Examples of functions of several aanctionota vari IT f function that a instead of 1 consists ie fix y of 2 variables ex 29 IE 5 a function of 3 variables fix y z A store sells a and margarine at sale of from the sounds of margarine Determine fix g 2 50 2001300 2 501200 interpret 1 409 at 2 50 perpound butter 1 to per pound pounds of butter and y The revenue 2 50 1 409 is given by ffx ge f 200 300 4 1140130 500 420 920 f2o the revenue from tmsm of butter and 300 lbs of margaring Thinkabout abuilding heat loss perday a A rectangular industrial building x y Z The dimensions given amount of heat loss per day by each side a E Net kg the total daily heat loss for This building Heatloss Area a Find a roof formula for loss sq foot fix y z of area roof in sq ft heat 10 xy loxy loxy f X y Z 8yz 6yZ 10 2 5 2 y 151zftotalda ly 1149t14yz heat loss b Find the total daily heat loss if the building has length 100 ft width 70 ft 14 70 50 1111007170 f 100 70 50 a height so ft 1511001150 75000 49000 77000 201 00014N as found A Economists f x y have Cx y p Cobb Douglas OLACI productionfunctiondxlab ir that the function A c constants a capital certain time period the number of goods produced when utilizing units of capital is y 60 be produced by of goods will labor and 16 units of capital units a During a units of of labor and y f x y a How many units units of using 81 60 81441161 6015817 To 60 27 12 3240 7181 16 3240 units of goodsprodu show whenever b capital being used production let a b ft A b units of labor units of capital 60 a b 2 4 y the amounts of labor a the so is 20 2b 2 f a b doubled are show f 2 60 29 go 2314944 2 60 2314 14 b 2 a b b I 2160846 I 2 f a b If a b t F de fxyinaa y dimcarve I fly y a o cure y plane called in Icurethight curve function a Determine the level the production for 60 4y 600 fly y 4 solve for y y 10 114 1 4 x ty fix g at height 600 y 3 fixy 60 it na 9 144 1 ex a PartialDerivatvessarti at derivative symbol diff The partial derivative of written treated as alone f f x y with respect to x a constant and fixy is considered a function the partial derivative with respectto g Similar the derivative of flxy where y is for is x Let fix g 5 392 Compute b If fix y Ey 3 5 y That 5 5 29 10 variable Any 5 32 Sy x cons 7 variable 594 3 2 1546 ex Let fly g I ix g muariables 3 2 2 5 29 0 6 624 stants 3 72 9 5g Compute b ariable If 5 2X 0 256 ex compute It and If for each b fixg e c fix g y y fix g 4 39 518 8 4 39 51714 32 4 39571 814 34 517 3 2414 34550 y Ey 2xye 1 3 4411 i 1 344 41 1 Fy ex Let f x y a Calculate If 6 29 1 4 611 214 6 8 3 2 2xy 59 1 4 b Evaluate at x y 1 4 Fy 2 5 1 4 5 2 1 2 5 to Vow that we understand the algebraic way find partial derivatives what does this look like siiiiiii iiiiiii is the ordinary derivative with y held constant Since ate of change of flx y with respect to x I interpret the partial derivatives from ex 4 above for y held constant It gives th ixg 3 42 9 59 1 4 14 varies if y is held constant at 4 nd hanges at a rate 14 times hange of X then fix y the if x increases by h units 14h units e flag increases by 7 varies then fix y 511 4 if y is held constant at 4 and changes at a rate 7 times change of X ie if x increases by h units 7h units flag increases by the ft I small we have function of 2 variables Then if h and K are f f ath b a btk f a b f a b If a b a b K t ex Let fly y z compute x yZ 32 If and 2 2 42 Rz Ry 3 Fy 2 b calculate Fz 2 3 1 Ry 3112 3 1 22 3 46 3 12 3 3 ex Let x y z It Ily 152 10 7 5 fly g 2 be the heat loss function from 7 1 Hxy 15 2 Calculate interpret 14yz 10 7 5 1515 75 11 7 77 1520 loss with marginal heat respect to change in x x is changed from 10 by h units if and the values of yo 2 remain fixed the amount of heat loss at 7 and 5 will change by 152 h units ex consider the production function fix g 60 y which units gives the number of units of goods produced when t a find labor and y units of capital are used 601 45 Evaluate II 81 16 and If x y y It and If 45 81514 16 YETI 4552 If 6014 y 314 3 4 15 at 81 y 16 81 167 15 8154 165 15 1 40 50 c Interpret the numbers computed in part b marginal productivity of labor and marginal productivity of capital Just higher order variables like computing second derivatives and derivatives of two we can do the same with variable in one differenttypes partial derivative of It partial derivative of partial derivative of partial derivative of It in respect to in respect to y in respect to in respect to y IF f If Almost all functions hold true If ex t Let fix g ex I Find 2x yex 2 b d y Ey 2xye 9 12 9 Re Y 2 e 9 iy 2x x e x 2 ye9 e ezxefylfy tl 2x Maxima a minima of functions of severalvariables EHI IM If x y then fix y has either a relative maximum or minimum at la b o and aib 0 f x y 2 2 2xy 5y 6 ex The graph of inimum point Find its coordinates 4 29 6 5 shows that flx y has a Hy 2 109 2 109 0 If 11 514 5 there is f 4 29 6 0 415 29 6 20g 2y 6 1815 If Y z My 14 14 119 y YE 60 44 ff 15 15V tix 11 1 y ixfff tt III 14T o Isr IN 3 5019100 1442 151111 147 8406 FIFTEEN 3775 616 56 i The building should …