In the implicit function exy² + x² = 1, y is a function of x. Find an explicit formula for y' and evaluate it at (x, y) = (0,0).

The sales function for a product is given by S(x)=119+40.5x2−0.5x3 where x represents thousands of dollars spent on advertising, 0≤x≤54 and S is in thousands of dollars.x Find the point of diminishing returns. Enter the amount spent on advertising as well as the sales in dollars. Amount Spent on Advertising:$ ____ total Sales: $___

A T-shirt screener can screen t-shirts (q) in two different ways. He can either use a fast screening machine (F) or a slow screening machine (S). Screen use is defined in terms of ”hours” running. His production function is f(F, S) = 10F + 6S. (a) The screener wants to be able to produce 120 shirts. List three feasible and efficient production plans (combinations of inputs) for doing this. (b) Graph the screener’s isoquant curve for q = 120. (c) The hourly cost of using the fast machine is $800 and the hourly cost of the slow machine is $200. What is the cost minimizing (optimal) combination of inputs for producing 120 shirts? (d) Suppose the screener must now produce 400 shirts. What is the cost minimizing combination of inputs?

Given the function f(x) = ln (x+1), x>0   In applying mathematical methods to economics/business problems, we use a broad menu of functions for the purpose of accurately specifying the properties of the economic variables in question. Find an example that you can use the above function to describe certain economic or business activity.

Which of the following is true? OA. Quantity rises and the effect on price is unknown B. Quantity is unknown and price rises OC. Quantity rises and price falls OD. Quantity and price rise q

f(x1,x2) 4 2 0 -2 -4 −1 2 0 1 1 0 2 -1 X1 3-2 x2 Figure 2: Graph of the function f(x1, x2) = 2x1 = x²-x+3 Consider the following function: f(x1, x2) = 2x1 - x-x+3. - For your visual convenience, a portion of the function is depicted in Figure 2. (a) Compute the gradient vector of f. (b) Compute the Hessian matrix of f. (c) Determine all points that yield stationary values of f. (d) Use the second-order conditions to determine whether each of the stationary val- ues of f is a maximum, a minimum, or neither.

Differentiate the function. g(x) = (x⁹ - 8x)-2 g'(x) = Differentiate the function. 5 f'(s) = f(s) = (2s7 + 1)4 Differentiate the function. 1 g(x) = (2x+5x+1) 3/4 g'(x) = Differentiate the function. dy y = √√√3x² + 6x + 2 dx

To produce the next popular toy, a company has to pay a factory $250, 000 to set up the production line. They also have to pay $25 per item for the raw materials and labor. Write  function for the average cost to produce x items. Then describe what happens to the average cost as the factory produces a large number of toys.

Consider the linear spline function f(x) = 1 + x + h1(x, 2) where h1(x,2) = x-2 if x>2 and 0 otherwise. What is the slope of f at the point x=-1?

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.