sales Remks · num . Addition & substraction Special cases Row Then order man . and A = (550) matrix) matrices , it is necessary for them to have the same order . Scal multiplication 10 3) a = Suppose year sales sales same the are be will each Je by given . month A = . Then whole the , B 1 3 YI B = Matrix multipleication Remks · o · A A . 0(zero = (A + B) (2 A) a · A = . (n = - matrix) 1) Amxs Let AB + A compete Born and B . = C Then . where , A , it is c = possible to matrix man EXAMPLE : Remks A(B · (A · + C) B)C + A = 2x3 AB + AC = Ac + = · ( 200 04 · 3 + I =312 B 4 BC (BC) (AB)C AB BA A · I 5418 ↑ = A2x3 Bixn · Cara = > - (Cos Co2 Ca S Seion 1 - Lecture A is matrix simply a rectangular of array considered numbers as one mathematical . object When there · as · If m = m=n , In a and n matrix has the in the array we , so that the same number of columns square matrix from the , top the left elements law) to an the , azz , . . . , bottom constitute right I = A 3x22x23x2 I . A = A 3x33x23x2 · /15 : :) the (ann) . EXAMPLE . m-by-n an matrix (written n) (running A have and rows , it is called a . matrix square · rows rows are m main diagonal of the matrix Seion 1 - Exercises QUESTION Fit - Hip (three gyms 20 % retains -Gymie - 1 retains 10% retains its of the 20 % and Hip to (0 45 rivals 40 % and toF i t Gymie to moving equally rest moving 50 % %, with members lossing 10 %, Gymie Fit, : and rest to Hip (0 6) . each) . Gymil to 1000 x (current members) snow how work to 1) Coefficients out next Matrix of year's 2) Matrix A 0 . 20 0 20 0 . 45 . . 0 0 . . 800 mult years Ax = b through for from members of Fit now 50 mmult(Aix) > - . 0 45 0 60 . 0 40 10 -200 > - membership 2 0 1T 11000 , 1200 , 800 = . (year 1) 10 mmult(A ; year 1) 140 648 1426 > - 1220 780 1094 QUESTION Z A = 3x2 A = B = , 2 x 3 2x3 B' = 3x2 , , , QUESTION 3 profits per unit C = C 3x3 = 3x3 /Frames LTD : XA + BV(xC vCXA ↓ VANB VB'x Al V3XA reputed unit per hard disks >PF/Y(mt0 b) 140 500 -00a 1000 = - (+x costs AXA Y > c - = brand , Ma ( + x 3) M3 M2 120 , M2 3 product sold 150 Q valid mult but no econ sense VQ'XP' -Q X QXP X QXP' ~ Q'x P X QXP has gives is a gives no the profit per valid mult the profit econ sense V region a sum be singular (mdeterm model per is cannot = a R3 400 150 M2 200 50 50 M3 58 (3x3) M4 M2 108 200 50 calculated R2 400 50 200 matrix R3 150 50 75 of Qxp - total profits valid mult that gives profit per model X Qx(P-C) > revenue per region - 20075 R1 - 0) of elements XQXP is R2 + 00 (3x3) statements : is RX 7 M3 a VQXP portfolio M1 Quantities , M8 w BX A M3 Determinants + Matrix Inversion Seion 2 - Video Sque matrix A - matrix and Determinants I 95014 Identity matrix (sque) po = I1 = · n . = · Let A · There A (I u)(t) = - Inl non-singular is = another only US = (7 + z) 20 - 25 square (1x 4) - - ( - 4) NON-SINGULAR 9 If IAI = 0 - > A is a b) If 1A1 +0 - > A a is singular matrix non-singular matrix (square) A. matrix (inverse of A I find to it , EXCEL use Inverse matrix and systems of line equations < + for matrices (a)-(b. = - SINGULAR & X rows 1 A) A 1 A-1 . = (90) main magora (A) A= elements all + det The inverse matrix · of O 3x3 1 number same columns I· ( A= has the that yy + zz = 2x + 6y + 52 8x + 9y+ 52 - = = 3 10 1 AX A = B- Any system I su I = 8 3x3 X of = linear equations I -I 3 XX B can be written I ( = I 3x1 like so Inverse matrix find to ". A AX INVERSE > x- = Of A A-%.B V I- IX > A = - " solved we B the using X A - = - B the SLE the inverse of coefficient matrix Excel formulas · · = mmult = transpose · = minverse · = mdeterm Seion 2 - Lecture · Determinants can calculated be only of square matrices EXAMPLE 124) A = (42) B = & 11 = + + u - = 00 ↑ I 1B1 + y = 14 - changes the sign the but 10 - = diagonal switching value stays the same Calculating determinants of 3x3 matrices = 3 + 0 18 + 0 + 0 . - 2 6 . - + +. 20 - 5 0 . = 2- 18 - . 3 . 2-0 26 = - . 0 . -5 8 Systems of line equations 2x + y X- y = = Y - 8 p 15-i)(i) ( .) : = matrix of coefficients matrix eents o A . 1 A-" = = square I Quiz explained S = is I 60s . A O 25 B C ( I 1 = Te . 5 2025 30 A C B ( > transpose > look - - main at diagonal Seion 2 - Exercises QUESTION (FindX) 1 (2 -12 321 253 X QUESTION · Firm's · TR · Tr = 2) - * x) + (1 . + 15 0 . - x ) . = [(2 2 2) + (3 % 3) + (5 0 - · · 3Pz y + 3P3 Show 360 of - = Q3 3rd x)] = 39 this into A is 1 - 1 3- - 5 3P- + (c = = , as expressed d) " , b = (360 , 763) of Matrix A = = 1 + = (2]r [ ] 36c + 6d 49c + 7d y9Q-234 = International + 3Pj + of 3Pz - 39 Pz + + + 5Pz 2Pz YPz - 1Pz Reseller 3 of - demand prices of components electric Demand > coefficient matrix interdependent for each al prod is /Porto model where affected by the the three Equilibrium · prices : value) : 6P- + 3Pz + 2Pz 10 = be x 2x2 X - can where prod (absowte = . = 763 equation Determinant = . Coefficients QD2 + intro 1) + Q [] x B = 5 . . 39 = 20 , = 2) . and y + YP . planned 5 Q= 7 Qu = ↑ . 5x when Q QS3 - Q= 6 LTD : 11 9 when (Frames = - CQ QUESTION 3 = 2) + (3 8 Ax = b Demand QS2 . - 39 = revenue Matrix 3) ↑ total 763 A 3 2 360 = . 3x + 2 + 30 39 = ypz 5Pz + - YP3 UPs > [ & 5 - = A - 1 xB > mmuct(minverse/A) 4 2 - Equilibrium price pr 150 3 - Equilibrium price P2 > Equilibrium price P3 2 mdeterm = - Y 2) Absolute value of determ of inv matrix : - Mdeterm /minverse x B) (A)) = 0 607 . Systems or Line Equations + Cramer’s rule Seion 3 - Video AX = B A - AX > - = A - 1B - IX Cramer's rule (i) =- = is X A- "B an right 4x - + 2x 3x xz + 2x2 = Ai is hand 8 = Y = 9 ax 3 + . solving SLE matrix the side found by replacing ith the column of A by the vector PROBLEM 3x3 + 5x2 + xz + for === where PRACTICE alternative method I 103)3 (i) + = = - 25x 329 = 1/ Seion 3 - Lecture · Transformations · Cramer's a x 3x x rule y - + - y 2y z typically is an = the as A= 8 + 62 = 12 - z = 41 represented as matrix a with square real coordinate explicit formula for the solution equations many + are multiplying by applied of a matrices and system of linear equations unknown I 10 3 - - 1 16 12 - 1 S B = 12) are vector . x = (Ax1 A with Seion 3 - Exercises QUESTION 1 y = C+ 1 + 6 C = a(y - T + d = + y (Three-sector y T) + b C = - ay + c + aT = b - + y Cramer's to I I =1o7 A - 0 201 . mdeterm = Matrix = data 1) question as y-a for solve B b = I I 0 7 = . 660 = 1( b = 70 , ( - 2 wi + 7 3 wh = $109 + 1 wi + 4C + 5wh = $198 and + wh = 701 40001 2 u wi z(3C) + 9(3() + 2 wi + 1 wi = + + y 483/168 = mdeterm = QDe = - 16 20 + - Wi = 3 QUESTION Y = = 200 , 6 = 400 , I d = 400 390/0 44 y = . = 886 36 . 390 + 132 + 14 Wi + 31C 3(4wi) + + 5(4wi) + A= > 2 + Wi 113160] & mdeterm = 483 Q Du 1 , 600-o - c = . mdeterm = . + 10914 0 2 += , 0 44 zB 19824 X = I + G 9B [ where 3 QUESTION B = 3C B = T= d /same 2 Ax form in ly , C , TIT (:] I QUESTION use + Matrixy I& G Express G - A = Matrix 1 + model) macroeconomic 3 p1 QD3 = 5 + 39 + - + 805 and 5 = = 1P2 + yPzQSz = 5Pz - 293 QS3 - 3124 M + yp + 5 + 3p2 393 - u(5) = - + 3pz GP , + 3 + 31 - + + 2wi + 7 + 12wi & B= (10 161 20 interdependent for three 39 + = wh = M demand Q 2 Pp + 3Pz + 2 Pz 1344 6 27 + wityC + 20wi undeterm = 805/161 I supply [ + + commodities) 2pz = ypz + ypz = 5Pz = - 593 2 19 10 mdeter m = 135 Find 3 equilibrium Cramers with t &- I a 510 m determ = 945 > - Equilibrium price for good 3 = 945/135 = 7 Ya Yz = c + = (z 11 12 + Only two V + - (4) = [ (0 74 s + . 0 5 -0 2 -0 2 0 . . . . 5 + 200 Mr 0 742 + 300 Me = . . > - + 7 + 300) X - M2 = expression =(20 200/ between trade other 300 + 12 Yz = (3 7 yz . 0 74 , = which of of Cz general and model a Ci = M2 of the coefficients Yx Mr countries Considering y = - x2 + imports the (Equations defining 5 QUESTION of trading 0 24 ( imports national * . nations incomes 0 24z exports . themselves and the = two so of one is Ay = 242 + exports country Xz = My system of equations < where S Matrix A = 1 + + 1 0 + . 242 0 . - 245 0 - . 24 - 6 2427 . Y = 42 = 0 0 545 + 0 + . 541 . 0 542 . . 542 -0 242 . - 0 244 . 0 . 0 161200 240 + = 11 + 208 = 11 + . 1 300 . + 360 are Graphs of line equations Seion 5 - video Line equations The coordinate system I - vo 10 > straight une) dx + - & a y- a x i s #10 > [ - o F v ey y Remks line have 1 we , = f - change in change in 12 , 5) y can n point) find the Y2-y = + x (3 7) , . Hence , if slope of z - : X ax SLOPE-INTERCEPT dx a= = -G . g : sign of straight the line-whether by 3x + 2 3(b) y = we I · point at intercepts + 3(2) x = 1 = 5 + 2 = 2 which the the y-axis m= 2 * X = 1 slope it -y 8 = 3 line (y-intercept) line does not m= z change - it's as O constant defines the increases or orientation of the a < Thechange in (positive) 0 O(negative) decreases the : when unit) 1 X =2 y = Horizontal intercept Thepointwhere Slope in y change increases - E (gradient) slope I measures a = The b + - +52 a = equation . m= 2 The Y =- b = m= f 11 goes through two = , z A ey (d e and f are numbers) the y-coordinate ( ? 4) divided line by cros a the change a in The point-slope formula (x0 1 Find . 2 , (y . EXAMPLE : 40)(X+, ys) the slope y0) - 1 m m(x = - - 1 . y z - E x0) (7)) (7)2 y = = = 3)(4 7) + - u) , 4x y + = - 16 + 7 19 5 Seion 5 - Lecture line straight A · distance slope is a se = = A function formed change in weight change in distance transforms an the input two when and run · figure them, between line ends points a re are extended connected to with the shortes t infinity. Y = into UX output an Shifting aphs Seion 5 - Exercises QUESTION Telsys- 8 > I cents 160 Graph 80 min Telteam > - 3 cents + 45 cents "connection 8x 3x fee" - telsys : 120 per How long call be - - must for = + 45 5x = 45 Telteam same 40- price : x = 9 > so it is Is - QUESTION 2 6 - 37 is monthly subscription Unit price = 3 per film monthly cost = 1 bill is 5 : 15 = 3 + x-x = 12 Demand and Supply analysis: Mket equilibrium Seion 6 - Video FUNCTION : number le A which assigns uniquely a X , number incoming - to DEMAND : SUPPLY : Qp = a-bp Q QD P = p = a - C + dP = QS each relationship negative defined - C & y positive relationship y - add > 13 - 3 IN y OUT -17- double and add > zx + 3 = outgoing y = or f(x) incoming value X = dependent The demand function Quantity = f /Price) pa Usually the (MOROn from when the price the graph pou) of decreases. demand function Inverse Demand Function > - P = f(Q) ASSUMPTIONS & Q = f(P) 1 . factors · · v = Ps > - Not income = realistic affecting prices substitute al there are other P Q Y · of x) rises, Q P mathematical SOME on decreasing > > (correct setup depends quantity Then - - > independent Ly = - value Q = f (P) > - an (Demand ( 2x + 3 = - ↓ -31 - 3 NOTATIONS : DIFFERENT double and 5- Pccomplementis goods. A = · T= advertising - tastes Ps - > - > - ↳ > - f > - Q is f 2 . is a linear function p = > + + a PROBLEM PRACTICE b Q - p = 3Q + 15 -x intercept - LINEAR 0 DEMAND a) Graph FUNCTION 2 = - y = 0 30 + 75 3Q = 75 Q = 25 b 0 p & Y = - 75- The supply function ASSUMPTIONS · > - · depends supply Price - on factors other many > when Q x = Q = 23 3(23) + 75 Fixed - 69 + 75 = 6 function supply Linear P = b) is aQ + b as c) 0 b > 0 Quantity % = 3x = - 3 x LINEAR x - 75 = = P = 18 when 3x + 75 - 18 57 19 SUPPLY FUNCTION DEMAND : In ter ms = (pricedecreasea es & p of e In Mket equilibrium ter ms of a-bD(quantity decrease Q = SUPPLY : p=(priceincrease es) terms off In In terms of Q= c + dP) increases) quantity as price increases Taxes & subsidies (Changers TAX : financial Compulsory -cre ate a gap charge by the government between what -Generally increase the price Typically decrease the quantity - SUBSIDIES : Financial of - - a of consumers consumers assistance sold by pay in the pay and decrease and the what market on producers the price a equilibrium) good or income receive producers receive market gou to encourage production/consumption good Typically Increase decrease the the price quantity sold consumers in the pay and market increase the price received by producers PRACTICE D a) PROBLEM 3 p = > = Find uQ + 120 - p = S + Qo , Po (market 1/3Q + 29 eg) or · - y(21) - - 13/3Q Find equilibrium new 84 + 120 = 36 Ps + tax Ps = 1/3Q + 29 + 13 Find ye + = = -3 Po . a fixed = = - p $13 · Who pays the tax ? Pst > = Pd ya + 120 > consumers Pot 18 = y(18) - of 1/32 + 42 = ↳ + 120 Pot pay the Po 48 Producers P(1 % ) tax 36 013 = - 3 of = 29 + +ax 7 70 Q = + equilibrium - the ↑ /30 1/3Q = 91 P -> imposition after - + 120 Q = 21 b) a = = = price 36 Pot-tax perceive will 48-13 = new 35 72 + 170 48 = Seion 6 - Lecture · Demand with · and the supply correlate the functions quantity demanded two variablethequantityandprice canexchange , . 1 Quantity a 2 . = as a function of the price f(P) price p= (or supplied) price as f(Q) a function of quantity /Q = their respective dependent/p /P-depent/Q = = lol and independent) independent ( 400? Xm = %z negative scope - (DEMAND FUNCTION) P P - m = 150 S - Pa = Q - = S 1/2Q + 400 1/39 - = + 100 Ps 1/2Q + 150 = Q Ideal mket (no other) Finding inverse function Pd (SUPPLY FUNCTION) - 1/2 slope positive pd > - - - 100 forcel A PS = - 1/3Q = Q= Q= 1/3Q pd 3(pd - - EQUILIBRIUM 100 Point 100) Pd - - PE 3pd + 300 - Ps Pd = X QE Cost Revenue TR a = P x TR(Q) = FIXED P(q) +O VARIABLE Or M = TR M > - ↓ TR cost : Profit & VC = TC - TR (FC + VC) - TC = FC + VC y=8 oFc constant = S C 9 Q Seion 6 - Exercises Ps = + 40 GQd + 232 8 - 9 3Q + 40 . 15 3Q . Pa = 3Qs 9 3Qs . = = QUESTION /Find Q) QUESTION Pd = Ps = . - GQd + 232 8 - + 2 (FindP) QUESTION 40 Ps GQd + 231 7 = Pd = - . (FindQ) 3 6Qs + 216 7Qd 325 + 3Q + 6Q = 199 Gov deducts Q = 21 the market 7 4% . as tax of . 192 . 8 - 12 6 . . 222 ... price (suppliers receive P= 3(An)) + 40 P = 103 67 0 . of each 7 . good Less) 4% 976P = 7 60Q + 216 . . p = - 8 2073 . ... a + 233 2613 . 7Q + 325 = 8 2073 . 9 0 . 739 = Q = 6 03 . 15 . ... a + 2073Q 233 2613 . P, of two Find QUESTION Y Ps = 8Qs (FindP) 216 + Pd = -7Q + 325 + Gov deducts t ax of of the 35 % market C1 : Q as 65P . -2 . price 12 . 30779 + 332 3077 Q= 0 = . 3785 7) p = = 0 = QUESTION -8 = + 3Pz = = 5Pz - - 30 + 2P . P = - 8 + 3Pz 8Pz = . 3/2 ... 3785 . 30 + 2P , - = 5Pz/QSz GP = 12 + yp1 7 3077 - - YP1 + yPz - y + yp + 79 + 325 - = . 3077Q YPz/QS + 6 + 2/3Pz . . YP + - y + UPs = 36 + 4Pz 3077Q + 332 3077 + 9 G commodities good each 8Q5 + 216 = = C2 : QDz 6 0 (interdependent) QUESTION G ) ... 325 - + = 312 + 3 p, of two Find 9/2 P2 = 1/2(6 + 213P2) 3(2 + 328 (interdependent) 5 1/2p1 + 1/3Pz + 2/392 = = = P2 P2 6 75 - . commodities QUESTION = 6 -YP , C2 : QDz = y + yp1 C +: QDs + - YP2/Qs = 5pz/Q = University allows 8 + GP2 organize a on 6 - yps yPz + 4Pz 36 + - = enterprising an take-away 42 /P2 Q1 = P. = 0 80 . 12 yp- + y(3 17 12 + zu = 11 2u = - 2 +/bPz) + 4/3p2 + = = 42 = - 9 15P2 42 - 7 3336 + 1192 49 3336 - · 9 167P . p- = . sets good - P+ - YP1 120 Pr + + - + 3 . 16 7 . = b b = z . + - new producing good - company demand function for 58 ypz = - 31 + 66 58 31 + 66 ypz = - 162 pi - + 49 = . 3336 = 23p - 26 32 167P . 34 83 good 1 and (substituted 2 3(p1 + 22) 170 . (FindP) a up and supply Q = 3 167/0 80) + b . 48 1/3(2 48) Harry . . 75 3336 QUESTION8 · = . p 3 + 00 4/3P2 - . . 26 9 = 8 + GP2 29/3P2 = P2 = 2 14P2 = -2 + = P, 5Pz - to service Campus Supply-z3D-Q y + ypi student food delivery 30 + 8P 12Po = 1/3Pz 3 + = 7 -30 + 8P , 5pz = zPz =Pz = - 201 - - 84 = 170- P , + YP2/Q = 3Ps-58 Qbz = + 20 + subsidy P1 of - 5Pz/QSz $22 only for = 2P2 ( m -1024 201 - determ = - 1 ( + 1950 - good 58 & - I Q1 50 81 25 . 84 Non-line equations Seion 7 - Video Types of nonline functions Quadratic - · f(x) = Quadratic equations i) ax2 + bx + Cubic Polynomial constants Experimental · logarithmic = 0 - c = 0 = 0 b c · · = b) + = 0 ax - = - u + b = - 3/a ↑ ↑ · Hyperbolic bx + X(ax ↑ · ax ii) c = 0 ax2 + solve for iii) axz + > x - + ax = b = 0 v + -c = - c(a - + X = =) - ca bx + c = 0 - x= b & 19x - 18 - 361 a X + Y - +un = = 144 > - Shape using No solution sign > > - 4 . the sols Determine - of of the the b2 - and 2 yac a b2 iii) b2-yac < O vertex yac = 0 2 solutions > 1 solution > No/real) solution - - Graph of a quadratic function called y =x PARABOLAS > U-shaped - + bx + 0 - max or equation a in if x= 0 then y ly-intercept) I quadratic ( - > - ii) "" -intercepts (if any intercept b = i) b2-yac > 0 1501 > - To sketch e aph of a quadratic function... . , 2a 0 x-4(1)(1) 1 - 0 [ + + O 4 2501) u(u)(a) - - have can solutions X = a 144 c)22 441 = Quadratic equations C + 12x + = - 80 + Y b)yx y(2)) 10) - yaC 0 = > - - 2a - - = 1 na) vb b + PROBLEM 2 PRACTICE a)2x2 - so U if a > 0 it a = c EXAMPLE Sketch the of graph f(x) = 2 + x- -max e · I 2 - ⑨ 8 = v6u - 2 - if 2x6 (x) if x 2 > 6 - 0 1)) -2) x - or 2 - i mmm - y( - f(x) > 0 - I 64-28 +2 --8)-z vertex & = vertex 2 1 O sols Sol sols Quadratic inequalities ax + bx For is which the the + c > 0 values of function EXAMPLE X b a above i) x 2 - 5x C + I 670 X-axis U A For 8 quadratic solving inequalities s ... 5 = (( 5)2 + - mo u(t)(6) - 2 1 2 # nunns Y Sign . . of root) a t 2 = = 2 nu · 2 ? X = 2 X23 Economic application (mket equilibrium) Ps Q2 = Pa Q- - = + 150)P= Q2 + 14Q 14Qs + 22 + - 0Qb + = Q + 2Q2 + 24Q % 6 Ps(y) = y + + y(y) + 22 = 94 Pd(y) - 16 = - - 2z (4) 10(4) - 40 + 150 = Q2 + 10Q + 150 - 128 - = 0 Q = y or 150 + 94 = Seion 7 - Lecture representing when · a parabola predict the parabola y x2 + 2x = v = b = - vbz - 6 - nac + - 0 by = , by knowing plugging x + 2 the direction and the y-intercept , you can numbers -= 6 - in = - = v 2 += 2a Ver tex formula Seion 7 - Exercises QUESTION 1 Ps = 13Q3 + 12Q5 + 50 Pd = -2Q2D - QUESTION 3 QUESTION 2 /Find PS Ps = Q2s + 12Q + 50 Pd = 3QD + 302 - 5 . 82 - Fred · · 130. = = 2 + 2 - 12Q + 50 Q2 + 12Q + 50 3Q + 302 = 15Q2 + 15Q - 252 - 5 802 . - shewes up his as but progresses tires and N - 0 002 + shift eventually down slows 0 = . 15 = speeds 30 + 302 6 80" + 15Q - stacks supermarket in 3Q + 302 - 252 = 0 V(15)2 y(15)( 252) 30 2 + . 0 3 + + 16 . litemered) - - = - 15 = /(15)2 - 4(6 8)( . - 252) 2(6 8) Find time when . 1823 87. 15 rate of work . 13(3 63)2 + 42/3 63) + 50 man - . . = I 264 77 - . 0 . 22 002 + 2 + 0 3 + + . 0 002 + 2 0 002 + 2 . . = V(-0 3)2 4(0 002)(6)2 2(0 002) - . . . - . . 6 3 16722 0 002 +2 + 0 3 + + 16 . - his is more + 0 - . 3 + 0 . - 37 22)0 60 + 60 Revenue, cost and profit Seion 8 - Video S TOTALREVENUE iR Q = (a < 0) TR = P . received money P = 100-2Q Express revenue total as IR = 10 S function by firm a function a TR is a from the sale o tas of Q quadratic function M I aQ + bQ = of EXAMPLE : then , Q amount The P(Q) demand the : aQ + b P= Let D - . (TR) y-intercept C = 0 & TR S ⑧ Q TR = 0 X- intercepts = 100Q-1Q2 = + 0 29(50 2Q = 0 - Q) or = 50 0 - Q= 0 Q = 50 - b/za Vertex = max iR > Q - TOTAL COST I Depends Fixed Costs unity TC = : (FC) Variable Costs (Depends on (TC) Relates the Q-rises on the : includes (VC) number when of : Q aspects includes aspects > - like raw labour . (VC/Q linear - cost - 100/2)-2) > - costs function 25 = 100(25)-2(25)2 the to increasing level function land, rents , equipment , still produced FC + 25 production increases like = + labour of = 1250 . output ( * material) , components , tool , unskilled COST(AC) : AVERAGE + /Q (Fc + vC(q))/Q = = (FC(Q) + vC hyperbolic function Ac L ra slope - a = 0 - E 00 (horizontal asymptote) = > - VC Q Q T = REVENUE TOTAL COST GRAPH TOTAL Between Before A and A A and and B & B - TR > TC aferB- Break-even TRLTC points TC = TR PROFIT(N) 25 Tr = Q . VC = 2 p 15 M= 20Q 1 - Q2 18 = /182 _4( - ) v ( +z + = = Q) + - - - find the 20Q - 25 2Q - 1)( 25) - - - 1) - = Q2 = 16 49 . . + profit Q 1Q 18 = V182 -100 - 2v 1 52 - max [ Y a -9 + 2v 9 - 2Q + 2) P = 20-Q1 TM = Q(20 > - +C = - TR-TC PROBLEM PRACTICE FC = : - 25 My-intercept + A= = -25 4VT 2 Matisobtained atte ver ted Summy · TR = Po Q · +C = FC M TR · · = + Y (VC) & · If PIQ) is If P(Q) and TC Breakeven points . IBEPS) TC - TC = TR TR be can There function linear TR - linear are is a M is > - a quadratic quadratic 0 = ↓ AC = TC/Q a # more than one Seion 8- Lecture & EXAMPLE S D P= 100 TR = . - Q Q = (100 Q) Q - 100Q-Q2 = -- p Q= 0 180 = Q or Substitutable and Complementy Goods SUBSTITUTABLE GOOD consumed be can good that another good A : instead i m of = Difference between gradient TR-TC INTER DEPENDENT COMMODITIES = markets If price increases one's increases E . (positive : Butter g Q = and 30 - Q , COMPLEMENTARY If the price demand with also GOOD other both . g : Bread a = 30 and - 3P IPA increases that price increases decreases & at cost , the Practice M = - - 493 + 9(492 3 = Vz2 - 392 + 39 - - xx = xz = 0 1 . . 49 4) 4(3)) 4) 28 decreases 433 693 for and equilibrium used is PA , lowering by producers , increasing supply One butter - affect Directly affects = curve lowering = (Por Q) one in whose others supply goods goods changes the does (negative relationship) E other Goods linked are SUBSIDY complement a of for for the PA substitute + IPA increases conjunction demand margarine 3P in , relationship) of - Seion 8 - Exercises QUESTION QUESTION I Find Calculate function demand given : the Q which TR being given Demand > P= - -gQ + 40 - > 8Q + a = 5 40- D = - TC : 5 96Q-7Q2 Tr 96 - - = 140 Y . 7 QUESTION 5 2Q2 + 34 4Q Tr = 5Q + 70 Find the min value of Q Tc - = for breaks even 2Q2 + 34 4Q + 70 the the firm 22 + 51 11 = . = FC = a # students + - value AC below y Q of Q Tr = 17a Q2/ N TR = - - Tc +c + = 179 - keep - 4Q = VC = 4000Q + 400Q2 to 51 . company undergrads - - for which the profit 8 + 4Q Q2 - 8 Q2 + 13Q (13) - 4Q - - 8 2) + x = 6 5 . $35299 => +ua 338000 46 uni 338000 70 maximizes 7 of = - 70 P = 17 > - calculate Q = # thousands profit G 8/vc Demand = 59 - which for 625 QUESTION attending 5Q . (b / 2a) X value maximizes . FC 5Q . max 202 + 46 50 - - QUESTION 51 + Find - . 5Q 2Q2 - 5Q + 70 TC = which the firm Tr = 500)/Q) 6 506 - . TC = + 149 = 0 = = (5Q 500/Q + P18 - 96 QUESTION 5Q 5Q + 500 = 7Q Tr = = VC = maximizes Ac = + c(Q = p = 96 Demand and FC = 500 value of - 8Q2 + 40Q TR = TR - QUESTION 3 2 400Q2 + - 31299Q + 3380000 31299[V/ - 30299(2 - 4(406)/368000) 2(400) 4009352999 X1 = 52 . 93 = 13 Xz = 65 31 . = 65 Indices, logims and exponentials Seion 9 - Video Let M b = b is & ↑ base I n is number whole a Integer (whole) number any power , exponent or a powers fraction n)0 index or n Fractional powers bn + n baln - "Ju + roo + of b + y = PROBLEM 0 n10 16 % . 2 27 % 3 . 3 . 45 VT = = = "Vez 8 5 1 - 23 - = 8-23 07/25 = v5 risen to = 11582 - 532 = 1122 14 power is itself) : ECONOMIC GROWTH Q = m > - b) ↓ y b - * b b = = . b /b = - b . " bm = - Inc - Q = . b . b n m factor productivity A , & , B are PROBLEM 3 = X 3/4 8 . d) V= (x5(2 * 12 + = y(2 - 3 + 0 y3) + 5/2 x + x 24/4 + = x x42 = VE = yez = + + 12(x5 y + x5/243 y12y3 parameters Cobb-Douglas Apple b . anbn = a)(x3y4)3 Production Function Q = A "where n . =/x32X2 312 x X total 1/b" = bm = a) ( x 3/4)8 labour f( ,) - ... simplify = Economic applications of indexes PRODUCTION AND capital b . butm = PRACTICE 1 1 any " b 16VT + 32 25 . 1 I b . = = > - (a b/h :V = ↳ . 4 . 3 = Vu5 b = bo = a yu/bV 2 Y = b" > - bu = . (b) 1 > - Rules of indices v5 b " PRACTICE = 100 #13 (12 N = 27 L = 100 Q (27 100) , = 100(27)13(100)" 100 . 3 . 10 = 3000 The indexes ( & p ) . B) 1 &+ &+ B = - + - Increasing constant returns to > scale - When allinputs areincreatedbycertain proportion , out a - returns scale -> to Whenalinputsincrease by a certain proportion output increases , ~ when B < 1 &+ 2 - Decreasing returns i) Q increases by a scale -> to all inputs increases are increased the output greater than 2-a factory factor returns 24 ii) Q increases by the iii) Q increases by a by same factor lower factor /doubles) than a cer tain proportion output , by a lesser proportion > - doubles more its than capital and labor , doubles , i t exhibits to scale Doubling inputs leads inputs results in less to exactly double a) = returns 2-doubling the than output > - of : scale 7k(2 Y K to +B IK Q (2k , 2)) 3 = 2) > - = = - After K doubling /2x)(22) = 7 . 2 . 4 KL + 174L)2 Logims logbm = sometimes a unknown is and L , the output . 8 multiplied = by 8 (778 PRACTICE a) 109 , 1000 > - Find 32 = n 25 - 109 232 if 32 n = 5 = = PROBLEM = 5 3 2 h b) to log10 = 189 , 010" = 5 n Rules of logims i) 109p(x y) = - ii) log , (f) iii) 109px" = = 109b + + 109yY 10gyX-logb Y n . log by the output Practice problem 4 Analyze and increasing logb(X + y) =109bx + 109bY -A double 6 PRACTICE PROBLEM Express a as a) (109b + single log 109bY) + 10902 - logs y + 10902 b) 4109yx solve 1993/7 2/ : 2109 by + PRACTICE + logpt" = 109pyz = 109p(x4 . y7 PROBLEM 7 equations the % ) 10 = * - both to log apply b) 5(2 + log(5 2) . 1095 maintain to log10 = x 10g2 + 5 = log sides * - balance log5 log2 + * = log1x xlog10 = X log 2 x 10g 10 x(log 10 - 1092) - 109 5 = log 5 = log 5 = X = x x log 10/2 log5 4 Seion 9- Lecture -" a" a = ao = · · a . exponentials don't They (a Instead . b (2) a .. 1/an-natural = 0 °= undefined 1 All pure · a - c . = . have their , d)r ar . = b a 0 a =a have the same (0 1) intercept , horizontal intercepts asymptote ar number . br . cr is the horizontal line dr . - EXERCISE -Re-eeze 35+ + 3 = + 3/e - 35 + 3 Seb + +3 = . e+ = 3 y - eb = + + 3 = + 0 - 6+ + 3 6+= - = 0 3 - += - 1/2 vs)1 + = f(b + + ) b+ = fb + + + = f+ b+ + = - b += fb - + + (f b) += 16) 452 = 36 452 36 + 0 - - - fb/f -b - = b - 0 = -(4) $2100 GDP = GDP 2019 in according grows 9 2000,0 05(0) -e . b) 2029 : 2100 2039 : 2000 , 0 = 1 e % 05 (20) - . GDPA to : 160) - = 2100 e 2000 + Sort year 2019 $3462 315 - . $5708 392 . Seion 9- Exercises QUESTION Rewrite to I an QUESTION 2 equivalent Identify expression un p - y the = 16 - - y = 391(1 3)" = v 1091 3 . = log = + 31 = 1 . 3 % + 131 6957 109131 6957 I . . 6957/109 1 3 . - A(1)w(x) 3263 A(5)u(4) 785 > - > 3463 = 3263 = = constant = >0 701(5)m(4)8-m m+ n n 1 = . = . + scale) to by modeled factory car a - = ↓ Q = 701 function : ( 47 - m 6679 = m/1095) . m(0 6 . 6 . . . 6990 0658 = + · 0 6024 : - - . m(0 . 6021) 0969) . 0969) . = Y 6024 m/logy) m(6 6021) . . 0 - L = K = 5 , L when 0 6990 m(8 6990) + m K= 1 , when Q =3263 0 . 6679 = m(0 109(3263/701) log (5m 6679 109(5m) 109 (45 2) 0 m - Y returns Akm2n - A 701(5)w(4)1-m = my . = - - 18 60 ↑ = mys)mp - Cobb-Douglas production a . Q 701 to : expressions log10) Output , Q , of 56493/391 mp + QUESTION 3 56493(use . /plus) 10gp46 QUESTION x equivalent logarithms/exponent with + + 0 0 . + . 6025 6021 m = 0 . 68 The exponential and natural log functions Seion 10 - Video M b = Number " 109bM - e 2 = . 718 = The exponential function x [eexponent y = ex a ... logst x Types of ow Unlimited : Population , * is e > 0 10 , any real y 1) - = number e + 0 e= 1 e-x pandemics , social indeed However, it will x reach a network point where users , e ) etc i t stops growing 10 , %) · - y Limited = f(x) growth : ex = consumption , changes g(x) A g(x) Be - 1 model to possessing + microwave A It income (logistic growth) Be PROBLEM explain the in ko-decay x A = PRACTICE kotgrowth - x 3 proportion of ovens years households after they were launched a) % at +=0 launch - Logistic growth example 55 y = 1 + 800e - 03 + y = . 551801 t = 10 + = 20 + = 50 0 07 = . 35 y = 1 y = 18 44 . . y = 55 . 99 S saturation level is 55 PROPERTIES The natural logim M= e > - m(M) = n simply rules a are log · m 1 = 0 Wee · to the base same e so Y PRACTICE PROBLEM the use as ruses of un unlazbe > at + in - (a) express to m b2 (b) as and 2 ma = + singleIn 2 mb 2(ma + mb) = 2(mnab) = 3) dwx 3 - m = + my 2 PRACTICE PROBLEM A firm's revenues a) Tr when decline += 0 += ↓ ↓ After 2 7 = . how 0 . 5e-0 157 . = 2 . 15 + 2 - . $2 7 TR = years many - 5 TR to according TR = 3 70 TR = 5 b) 5 7/5 . = e - 0 . 616 = E 157 . . - . We-8 15) 10 54) = un 0 -0 . 15t u10 Seion 10 - Lecture e g(x) f(u) = ema If (0 , 0) [ [ > a) ym(x + a) = m(x + a) m(x + a) x + a - = um(x 1) - 2m(X = un(X = = 3x 4 x2 - 8 - = 2x - + - 1) 1)2 b) 17 2e e - 107 = - 3 + 3+ in = = - 2e = = - 3 4 > e-3r -3 my OPERATIONS c) > mn45 in a ROUND DECIMALS 9v = ma x = = x = - - y5 v a DO NOT MID 90 1 0 70(x7 . e = + > Exercises x mx - ev (0, 1) x2 = a = = = 1 . 1 * = un +1 m+ 09 1/una Seion 10 - Exercises QUESTION I value of -at y = Ae 45464e e a(3) - a(3) with exponentially reduces product a age -1187 = 1187/45464 = New-busugy-as (m (0 2461))13 = $1487 + = 3 - a 0 = . . 4674 QUESTION Z P QUESTION 3 P= A = 1 be + - % rt of signed 10 90 local - V( + ) gamers up to 200 - 0 . 005 + un(u( + )) game new = 100( - .. A = 90 (max p = 30 when value ) + = 0 30- If 76 % of after signed up gamers * V(t) ↓ un ~ v (t) days + is be 90 = + - an 90 = 1 + = 2e - r (8) 30b 76(1 76 + + = (8r) r r QUESTION The amount P( + ) = 90 - = - 28 X = = Ae debt owed -0 2 + t = . percentage - - function vertical intercept un 200 + 0 0055 . 60 = . - r8) (8r) = 90 - (200 e un 200 + un 200 + 0 . 0057) un es . 0 . = 005 + = 0057 98 = = un 90 76 = 14 un( 4/152) + 38482/--8 . Y of 90-A 90 function increasing 0 298 = by 62e 0 . - 2115) in a years 28 % what 18 % = 2 = ze 152e -52e - intercept 30 + 3 = = b = 2 76 decreasing a = 30 horizontal a is ~ Ust) has a ~ un(V ( + )) days 8 has (V( + )) in country from ? A = 90-28 A 62 = 2007 2007 2022 = as 86 9132287 . a percentage of gross domestic product (p) Seion 11 - Exercises QUESTION /Frames LTD profits costs reputed >D(mtD (0xb] per unit -C200m hard brand disks 150 600 = - unit per : -c = My M2 (200 , Ma ( + x 3) S Q (3x3) X Q'XC' V The is is valid mult a of the sum equal gives X is QXP the is a valid 16 - 5 - = + 3 + - 3p1 32 = 31 - 5pz + GP1 = + demand and supply Q is M4 M2 M3 R1 300 200 150 R2 400 so 260 R3 15058 150 sold matrix singular a - 293QS3 3P2 + - yp2 52 = 27 + 3P3 6 - 5P3 + - 3 - 2 - Y - 5 - 5 + 31 - - 3 3 2P1 + 3pz - 20 + 31 392 YP3 16 - 43 + 2P3 - - of three = undeterm = 23 5pz + = - - = 2p3 32 + uP 43 + 392 = 27 + EQUILIBRIUM 393 PRIZES Y minverse 5 (minverse (A)) + 1P2 + MP3 135 0 x(:%6) & 0074 . Equilibrium quantity Q = y3 + 3(8) + Statements : supply equation of = 3 goods will of goods 2 the (good quantity Equilibrium The 200 calculated be cannot not mdeterm = X 50 150 region Qu = -32 + up , 2 P3 1Pz + 493QSz - 23 - ~ 50 gives profit per model IQ'XPI result M3 150 (interdependent commodities I 2 -16-2P1 + 3P2 = The . per profit per model QD2 = 20 + 39- Qd3 mmult gives the TR per VQXC gives iC of products profit selling price per model QUESTION = total 200 - region QX C X QXPI matrix the to XP-C QD XQXP-C (P-C) gives the reuperregion M2 (3x3) statements : Q' R3 150 M3 M2 a X R2 M1300y00 RX 7 sold 220S portfolio product Quantities , My 3 w 43 + 24 good b 2) 2 + = 67 affected because the be for goods are interdependent X The V The at price equation supply QUESTION equilibrium (supply 3 Ps = 10Q -s + 12Qs PD - = - 2Q2d 15 = - 3Q v(15)) - 24 of + + 50 good and - 302 y(12)( 252) - I demand and be will , > - 15 + - = - 252 12325 will only not one change by the subsidy affected equilibrium price ( find 10Q2 + 129 + 50 = ↑ 202 3 the = = 2Q2 - 32 + 302 0 · 0(y(2 - VX = 141 - 258 + 12(y) + 50 = QUESTION Y V Taxes X Taxes reduce X Taxes increase X subsidies market the change equilibrium r subsidies X subsidies X subsidies X Taxes equilibrium the price of a good quantity the at both P and Q and equilibrium QUESTION V( + ) increase = 200 - 0 . value of un(u( + )) * V(t) ~ v (t) X when ↓ un has ) . (Multinational /FC VAC 800/Q V = 1 + function X Demand function > V + c = Ae and vC = Q = + - 5-pd/8 Pd = 8Q + 10 800 + Q QUESTION y 500 -at (value New After = of . 005 + = 0057 0 for year/ next + Pd = function - Pd = XAL = 800 + TC = expression with exponentially 0 13588 . 3371 = = e 40312e - = 0 . age) - 9(2) (2) . a 8Q-40 -Q + 40 10/Q Quadratic un(0 3371) QUESTION8 + function 13588 = 200 Demand product reduces years . 0 VDemand 40312 2 = un 20 + -Q X 7 v( + ) 200 Un = strategy planning X Demand = = company , function intercept . 0 005 + + = 0 When X 0 = 200 + 0 0055 un = intercept decreasing a vertical a (200 e (V(t)) un , is un Y0Q price function increasing += 0 (V( )) un TR = -8Q + good equilibrium at horizontal a an ~ un(V ( + )) G a quantity both the increase quantity has is + rust) QUESTION of investment ( growing exponentially S 5 005 + p increase the decrease the equilibrium at equilibrium at change the market = - a(2) 544 (considering previous exercise) X None X Function of the y answers has a are right horizontal intercept V Function V Function y decreases y has a when t increases vertical intercept
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