NYU Multivariable Functions & Constrained Optimization Example Worksheet

User Generated

1971_

Mathematics

New York University

Description

  • approximately 10 Multiple Choice questions and
  • approximately 4 Free Response questions with parts.

All the chapters covered are in the document and there are also some practice question

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Constrained Optimization Example A company produces the same product at three di↵erent factories. Denote these quantities (in kg ) by x, y , z. The cost of producing each is C1 (x) = x 2 + 4x + 5, C2 (y ) = y 2 , C3 (z) = (z + 3)2 1. What quantity should they produce at each factory in order to produce 28 kg of the product at minimal cost? 28 x y z : 28 - xtytz Constant cost : 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 farmer has 4000 ft of fencing and wants to fence o↵ a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field 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 more written homework. I Homework 11 is only for practice. I Final Exam: Friday, May 15th I It consists of everything we learned up till (including) Optimization of Multivariable Functions. I Last Quiz this Friday jealousy I Partial Derivatives, Linearization, Di↵erential, Elasticity, Min/Max, Critical 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 domain I Exponential functions, linear and quadratic functions, polynomials, rational functions I Cost, revenue, profit; Supply and Demand Important Topics (everything!) Part I: The Basics 1. Functions I Finding domain I Exponential functions, linear and quadratic functions, polynomials, rational functions I Cost, revenue, profit; Supply and Demand 2. Limits Direct substitution property; limit laws; divide & multiply by conjugate 3. Continuity Definition of continuity; intermediate value theorem 4. Derivative I Definition of the derivative in terms of limit I Interpretations (rate of change; slope; marginal ...) I Basic di↵erentiation rules I Product, quotient, chain rules 5. Higher-order derivatives and convexity Part II: More Fun with Derivatives 1. Derivatives of exponential and logarithmic functions 2. Implicit di↵erentiation 3. Logarithmic di↵erentiation 4. Derivatives of inverse functions: (f 1 (x))0 = f 0 (f 1 1 (x)) 5. Linear approximations; Di↵erentials 6. Elasticity 7. Single-variable optimization I Local/Absolute max/min; Critical number; first-derivative test; second-derivative test; Extreme value theorem; Closed Interval Method. I Applications to Revenue and Cost. I Constrained optimization; Method of substitution Part III: Functions of two (or more) variables 1. Domain of f (x, y ), graph of f (x, y ), cross-sections of f (x, y ). 2. Contour (level curve) diagrams of f (x, y ) Interpreting contour diagrams and table of values 3. Partial Derivatives; Higher-order partial derivatives I Limit definition; Interpretation I Computing partial derivatives using di↵erentiation rules I Estimating partial derivatives from contour diagrams/tables of values 4. Partial elasticity 5. Linearization, Di↵erentials 6. Optimization I Definition of absolute max/min, local max/min I Finding critical points 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 following is the domain of the function 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 continuous, one-to-one function, and f (4) = 3, f (3) = 5, f 0 (3) = 4, f 0 (4) = 0.5. Let g (x) denote the inverse of f (x). Which of the following is the equation for the line 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 following shaded region is the domain of which one of the functions below? 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 following is the partial derivative of f (x, y ) = xe y with respect 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 table below to estimate the partial 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 find the partial derivative of f (x, y ) = (y 2 + 1)xy with respect 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 function f (x) whose domain is ( 1, 1) and whose first derivative is graphed below. y a b c d e f y = f 0 (x) x Which one of the following is true? A. f (x) has a local minimum at x = b. B. The critical numbers of f (x) are x = a, c, f (and nothing else) C. f (x) has local minima at x = c and x = f D. On the interval (c, d), f (x) is concave downward (concave) E. None of the above Back to board Optimization 200 A rectangular storage container with an open top is to have a volume of 10 m3 . The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $5 per square meter. Find the cost of materials for the cheapest such container. Optimization 300 Find all critical points 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) only C. (0, 1), (1, 0), (1/3, 1/3), and (0, 0) only O Back to board x (1-2×-4)=0 or -_ ( D. (1/3, 1/3) and (0, 0) only y = A. (0, 0) only E. None of the above g-Ogre ' D= -21 Zytl ) tf x I -2ft )tl= } x= 0,0 ) , ( ' so) ° ( ( t ' , , t) ) Applications 100 The demand 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 price of potatoes and m is mean income. Find the elasticity of demand with respect to price and the elasticity of demand with respect to income. Back to board Applications 200 Consider the Cobb-Douglas production function f (x, y ) = Ax 2/3 y 1/3 . Find the linear approximation of f (x, y ) at the point (1, 8). Back to board Applications 300 Use linearization to approximate the value of f (1.02, 1.09) where f (x, y ) = 3x 2 + xy Back to board y 2. Miscellaneous 100 True or false? The limit lim h!0 p 9+h h does not exist. A. B. C. D. True, and I know exactly why True, but I’m not completely sure False, and I know exactly why False, but I’m not completely sure Back to board 3 Miscellaneous 200 True or false? The equation 32x + x = 1 has A. B. C. D. a solution in the interval [ 2, 2]. True, and I know exactly why True, but I’m not completely sure False, and I know exactly why False, but I’m not completely sure Back to board Miscellaneous 300 True or false? ¥; ,fcH=¥ The function f (x) = ✓ ( ' =4 , 1×32+44=4 x2 4 x 2 x
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