Chapter 15 Multivariable Calculus

Section 1 – Functions of Several Variables

Section 2 – Partial Derivatives

Section 3 – Extrema of Multivariable Functions

Section 4 – Constrained Maxima and Minima

Chapter 14 Methods of Integration

Section 1 – The Substitution Method

Section 2 – Integration by Parts

Section 3 – Area Between Curves

Chapter 14 Practice Solutions (PDF) – 9/5/19

Chapter 13 Measuring Area

Section 1 – Antiderivatives

Section 2 – Approximating Area

Section 3 – Definite Integral

Section 4 – The Fundamental Theorem of Calculus

Chapter 13 Practice Solutions (PDF) – 9/5/19

Business Applications of Extrema

When we optimize business functions such as profit, revenue, or average cost, the functions are ultimately derived from financial data produced by the business. Ideally, these data would uniquely determine a function which could be optimized. However, conditions at a business normally do not allow the trends formed by cost and revenue data to be fit perfectly by a function. Costs vary due to fluctuations in the cost of materials, electricity, and labor. Prices may also change leading to fluctuations in revenue and profit. Functions like those in Section 12.3 are typically obtained through linear and nonlinear regression.

In this section, the objective functions we wish to minimize describe a particular variable cost and are constructed by examining the application. We’ll also look at designing an object with maximum volume subject to limitations on its dimensions. In each case, information about the application will be used to model the objective function. Once the objective function is in place, we’ll analyze the function to find its relative extrema using the derivative of the objective function.

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Section 12.4 Workbook (PDF) – 9/5/19

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Section 12.3

Optimizing Business Functions

In earlier sections, we utilized revenue, cost, and profit functions to help us understand the ideas behind marginal functions. In this section we examine the graphs of these functions to help us make decisions about production levels for a business.

A business’s total revenue is the total amount of money received for selling a good or service. In general, the letters TR is used to name the total revenue function. It can be a function of any number of variables, but in this section we’ll think of the revenue function as being a function of the number of units of a product or service produced and sold, Q, or the unit price of a product or service, P.

The total cost a business incurs is the cost of all inputs the business uses in production. Some of these costs are fixed and others are variable. By this, we mean that as production changes, some costs change and others remain the same. The costs that change as the production changes are the variable costs.

Labor, the cost of materials and the cost of utilities are all examples of variable costs. The costs that are constant as the production changes are the fixed costs. Lease payments for a factory and retail space or payments for fire insurance are examples of fixed costs. Since all of a business’s costs are variable or fixed, the total cost is the sum of the fixed and variable costs:

Total Costs = Variable Costs + Fixed Costs

In general, the name TC is used to represent the total cost function. In this section, we assume that the total cost is a function of a single variable representing the number of units of a product or service produced and sold, Q, or the unit price of a product or service produced, P.

The profit for a business is the difference between the total revenue and the total cost,

Profit = Total Revenue – Total Costs

The letters Pr are used to denote the profit function. Don’t confuse Pr with P. Remember, P corresponds to the unit price of a good or service.

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