Mechanics and Techniques Problems

13.1. Answer each of the following questions, given the function of two variables: f(x,y) = 8xy - 3x² + 2y².

1. Find the value of the function when x = 2 and y = 1.
2. Determine a value of y so that when x = 10, the function is equal to 124. You may use algebra, Goal Seek or some other method to find the answer, but explain your solution method.
3. Create a graph of the function of one variable g(x) where g(x) = f(x, 3).

13.2. Using the discrimminant identify the shape of the 3D surface plot of each function below. Describe the shape as being either: a bowl, a hill, a saddle, or impossible to tell.

1. f(x,y) = 2x² - 3xy + y² + 4x - 5
2. g(x,y) = 3x² - 2xy + y² + 4y - 5
3. h(x,y) = -3x² + 2xy - y² + 4y - 5x + 1
4. k(x,y) = -0.3x² + 0.2xy - 0.1y² + 4y - 5x + 1

13.3. Get Bent, Inc. sells assembled and unassembled recumbent bicycles. The estimated quantities demanded each year for the assembled and unassembled bikes are x and y units when the corresponding unit prices (in dollars) are

1. Find the annual total revenue function, R(x,y).
2. Find the approximate domain of the revenue function. That is, find the set of values of x and y such that the unit prices are all positive.
3. Create a 3D surface plot of the revenue function for all points (x,y) in the domain.
4. Create a 3D contour plot of the revenue function for all points (x,y) in the domain.

13.4. The revenue function below was developed as a model for the revenue data ”Shaken and Stirred” collected regarding its sales of gin (x) and vodka (y). The sales quantities of each are measured in liters. The company would like to know if the revenue function supports the notion that their products are complementary commodities.

1. Factor the expression to put it into the form below. Assume that the mixed term (the xy term) splits equally into the two demand functions.
2. From your factored revenue function, identify the demand functions for gin (x) and vodka (y) sold by Shaken and Stirred.
3. Analyze your demand functions and explain whether the products are complementary commodities or substitute commodities.

13.5. The contour diagram below shows the total revenue from selling two different products.

1. Give at least four sets of production pairs (q₁,q₂) such that the revenue is positive.
2. Give at least four sets of production pairs (q₁,q₂) such that the revenue is greater than 200,000.