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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

Graph the following functions. f(x) 2x + 2 x + 2 3 x/2 - if x < 0 if 0 ≤ x ≤ 2 ifx > 2
Sketch a graph of y = x5.
Draw a right triangle to simplify the given expression. Assume x > 0 and 0 ≤ θ ≤ π/2. cos (2 sin ¹x)
Find the domain and range of the following functions. a. f(x) = x² + √x 1 b. g(y) y-2 c. h(z) = √₂²-22- 2z - 3
Graph the following equations. Use a graphing utility only to check your work. a. 2x 3y + 10 = 0 b. y = x² + 2x - 3 2 c. x² + 2x + y² + 4y + 1 = 0 d. x² 2x + y² 8y + 5 = 0
Suppose a long-distance phone call costs $0.75 for the first minute (or any part of the first minute), plus $0.10 for each additional minute (or any part of a minute). a. Graph the function c = f(t)
Sketch the graph of a function f with all the following properties. lim_f(x) x →-2 =∞ lim_f(x) = 2 x-3¯ lim f(x) = ∞ x→-2+ lim_ f(x) = 4 x-3+ 8 lim f(x)= x->0 f(3) = 1 = 8
Evaluate the following limits analytically. lim 1877² 2 x-1000
Evaluate the following limits analytically. lim √5x + 6 x-1
Evaluate the following limits analytically. 3 - 7x² + 12x 4- x x3 lim
Evaluate the following limits analytically. lim x-4 x³ - 7x² + 12x X 4- x
Evaluate the following limits analytically. lim h→0 √5x + 5h h V5x where x is constant
Evaluate the following limits analytically. lim √3x + 165 x-3
Evaluate the following limits analytically. lim 1-1/3 t - 1/3 (3t-1)²
Evaluate the following limits analytically. 1-x² lim x1x²8x + 7
Evaluate the following limits analytically. lim pl P >1 p³ - 1 p-1
Evaluate the following limits analytically. 1 1 lim 3 x - 3 \ x +1
Evaluate the following limits analytically. 1 Vsin lim XT/2 x + π/2 x - 1
Evaluate the following limits analytically. √x - 3 lim x 81 x 81
Evaluate the following limits analytically. sin²0 lim 8/4 sin cos²0 cos 0
Without using a calculator, evaluate, if possible, the following expressions.cos-1 (-1)
Without using a calculator, evaluate, if possible, the following expressions. cos ¹(- /2 2
Without using a calculator, evaluate, if possible, the following expressions. -1 Sin 1 V3 2
Without using a calculator, evaluate, if possible, the following expressions.tan-1 1
Without using a calculator, evaluate, if possible, the following expressions. -1 cos¹ (cos 7π/6)
Without using a calculator, evaluate, if possible, the following expressions. cos-1 2
Evaluate the following integrals as they are written. 4 SS.CF 1. ху (x² + y²)² dx dy
Evaluate the following integrals as they are written. S.S. 0 In x x³ey dy dx
Evaluate the following integrals as they are written. 3 X - dy dx IJI Y y
Explain how cylindrical coordinates are used to describe a point in R3.
Let S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. T:x=2u, y = v/2
Suppose S is the unit square in the first quadrant of the uv-plane. Describe the image of the transformation T: x = 2u, y = 2v.
Let S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. T:x=-u, y = -v
Explain how to find the balance point for two people on opposite ends of a (massless) plank that rests on a pivot.
Explain how spherical coordinates are used to describe a point in R3.
Explain how to compute the Jacobian of the transformation T: x = g(u, v), y = h(u, v).
Assuming f is integrable, change the order of integration in the following integrals. [[ f(x,y) f(x, y) dy dx
If a thin 1-m cylindrical rod has a density of ρ = 1 g/cm for its left half and a density of ρ = 2 g/cm for its right half, what is its mass and where is its center of mass?
Sketch the following systems on a number line and find the location of the center of mass. m₁ = 10 kg located at x = 3 m; m₂ = 3 kg located at x = -1 m
Explain how to find the center of mass of a thin plate with a variable density.
Assuming f is integrable, change the order of integration in the following integrals. 1 f(x, y) dx dy 0 Jy-1
Let S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. T: x 2u + v, y = 2u =
Let S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. T:x= (u + v)/2, y = (u - v)/2
Sketch the following systems on a number line and find the location of the center of mass. m₁ = 8 kg located at x = 2 m; m₂ = 4 kg located at x = -4 m; m3= 1 kg located at x = 0 m
Assuming f is integrable, change the order of integration in the following integrals. S.S. 1-y² f(x,y) dx dy
Suppose S is the unit cube in the first octant of uvw-space with one vertex at the origin. Describe the image of the transformation T: x= u/2, y = v/2, z = w/2.
In the integral for the moment Mx of a thin plate, why does y appear in the integrand?
Let S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. T:x=u²v², y = 2uv
Find the mass and center of mass of the thin rods with the following density functions. p(x) = 1 + sin x, for 0 ≤ x ≤ T
Let S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. T: x = = u сos (TV), y = u sin (TV)
In the integral for the moment Mxz with respect to the xz-plane of a solid, why does y appear in the integrand?
Find the mass and center of mass of the thin rods with the following density functions. p(x) = 2x²/16, for 0 ≤ x ≤ 4
Let S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. T:x= 2uv, y = u² - y²
Find the mass and center of mass of the thin rods with the following density functions. p(x) = 1 + x³, for 0 ≤ x ≤ 1
Let S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. T: x = v sin (Tu), y = v cos (Tu)
Find the mass and center of mass of the thin rods with the following density functions. p(x) = 2 + cos x, for 0 ≤ x ≤ T
Choose a convenient method for evaluating the following integrals. 2y dA; R is the region bounded by x = 1, x = 2, √x + 1 R y = x³/2, and y = 0.
What coordinate system is suggested if the integrand of a triple integral involves x2 + y2?
Choose a convenient method for evaluating the following integrals. ff (x (x + y) dA; R is the disk bounded by the circle r = 4 sin 0. R
Choose a convenient method for evaluating the following integrals. JJx x-1/2e dA; R is the region bounded by x = 1, x = 4, R y = Vx, and y = 0.
Find the image R in the xy-plane of the region S using the given transformation T. Sketch both R and S. S = {(u, v): u² + y² ≤ 1}; T:x = 2u, y = 4v
Find the image R in the xy-plane of the region S using the given transformation T. Sketch both R and S. S = {(u, v): v ≤ 1-u, u ≥ 0, v ≥ 0}; T: x = u, y = v²
What coordinate system is suggested if the integrand of a triple integral involves x2 + y2 + z2?
Find the mass and center of mass of the thin rods with the following density functions. p(x) = {₁ 1 1+x if 0 ≤ x ≤ 2 if2 < x≤ 4
Find the mass and center of mass of the thin rods with the following density functions. p(x) x² (x(2 - x) if 0 if 1 ≤ x ≤ 1 < x < 2
Choose a convenient method for evaluating the following integrals. S 0 Jy1/3 x ¹0 cos (x¹y) dx dy
Choose a convenient method for evaluating the following integrals. (x² + y²) dA; R is the region {(x, y): 0 ≤ x ≤ 2, R 0 ≤ y ≤ x}.
Evaluate the following integrals in cylindrical coordinates. 2πT 1 SIL 0 dz r dr de X N -1 -+--- y
Find the image R in the xy-plane of the region S using the given transformation T. Sketch both R and S. S = {(u, v): 2 ≤ u ≤ 3,3 ≤ y ≤ 6 }; T: x = u, y = v/u
Find the image R in the xy-plane of the region S using the given transformation T. Sketch both R and S. S = {(u, v): 1 ≤ u≤ 3,2 ≤ v≤ 4}; T: x = u/v, y = v V
Evaluate the following integrals in cylindrical coordinates. p3 p√9. √9-y² 9-3√√√²+y² 3 X dz dx dy 2. 9 -3 3
Compute the Jacobian J(u, v) for the following transformations. T:x = 3u, y = -3v у =
Evaluate the following integrals over the specified region [[ 3x²y dA; R = {(r, 0): 0 ≤ r ≤ 1,0 ≤ 0 ≤ π/2} R
Evaluate the following integrals in cylindrical coordinates. 2 1 Ꮭ. 2 -3 J 0 / 1 + x? + y? X dz dy dx 31 2 -3 3 Ꭹ
Compute the Jacobian J(u, v) for the following transformations. T: x = 4v, y = -2u
Choose a convenient method for evaluating the following integrals. 2 SS x³yV1 + xy² dx dy
Evaluate the following integrals in cylindrical coordinates. (x² + y2)3/2 dz dx dy -7³²J-1 X -1 1 1 y
Evaluate the following integrals in cylindrical coordinates. •√2/2 VI-2² S.S. 0J0 X et-y dy dx dz
Compute the Jacobian J(u, v) for the following transformations. T:x = 2uv, y = u? – v2
Compute the Jacobian J(u, v) for the following transformations. T:x= = u сos (πv), y = u sin (Tv)
Evaluate the following integrals over the specified region 1 (1 + x² + y²)² JS R dA; R = {(r, 0): 1 ≤ r ≤ 4,0 ≤ 0 ≤ πT}
Evaluate the following integrals in cylindrical coordinates. 16-x 4 -4√ √16-2²√ √√√x+1² dz dy dx
Evaluate the following integrals in cylindrical coordinates. 1/2 V/1-y² -1J0 V3y (x² + y2)1/2 dx dy dz
Find the coordinates of the center of mass of the following plane regions with variable density. Describe the distribution of mass in the region. R = {(x, y): 0 ≤ x ≤ 4,0 ≤ y ≤ 2}; p(x, y) =
Compute the Jacobian J(u, v) for the following transformations. T:x = (u + v)/√2, y = (u - v)/√2
Evaluate the following integrals in cylindrical coordinates. √9-x² √x²+x² "T 0 0J0 (x² + y2)-1/2dz dy dx
Compute the Jacobian J(u, v) for the following transformations. T:x= u/v, y = v
Find the coordinates of the center of mass of the following plane regions with variable density. Describe the distribution of mass in the region. R = {(x, y): 0 ≤ x ≤ 1,0 ≤ y ≤ 5}; p(x, y) =
Solve the following relations for x and y, and compute the Jacobian J(u, v). u = x + y₂ v = 2x - y
Solve the following relations for x and y, and compute the Jacobian J(u, v). u = xy, v = x
Rewrite the following integrals using the indicated order of integration. S.S. V16- 0 V16-giả gử V 0 f(x, y, z) dy dz dx in the order dx dy dz
Solve the following relations for x and y, and compute the Jacobian J(u, v). u = 2x - 3y, v = y = x -
Solve the following relations for x and y, and compute the Jacobian J(u, v). u = x + 4y, v = 3x + 2y
To evaluate the following integrals carry out these steps.a. Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables.b.
Rewrite the following integrals using the indicated order of integration. 2 9-x² II. (5.5.2) dy dz 0 0 f(x, y, z) dy dz dx in the order dz dx dy
To evaluate the following integrals carry out these steps.a. Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables.b.
Evaluate the following integrals, changing the order of integration if needed. 0 1- -zJ-VI-2² dy dx dz
Evaluate the following integrals, changing the order of integration if needed. sin .x SS S 0 0 dz dx dy
Evaluate the following integrals, changing the order of integration if needed. √2-x²/28-1²-² Tim -√2-x²/2√x²+3y²2 dz dy dx
To evaluate the following integrals carry out these steps.a. Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables.b.

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