The productivity of a country is given by f (x, y) = 300x2/3y1/3, where x and y are the amount of labor and capital. (a) Compute the marginal productivities of labor and capital when x = 125 and y = 64. (b) Use part (a) to determine the approximate effect on productivity of increasing capital from 64 to 66 units, while keeping labor fixed at 125 units. (c) What would be the approximate effect of decreasing labor from 125 to 124 units while keeping capital fixed at 64 units?
> Give a specific example of data that might be gathered from each of the following business disciplines: accounting, finance, human resources, marketing, information systems, production, and management. An example in the marketing area might be “number of
> Refer to Example 3. If labor costs $100 per unit and capital costs $200 per unit, express as a function of two variables, C(x, y), the cost of utilizing x units of labor and y units of capital.
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Consider the Cobb–Douglas production function f (x, y) = 20x1/3y2/3. Compute f (8, 1), f (1, 27), and f (8, 27). Show that, for any positive constant k, f (8k, 27k) = k f (8, 27).
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (
> Both first partial derivatives of the function f (x, y) are zero at the given points. Use the second-derivative test to determine the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (x, y) = 1/x + 1/
> Both first partial derivatives of the function f (x, y) are zero at the given points. Use the second-derivative test to determine the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (x, y) = y ex - 3
> Both first partial derivatives of the function f (x, y) are zero at the given points. Use the second-derivative test to determine the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (x, y) = x4 - 4xy
> Both first partial derivatives of the function f (x, y) are zero at the given points. Use the second-derivative test to determine the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (x, y) = 2x2 - x4
> Both first partial derivatives of the function f (x, y) are zero at the given points. Use the second-derivative test to determine the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (x, y) = 6xy2 - 2
> Both first partial derivatives of the function f (x, y) are zero at the given points. Use the second-derivative test to determine the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (x, y) = 3x2 - 6x
> The function f (x, y) = 1/2 x2 + 2xy + 3y2 - x + 2y has a minimum at some point (x, y). Find the values of x and y where this minimum occurs.
> Find a formula C (x, y, z) that gives the cost of material for the rectangular enclosure in Fig. 7(b), with dimensions in feet. Assume that the material for the top costs $3 per square foot and the material for the back and two sides costs $5 per square
> The function f (x, y) = 2x + 3y + 9 - x2 - xy - y2 has a maximum at some point (x, y). Find the values of x and y where this maximum occurs.
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = x4 - 2xy - 7x2 + y2 + 3
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = x3 + x2y - y
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = x4 - 8xy + 2y2 - 3
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = 1/3 x3 - 2y3 - 5x + 6y - 5
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = 15/4 x2 + 6xy - 3y2 + 3x + 6y
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = 2x3 + 2x2y - y2 + y
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = -8y3 + 4xy + 4x2 + 9y2
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = -8y3 + 4xy + 9y2 - 2y
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = x2 - y3 + 5x + 12y + 1
> Find a formula C (x, y, z) that gives the cost of materials for the closed rectangular box in Fig. 7(a), with dimensions in feet. Assume that the material for the top and bottom costs $3 per square foot and the material for the sides costs $5 per square
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = x3 + y2 - 3x + 6y
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = 4x2 + 4xy - 3y2 + 4y - 1
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = 3x2 + 8xy - 3y2 - 2x + 4y - 1
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = -3x2 + 7xy - 4y2 + x + y
> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = x2 - 5xy + 6y2 + 3x - 2y + 4
> Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = ex/(1 + ey)
> Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = (2x - y + 5)2
> Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = 1/(x + y)
> Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = x/y + y/x
> Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = x exy
> Let f (x, y) = xy. Show that f (2, 3 + k) - f (2, 3) = 2k.
> Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = 2x2ey
> Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = x2 - y2
> Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = 5xy
> A farmer can produce f (x, y) = 200 √(6x2 + y2) units of produce by utilizing x units of labor and y units of capital. (The capital is used to rent or purchase land, materials, and equipment.) (a) Calculate the marginal productivities of labor and capita
> Let f (x, y) = xey + x4y + y3. Find ∂2f/∂x2, ∂2f/∂y2, ∂2f/∂x∂y, and ∂2f/∂y∂x.
> Let f (x, y) = x3y + 2xy2. Find ∂2f/∂x2, ∂2f/∂y2, ∂2f/∂x∂y, and ∂2f/∂y∂x.
> Let f (x, y) = x/(y – 6). Compute ∂f/∂y (2, 1) and interpret your result.
> Let f (x, y) = xy2 + 5. Evaluate ∂f/∂y at (x, y) = (2, -1) and interpret your result.
> Let f (x, y) = (x + y2)3. Evaluate ∂f/∂x and ∂f/∂y at (x, y) = (1, 2).
> Let f (x, y) = xy. Show that f (2 + h, 3) - f (2, 3) = 3h.
> Let f (x, y) = x2 + 2xy + y2 + 3x + 5y. Find ∂f/∂x (2, -3) and ∂f/∂x (2, -3).
> Let f (x, y, z) = xy/z. Find ∂f/∂x, ∂f/∂y, and ∂f/∂z.
> Construct angles with the following radian measure. -π
> Determine the radian measure of the angle shown.
> Determine the radian measure of the angle shown.
> Determine the radian measure of the angle shown.
> Evaluate the given integral. ∫0 π/4 (2 + 2 tan2 x) dx
> Let f (x, y, z) = xz eyz. Find ∂f/∂x, ∂f/∂y, and ∂f/∂z.
> Evaluate the given integral. ∫0 π/4 tan2 x dx
> Evaluate the given integral. ∫ (2 + tan2 x) dx
> Evaluate the given integral. ∫ (1 + tan2 x) dx
> Evaluate the given integral. ∫ tan2 3x dx
> Evaluate the given integral. ∫ tan2 x dx
> Find the average of the function f (t) over the given interval. f (t) = cos t + sin t, - π ≤ t ≤ 0
> Find the average of the function f (t) over the given interval. f (t) = 1000 + 200 sin 2 (t - π /4), 0 ≤ t ≤ 3 π /4
> Find the average of the function f (t) over the given interval. f (t) = t - cos 2t, 0 ≤ t ≤ π
> Find the average of the function f (t) over the given interval. f (t) = 1 + sin 2t – 1/3 cos 2t, 0 ≤ t ≤ 2π
> In Fig. 2: Find the shaded area A4. Figure 2: 1 Y 0 A₁ cosx TU2 A3 sin x X
> Let f (x, y, z) = zex/y. Find ∂f/∂x, ∂f/∂y, and ∂f/∂z.
> In Fig. 2: Find the shaded area A3. Figure 2: 1 Y 0 A₁ cosx TU2 A3 sin x X
> In Fig. 2: Find the shaded area A2. Figure 2: 1 Y 0 A₁ cosx TU2 A3 sin x X
> In Fig. 2: Find the shaded area A1. Figure 2: 1 Y 0 A₁ cosx TU2 A3 sin x X
> Evaluate the following integrals. ∫ 2 sec2 2x dx
> Evaluate the following integrals. ∫ sec2 x/2 dx
> Evaluate the following integrals. ∫-π π (cos 3x + 2 sin 7x) dx
> Evaluate the following integrals. ∫0 π (x - 2 cos (π - 2x))
> Evaluate the following integrals. ∫ cos (6 - 2x) dx
> Evaluate the following integrals. ∫0 π /2 cos 6x dx
> Evaluate the following integrals. ∫ (3 cos 3x - 2 sin 2x) dx
> Let f (x, y, z) = (1 + x2y)/z. Find ∂f/∂x, ∂f/∂y, and ∂f/∂z.
> Evaluate the following integrals. ∫ sin (π - x) dx
> A spirogram is a device that records on a graph the volume of air in a person’s lungs as a function of time. If a person undergoes spontaneous hyperventilation, the spirogram trace will closely approximate a sine curve. A typical trace
> A spirogram is a device that records on a graph the volume of air in a person’s lungs as a function of time. If a person undergoes spontaneous hyperventilation, the spirogram trace will closely approximate a sine curve. A typical trace
> A spirogram is a device that records on a graph the volume of air in a person’s lungs as a function of time. If a person undergoes spontaneous hyperventilation, the spirogram trace will closely approximate a sine curve. A typical trace
> Find the area of the region bounded by the curves y = x and y = sin x from x = 0 to x = π.
> Find the area of the region between the curve y = cos t and the t-axis from t = 0 to t = 3π/2.
> Find the area of the region between the curve y = sin t and the t-axis from t = 0 to t = 2π.
> Find the area under the curve y = 2 + sin 3t from t = 0 to t = π/2.
> Sketch the graph of y = t + sin t for 0 ≤ t ≤ 2π.
> Find the equation of the line tangent to the graph of y = tan t at t = π/4.
> Let f (p, q) = 1 - p(1 + q). Find ∂f/∂q and ∂f/∂p.
> The identity sin (s + t) = sin s cos t + cos s sin t was given in Section 8.2. Compute the partial derivative of each side with respect to t, and obtain an identity involving cos (s + t).
> If f (s, t) = t sin st, find ∂f/∂s and ∂f/∂t.
> If z = sin wt, find ∂z/∂w and ∂z/∂t.
