Find the average of the function f (t) over the given interval. f (t) = 1 + sin 2t – 1/3 cos 2t, 0 ≤ t ≤ 2π
> 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
> 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 effec
> 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
> 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.
> If f (s, t) = sin s cos 2t, find ∂f/∂s and ∂f/∂t.
> Show that y = 3 sin 2t + cos 2t satisfies the differential equation y’’ = -4y.
> If f (t) = sin2 t, find f ‘(t).
> f (t) = et tan t
> f (t) = etan t
> Differentiate (with respect to t or x): f (t) = tan 2t / cos t
> Differentiate (with respect to t or x): f (t) = sin t / tan 3t
> Let f (L, K) = 3√(LK). Find ∂f/∂L.
> Differentiate (with respect to t or x): y = sin4 e3x
> Differentiate (with respect to t or x): y = e3x sin4 x
> Differentiate (with respect to t or x): y = ln (cos x)
> Differentiate (with respect to t or x): y = ln (sin x)
> Differentiate (with respect to t or x): y = ln x cos x
> Differentiate (with respect to t or x): y = sin x tan x
> Differentiate (with respect to t or x): y = tan (sin x)
> Differentiate (with respect to t or x): y = sin (tan x)
> Differentiate (with respect to t or x): y = tan e-2x
> Differentiate (with respect to t or x): y = tan (x4 + x2)
> Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = √(x2 + y2)
> Differentiate (with respect to t or x): f (x) = tan3 2x
> Differentiate (with respect to t or x): f (x) = cos3 4x
> Differentiate (with respect to t or x): f (x) = (cos x – 1) / x3
> Differentiate (with respect to t or x): f (x) = cos 2x / sin 3x
> Differentiate (with respect to t or x): g(x) = sin(-2x) cos 5x
