Suppose you have been hired by a research firm trying to understand the market for Widgets (a hypothetical product).  Your analysis of the data indicates that the Demand curve for Widgets is estimated to be linear and given by equation Qd = 150 – 3P and the Supply curve for Widgets appears to be linear as well and is estimated as Qs = 2P – 20.   Graphically draw these two curves, labeling all relevant points (such as intercepts for each line) on the horizontal and vertical axes.

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: $___

M3

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

For a certain company, the cost for producing x items is 60x + 300 and the revenue for selling x items is 100x – 0.5x². - The profit that the company makes is how much it takes in (revenue) minus how much it spends (cost). In economic models, one typically assumes that a company wants to maximize its profit, or at least wants to make a profit! Part a: Set up an expression for the profit from producing and selling x items. We assume that the company sells all of the items that it produces. (Hint: it is a quadratio polynomial.) Part b: Find two values of x that will create a profit of $300. The field below accepts a list of numbers or formulas separated by semicolons (e.g. 2; 4; 6 or x + 1; x – 1). The order of the list does not matter. To enter va, type sqrt(a). x = Part c: Is it possible for the company to make a profit of $15,000?

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.

A lake has two species of fish: bass and catfish. The number of bass and catfish living in the lake, B and C respectively, depend on the temperature of the lake, T (in degrees Farenheit). The functions modeling this dependency are given below: B(T) = 8T2 + 976T – 28890 С(Т) 3D — 2T?+244T — 6893 What are the two temperatures at which the number of bass in the lake is the same as the number of catfish? Enter exact expressions or round your answers to the nearest thousandth. T : T

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

According to the equation, sales (Y, measured in $1,000 per month) and advertising (X, measured in $1,000 per month) are related as follows:   Y=40+3X Suppose you are told that the sales committee has set an ambitious goal of $64,000 for next month’s sales. The treasurer needs to know how much to budget for advertising, and you have to come up with the answer.