SOLUTION: NYU Multivariable Functions & Constrained Optimization Example Worksheet

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Constrained Optimization Example A troop effects the similar fruit at three di↵erent factories. Denote these quantities (in kg ) by x, y , z. The consume of submissive each is C1 (x) = x 2 + 4x + 5, C2 (y ) = y 2 , C3 (z) = (z + 3)2 1. What amount should they effect at each factory in ordain to effect 28 kg of the fruit at minimal consume? 28 x y z : 28 - xtytz Constant consume : C, t Cz C, t = = C (x = , y , z [y ] 'd [444×+5] = - - + + [ ( zest - I ] ) X2t4xt5tyZtf@g_x-y1t3TZ_1-7CCx.y ccx y , Gpiohd 28 , ) : Cx Cy = = 2x 2g - x - t ) = X2t4×ty2+ = c- y 4 2. 2/31 + ( 31 - x - x y) - C ( 31 - y) C- D= - - t ) = - x - 62 - y) 58 + 2+4 + dy 4x t t 2g 2x = o → *:÷÷÷f" 4×+29=58 " - ( (9,11 ) = 92+419 ) lit 131 Look at boundary Check + 2x y - 9 a) - - o " 2+4 xey=28 ytz - - 28 is ' = y=zs z -28 - - x y - :i%i 363 Fadden = -12=28 =o " constraint - xty " get to: 28 Iµ xh - 28 -2=28 " at I ¥j¥ µ¥→× → xez log! ] test y " × , Y • 2- x and x=o CCO , y) 02+4 - fly )=4y ft 't ) f- ( ) o few) = = = - o ② 28 gey EE = fly )=2y2 - EYE 0 ① 62g - 2. co> 965 797 2- ( tyZ ) 31 - o - 4=0 01×128 ? g) 2+4 =2y 9615 62g - +965 y 62 (E) 28 y=zsµg¥g×l¥8 xey= =o y③y=o2! = ¥= Iz 6431) +965=484.5 = = Cco , C ( o o , , 28) 28,0) = c (o , 15.5, 12.5 ) ② 4=0 and CCX , o ) gcx) - - X = 28 E ex o 74×+02+131 g 14.5) ( o) ) gL28 x - 032+4=74-4×+961 2×2-58×+965 X 84×1=4×-58 g( - = = = 544.5 965 909 = = . = c. C c ( ( ( 548=14.5 -_ 14.5, o o , 25, o zsg , o 13.5) , , o ) - 62×-1×24 ③ (( 28 y= x , 28 - - and X ) =x44x X + E o ( 28×12 = hcx )= 2×2 - 52 X t X E 28 + ( 31 - x - 784 2144×-1 - 797 13h43 ( h' 1×1=4×-52 ) h ( 07 h ( 28) = = = 459 X= = 797 = 909 = . ( C C = 15 13 ( o ) o , , , 28,0) ( 28 , O , O ) [ 28×3774 56×-1×2-43174 Constrained Optimization Example A fellow-subject has 4000 ft of sheet and wants to stave o↵ a across ground that borders a undeviating large stream. He needs no stave parallel the large stream. What are the dimensions of the ground that has the largest area? - - - - 4000 { = 2x + T A = x. y y y - = A' txt " A G) = 4 ooo -4 - 4× Lo x = [ 4000 Ly ooo x critical concave 4000 # X= down - - 2x - 2X ] 2×2 1000 everywhere . Abs.maxareaBwhenX=coooardy=2oo+ Announcements I No further written homework. I Homework 11 is merely for performance. I Final Exam: Friday, May 15th I It consists of everything we perceiveing up tend (including) Optimization of Multivariable Functions. I Last Quiz this Friday jealousy I Particular Derivatives, Linearization, Di↵erential, Elasticity, Min/Max, Discriminating Points, Second Derivative Test I Recitation Handout 13 I WebAssign WA16 due May 11 Important Topics (everything!) Important Topics (everything!) Part I: The Basics 1. Functions I Finding inclosure I Exponential offices, straight and quadratic offices, polynomials, rational offices I Cost, retaliate, profit; Supply and Demand Important Topics (everything!) Part I: The Basics 1. Functions I Finding inclosure I Exponential offices, straight and quadratic offices, polynomials, rational offices I Cost, retaliate, profit; Supply and Demand 2. Limits Direct supply property; name laws; divorce & expand by conjugate 3. Continuity Definition of continuity; included compute theorem 4. Derivative I Definition of the derivative in stipulations of name I Interpretations (trounce of change; slope; final ...) I Basic di↵erentiation rules I Product, quotient, security rules 5. Higher-ordain derivatives and convexity Part II: Further Fun delay Derivatives 1. Derivatives of exponential and logarithmic offices 2. Implicit di↵erentiation 3. Logarithmic di↵erentiation 4. Derivatives of inverse offices: (f 1 (x))0 = f 0 (f 1 1 (x)) 5. Straight adits; Di↵erentials 6. Elasticity 7. Single-variable optimization I Local/Absolute max/min; Discriminating number; first-derivative test; second-derivative test; Extreme compute theorem; Closed Interval Method. I Applications to Retaliate and Cost. I Constrained optimization; Method of supply Part III: Functions of two (or further) variables 1. Inclosure of f (x, y ), graph of f (x, y ), cross-sections of f (x, y ). 2. Outcontinuity (flatten deflexion) diagrams of f (x, y ) Interpreting outcontinuity diagrams and consideration of computes 3. Particular Derivatives; Higher-ordain particular derivatives I Name definition; Interpretation I Computing particular derivatives using di↵erentiation rules I Estimating particular derivatives from outcontinuity diagrams/tables of values 4. Particular elasticity 5. Linearization, Di↵erentials 6. Optimization I Definition of absolute max/min, topical max/min I Finding discriminating aims I Second derivative test MFE1 Jeopardy! (Final Exam Review Edition) Functions Derivatives Optimization Applications Miscellaneous 100 100 100 100 100 200 200 200 200 200 300 300 300 300 300 Functions 100 Which of the subjoined is the inclosure of the office p 4 x2 f (x) = ? ln(x + 1) A. ( 1, 2] B. [ 2, 2] C. (0, 2) D. ( 1, 2) E. None of the above Back to board Functions 200 Suppose that f (x) is a natural, one-to-one office, and f (4) = 3, f (3) = 5, f 0 (3) = 4, f 0 (4) = 0.5. Let g (x) personate the inverse of f (x). Which of the subjoined is the equation for the continuity that is tangent to the graph of g (x) at x = 3? A. y = 0.25x + 4.75 B. y = 4x + 17 C. y = 2x + 10 D. y = 4x + 10 E. None of the above Back to board Functions 300 The subjoined hidden country is the inclosure of which one of the offices underneath? 2 1 -2 1 -1 2 -1 -2 p p 4 x2 B. f (x, y ) = ln(x 2 + y 2 4) C. f (x, y ) = ln(x 2 + y 2 p D. f (x, y ) = x 2 + y 2 4) A. f (x, y ) = y+ E. None of the above Back to board 4 y2 ln(x 2 ) Derivatives 100 Which of the subjoined is the particular derivative of f (x, y ) = xe y delay regard to x? A. fx (x, y ) = lim h!0 B. fx (x, y ) = lim he y h he y xe y h (x + h)e y C. fx (x, y ) = lim h!0 h h!0 (x + h)e y +h h E. None of the above D. fx (x, y ) = lim h!0 Back to board xe y xe y Derivatives 200 Use the consideration underneath to estimate the particular derivative of fx (0, 1). y= x= 1 x =3 9 5 9 E. None of the above Back to board 4 11 C. fx (0, 1) ⇡ 1 y =1 2 6 3 D. fx (0, 1) ⇡ 0 3 x =0 A. fx (0, 1) ⇡ B. fx (0, 1) ⇡ 4 1 2 y= 9 Derivatives 300 Use logarithmic di↵erentiation to meet the particular derivative of f (x, y ) = (y 2 + 1)xy delay regard to y . A. fy (x, y ) = 2xy 2 (y 2 + 1)xy 1 ⇣ 2 ⌘ 2 B. fy (x, y ) = (y 2 + 1)xy y2xy 2 +1 + x ln(y + 1) C. fy (x, y ) = D. fy (x, y ) = 2xy 2 + x ln(y 2 + 1) y 2 +1 (y 2 + 1)xy ln(y 2 + 1) E. None of the above Back to board Optimization 100 Consider the office f (x) whose inclosure is ( 1, 1) and whose first derivative is graphed underneath. y a b c d e f y = f 0 (x) x Which one of the subjoined is gentleman? A. f (x) has a topical insufficiency at x = b. B. The discriminating aggregate of f (x) are x = a, c, f (and dot else) C. f (x) has topical minima at x = c and x = f D. On the space-between (c, d), f (x) is excavated downward (concave) E. None of the above Back to board Optimization 200 A across storage container delay an perceiven top is to bear a volume of 10 m3 . The diffusiveness of its sordid is twice the width. Material for the sordid consumes $10 per balance meter. Material for the sides consumes $5 per balance meter. Meet the consume of materials for the cheapest such container. Optimization 300 Find all discriminating aims of f (x, y ) = xy (1 ↳ § y = y - x - = ( tx y) - ( i - x - g) + t xy C- c) xyc - - - y ). = = 3y y - X zxy - 4g = - I - XZ - - y Zxy = G - x - 2g) so y= -2×+1 X=o 4=-2*1 I y B. (0, 1), (1, 0), and (0, 0) merely C. (0, 1), (1, 0), (1/3, 1/3), and (0, 0) merely O Back to board x (1-2×-4)=0 or -_ ( D. (1/3, 1/3) and (0, 0) merely y = A. (0, 0) merely E. None of the above g-Ogre ' D= -21 Zytl ) tf x I -2ft )tl= } x= 0,0 ) , ( ' so) ° ( ( t ' , , t) ) Applications 100 The insist for potatoes in the United States from 1927 to 1941 was estimated to be q(p, m) = Ap 0.28 m0.34 , where p is appraisement of potatoes and m is average allowance. Meet the elasticity of insist delay regard to appraisement and the elasticity of insist delay regard to income. Back to board Applications 200 Consider the Cobb-Douglas fruition office f (x, y ) = Ax 2/3 y 1/3 . Meet the straight adit of f (x, y ) at the aim (1, 8). Back to board Applications 300 Use straightization to approach the compute of f (1.02, 1.09) where f (x, y ) = 3x 2 + xy Back to board y 2. Miscellaneous 100 True or bogus? The name lim h!0 p 9+h h does not await. A. B. C. D. True, and I perceive precisely why True, but I’m not thoroughly sure False, and I perceive precisely why False, but I’m not thoroughly sure Back to board 3 Miscellaneous 200 True or bogus? The equation 32x + x = 1 has A. B. C. D. a key in the space-between [ 2, 2]. True, and I perceive precisely why True, but I’m not thoroughly sure False, and I perceive precisely why False, but I’m not thoroughly sure Back to board Miscellaneous 300 True or bogus? ¥; ,fcH=¥ The office f (x) = ✓ ( ' =4 , 1×32+44=4 x2 4 x 2 x Purchase solution to see full attachment

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