By Daniel A. Murray
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Thus this function is likely to have an absolute minimum but no absolute maximum. Begin by calculating the first order partial derivatives: fx = 2x + 3y 6 and fy = 3x + 6y + 3 Then set both partials equal to zero and solve the system of equations. There are several approaches one can take to solve a system of equations such as this. Some methods will be described in detail in Chapter 6, but for now, begin by setting the two equations equal to each other and manipulating the terms: 2x + 3y 6 = 3x + 6y + 3 Þ 0 = x + 3y + 9 Þ x 9 = 3y < previous page page_51 next page > < previous page page_52 next page > Page 52 After setting the two functions equal to each other, subtract 2x + 3y 6 from both sides of the equation.
4. As you can see from the graph, the values for f(x) continue to get larger and larger as x approaches 1. This should not be a surprise. As x 1 the value of (x 1)2 gets smaller and smaller, which yields larger and larger values of f(x). Thus we say that = +¥, or that the limit of f(x) grows without bound. Note: As mentioned in Chapter 1, it is not always easy to determine the overall shape of a function's graph from just a few points. Although you should have an idea of the shape of the graph before you plot any points, this knowledge comes primarily from practice and you should be able to get information on the characteristics of the graph just from looking at the function.
The marginal cost is the change in the cost as we move from one particular level of production to the next. Similarly, marginal revenue (benefits) is the change in revenue from one level of production to the next. , where the slope of the cost function equals the slope of the revenue function). As another rate of change problem, consider Carl Lewis again. Suppose that d = f(t) tells us how much distance Lewis has traveled at a particular point in time. We can use this function to calculate his instantaneous speed at time t: where v = velocity, d = distance, and t = time.
A First Course in Infinitesimal Calculus by Daniel A. Murray