Business Calculus

1. Business Calculus Total Accumulated Change

2. Terminology In previous chapters, we discussed cost, revenue and profit as a function of x items produced and sold. In this context, we found that R(x) would represent the total revenue for selling x items. The revenue could be measured in dollars. We would then use the derivative of the revenue function to say something about the additional revenue brought in by selling one more item. In this case, the derivative is measured in dollars per item, but we would say that the additional revenue for one more item is measured in dollars.

3. 4.1 Total Accumulated Change In chapter 4, we will be given information about the marginal cost, marginal revenue, or marginal profit, and asked to determine information about cost, revenue, or profit. Since marginal revenue is a derivative, we must find a way to get original function information from the derivative. To see how this can be done, we will consider area under a curve. We begin with the area of a rectangle: Area = base * height The base of the rectangle will lie along the x – axis, and the height of the rectangle will be determined by the height of the function.

4. Consider a company with a marginal revenue given by the graph below, measured in dollars per item sold. Using this graph, we can find the revenue obtained from selling the first x items. If x =10, we have is the total accumulated revenue from selling items 1 through 10.

5. Notice in this example, 30 represents the height of the function andthe height of the rectangle, and 10 represents the number of items and the base of the rectangle. We can then say that the area of the rectangle gives the total revenue for selling items 1 through 10. If we wanted to find total revenue for selling items 1 through 40, we would add together the areas of 4 different rectangles. Total revenue for selling items 1 through 40: 30(10) + 32(10)+34(10)+36(10) = $1320 .

6. It is important to note that this dollar value represents total accumulated change in revenue. For example, if we wanted total accumulated change in revenue from selling the 41st through the 60th item, we would add the areas of the 5th and 6th rectangle. 38(10) + 40(10) = $780 . This number does not account for any revenue gained before the 41st item sold. • More Terminology We will call the base of each rectangle ∆x, and the height of each rectangle R(x), or more generally, f (x).

7. Riemann Sums If the marginal revenue is given by a continuous function R′(x), we will use rectangles to approximate the total accumulated change in revenue from marginal revenue. Depending on the function, this use of rectangles could result in an overestimate, or an underestimate of total revenue. The number of subdivisions creating the rectangles will also affect the accuracy of the total accumulated change.

8. 4.2 The Antiderivative Is it possible to get an accurate value for total accumulated change for a continuous function? This question could be phrased: is it possible to find the revenue function from the marginal revenue? This process is called ‘finding the antiderivative’. If we had the original revenue function, instead of the derivative of revenue, we could simply find total revenue at x = a by evaluating R(a). We could also find the total accumulated change in revenue from x = a to x = b by finding R(b) – R(a).

9. The Antiderivative The notation for the antiderivative incorporates the idea of summing rectangles. The elongated ‘S’ is called the integral symbol, and represents the sum. The dx is the differential, and tells us the input variable for the function f. The antiderivative of f with respect to x is . In specific examples, the function f (x) can be a marginal cost, marginal revenue, or marginal profit. When an antiderivative is found, we are ‘undoing’ the derivative and finding information about cost, revenue or profit. However, the antiderivative is not exactly the original function. Recall that the derivative of a constant is 0. It is impossible for an antiderivative to retrieve the original constant.

10. Basic Antiderivative Rules as long as n ≠ -1

11. Basic Antiderivative Rules Coefficient Rule: Sum/Difference Rule: The constant of integration can be found for application problems, if additional information is supplied.