Search Results

Math Topics

Applied Calculus

Calculus of Functions of Several Variables

##
1
Functions of Several Variables

The manufacturing process of a company is described by the Cobb-Douglas production function \[f(x, y)=Cx^ay^b\]

Approximate, to two decimal places, the number of units produced when \(85\) units of labor and \(20\) units of capital are used.

Find the change in the level of production if the number of units is decreased by \(1\) from \(85\) units to \(84\) units, and the number of units of capital is increased by \(5\) from \(20\) to \(25\).

Notice that the exponents on the variables \(x\) and \(y\) add to one. Show that this function exhibits constant returns to scale.

##
2
Optimization of Functions of Two Variables

A manufacturer markets a product in two states, \(A\) and \(B\), and, because of the different economies of the states, must price the product differently in each. (Such pricing is called *price discrimination* or *differential pricing*.) The manufacturer wishes to sell \(x\) units of the product in state \(A\) and \(y\) units of the product in state \(B\). To do so, the manufacturer must set the price in state \(A\) at \(86-\dfrac{x}{18}\) dollars and in state \(B\) at \(122-\dfrac{y}{28}\) dollars. The cost of producing all \(x+y\) items is \(45\text{,}000+4(x+y)\) dollars. How many items should be produced for states \(A\) and \(B\), respectively, to maximize the manufacturerâ€™s profit, and what is the maximum profit?

##
3
Constrained Maxima and Minima

The production of a manufacturer is given by the Cobb-Douglas production function \[f(x,y)=30x^\frac{1}{5}y^\frac{4}{5}\] where \(x\) represents the number of units of labor (in hours) and \(y\) represents the number of units of capital (in dollars) invested. Labor costs \(\$15\) per hour and there are \(8\) hours in a working day, and \(250\) working days in a year. The manufacturer has allocated \(\$4\text{,}500\text{,}000\) this year for labor and capital. How should the money be allocated to labor and capital to maximize productivity this year?

##
4
The Total Differential

The productivity of a company (the number of units a company is able to produce) is described by the Cobb-Douglas production function \[f(x,y)=42x^\frac{3}{7}y^\frac{4}{7}\] where \(x\) represents the number of units of labor and \(y\) the number of units of capital utilized. Approximate the change in output if the number of units of labor is decreased from \(2\text{,}187\) to \(2\text{,}183\), and the number of units of capital is increased from \(128\) to \(130\).

##
5
Double Integrals as Volume

Find the volume of the space under the surface \[f(x,y)=x+4y\] and over the rectangular region bounded by \(2\leq x \leq 5\) and \(3\leq y \leq 10\). The rectangular region is pictured in Figure 6. The view is from above looking down onto the \(xy\)-plane from the \(z\)-axis.

Find the volume of the space under the surface \[f(x,y)=3x^2+6xy^2\] and over the region bounded by the curve \(g_1(x)=x^3\), the line \(g_2(x)=4x\), and between \(x=0\) and \(x=2\). The region is pictured in Figure 7. The view is from above looking down onto the \(xy\)-plane from the \(z\)-axis.

In Example 3 we used the double integral \[\int_a^b\int_{x^3}^{4x} \left(3x^2+6xy^2\right)dy\,dx\] to determine the volume of the space below the surface \[f(x,y)=3x^2+6xy^2\] Show that the same volume can be found by changing the order of integration to \(dx\, dy\).

A manufacturer produces laser-cut glass rods that are a radius of \(1\) inch and are to be \(10\) inches long. Because of some variability in the production process, the glass rods have radii that are uniformly distributed between \(0.995\) and \(1.005\) inches and lengths that are uniformly distributed between \(9.995\) and \(10.005\) inches. The cuts for the radius and length are independent of each other. What is the probability that a randomly selected glass rod will have a radius between \(0.999\) inches and \(10.001\) inches?

Molds are microscopic organism found both outside the home and inside the home. Outside, they help to break down plant and animal matter. Inside, however, they can cause health problems such as skin rash, eye irritation, or chronic cough, and they can put people with respiratory disease at risk for lung infection. Imagine a wall covered with a population of mold. Take the lower left corner of the wall as the origin of a Cartesian coordinate system with the \(x\)-axis running along the bottom of the wall and the \(y\)-axis running along the left side of the wall. See Figures 8 and 9.

##
6
The Average Value of a Function

The weekly revenue realized by a company is approximated by the revenue function \[R(x, y)=350x+600y-4x^2-3y^2\] for the sale of \(x\) units per week of Product \(A\) and \(y\) units per week of product \(B\). Over the year the company produces between \(20\) and \(30\) units of product \(A\) each week and between \(80\) and \(110\) units of product \(B\) each week. Approximate this companyâ€™s average weekly revenue from products \(A\) and \(B\) over the year.