Consider the Lagrangian function L=q%q,+(Y-p,9, -P292)- Which of the following expressions is NOT a first-order condiction of this function: OA (1-a)a,-ap2 =0 O B. aq a-Ap,=0 OC. (Y-P191 -P292)=0 OD. (1-a)aa, - Ap2 = 0 OB. a-1,1-a
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- It is estimated that a grocer’s daily profit from the sale of two brands of cherry juice is given by thefunction:P(x,y) = (x - 30)(70 - 5x + 4y) + (y – 40)(80 + 6x – 7y) centswhere x is the price per can of Brand V Juice and y is the price per can of Brand Triple V Juice. CurrentlyBrand V Juice sells for 50 cents per can and Brand Triple V Juice sells for 52 cents per can.i. Use marginal analysis to estimate the change in the daily profit that will result if the grocer raises theprice of Brand Triple V by one cent per can while keeping the price of Brand V unchanged.ii. Find a stationary point of the daily profit function. Classify this stationary point.The first order, and the second order, partial derivatives of the following function are, respectively z = r°y² + 15ye* Select one: = 2r*y O a. 2x3 + 15e" and Ob. 3x + 15e* and 2x * = 2y + 15e" and = 2x Oc. Od. = 2x®y+ 15e* and * = 3.x?y+ 15e" and 6ry e. Clear my choiceWindies Cricket manufactures Windies supporter jerseys. The quantity q, of thesejerseys demanded weekly is related to the wholesale price per jersey p, by thefollowing equation:? = −0.006? + 15The weekly total cost incurred by Windies Cricket for producing q jerseys is:?(?) = 38? − 0.02?2 + 30,000a. Determine an expression to represent the weekly Revenue functionb. Determine an expression to represent the weekly Profit function.c. Calculate the weekly Revenue when ten dozen jerseys are produced and sold.d. Will Windies Cricket record a loss or profit if they were to produce and sell 2450jerseys?e. If Windies Cricket wishes to maintain a total cost less than $42,000.00, whatrange(s) of Windies jerseys should be produced. f. Calculate the equilibrium price and quantity for Windies jerseys.
- (i) Find the rate of change of the function f(x) =x + 2/ 1 − 8x with respect to x when x = 1.(ii) The number of units Q of a particular commodity that will be produced with Kthousand dollars of capital expenditure is modeled by Q(K) = 500 K2/3Suppose that capital expenditure varies with time in such a way that t months from nowthere will be K(t) thousand dollars of capital expenditure, whereK(t) =2t4 + 3t + 149 / t + 2 (a) What will be the capital expenditure 3 months from now? How many units will be producedat this time?(b) At what rate will production be changing with respect to time 5 months from now?Will production be increasing or decreasing at this time?Q4 A business manager determines that t months after production begins on a new product, the number of units produced will be P thousand,where P(t) =6?2 + 5? /(? + 1)2 production in the long run (i) A ruptured pipe in a North Sea oil rig produces a circular oil slick that is y meters thick at a distance x meters from the rupture.Turbulence makes it difficult to directly measure the thickness of the slick at the source (where x = 0),but for x > 0,it is found that y =0.5(x2 + 3x)/ x3 + x2 + 4x Required:a)Assuming the oil slick is continuously distributed,how thick would you expect it to be at the source?Differentiate the following function: y = (10x – 3)? O Y'= 20( x - 3) O Y'= 20(10x+3) O Y'= 20(10x-3) O Y'= 2 (10x-3)
- Find the derivative of ln(2x2)+ln(2x)+13 where x=1 Select one: a. None of the above b. 1/3 c. 3 d. 1(1) A course repeater claims that his brother who did some Calculus at college years ago, but currently on some cough medication, believes that the first derivative of (2x² – 1)(x² + 3) [(2x? – 1)(x² + 3) 2х + |2x2 -1 х2 +3 4x 2x | f(x) -is f'(x) x? + 1 x2 + 1] x2 + 1 and (x² + 8)7 14х (x2 + 8)7 30х + 2 – 3x2] |(2 – 3x²)5 that the first derivative of g(x) is g'(x) = (2 — Зx?)5 lx² + 8 ' Using the product, qoutient and chain rules you learnt in Unit 7 of this Course, could you confirm whether his brother is right, wrong or a little tipsy.The production of a certain good in terms of the amount of labour invested L and the amount of capital invested K, is given by the production function Q = 5 L²/3 K ¹/3 At this moment 512 units of labour and 1 000 units of capital are invested daily. This means that at this moment 3200 units are produced daily. a) The company receives an order for 3200 units. Give an implicit equation for the combinations (L, K) resulting in a production of 3200 units. b) Determine K'(L) for L = 100 using implicit differentiation. Interpret your result in a sentence in the context of the word problem. c) Determine the following expression and interpret your result in a sentence in the context of the word problem: 100 K'(100) K d) Set up the graph of K as a function of L using a logarithmic scale on both axes. Interpret the slope in L = 100.
- Anation with fixed quantities of resources is able to produce any of the following combinations of carpet and carpet looms: Yards of Carpet (Millions) Carpet Looms (Thousands) 75 60 20 45 35 30 45 15 50 50 These figures assume that a certain number of previously produced looms are available in the current period for producing carpet. To answer this question, assume carpet looms is on the y-axis and yards of carpet is on the x-ais The plot of these points that make up the PPF has a vertical intercept of and a horizontal intercept of This PPF would be (Enter your responses as a wvhole number) bowed out a straight lineDoubt: Calculations obtained from TC come from function TC = 80 + 3q + 2q2? For example, for q = 1, for me it is TC = 85, not 83 and so on. Please check it. Thanks10K L%. 1). Consider the production function Q = (a) What is the output when K = 100000 and L = 243? (b) Use the marginal analysis to estimate Q(99995, 243) and Q(100000, 245). (c) Use a calculator to compute these two values of Q to three decimal places and compare these values with your estimates in (b). (d) How big must AL be in order for the difference between Q(100000, 243+AL) and its linear approximation Q(100000, 245) + -DAL, to differ by more than two units?