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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

Find the absolute maximum and minimum values of the following functions on the specified set. x3³/3y³/3 + 2xy on the rectangle f(x, y) = {(x, y): 0 ≤ x ≤ 3,-1 ≤ y ≤ 1}
Find the absolute maximum and minimum values of the following functions on the specified set. f(x, y) = x² + y4 - 4xy+1 on the square {(x, y): -2 ≤ x ≤ 2,-2 ≤ y ≤ 2}
Identify the critical points of the following functions. Then determine whether each critical point corresponds to a local maximum, local minimum, or saddle point. State when your analysis is
Find the absolute maximum and minimum values of the following functions on the specified set. f(x, y) = x²y - y³ on the triangle {(x, y): 0 ≤ x ≤ 2,0 ≤ y ≤2-x}
Find the absolute maximum and minimum values of the following functions on the specified set. f(x, y) = xy on the semicircular disk {(x, y):-1 ≤ x ≤ 1,0 ≤ y ≤ V1 - x²}
What point on the plane x + y + 4z = 8 is closest to the origin? Give an argument showing you have found an absolute minimum of the distance function.
Use Lagrange multipliers to find the dimensions of the rectangle with the maximum perimeter that can be inscribed with sides parallel to the coordinate axes in the ellipse x2/a2 + y2/b2 = 1.
Use Lagrange multipliers to find the dimensions of the right circular cylinder of minimum surface area (including the circular ends) with a volume of 32π in3.
Find the point(s) on the cone z2 - x2 - y2 = 0 that are closest to the point (1, 3, 1). Give an argument showing you have found an absolute minimum of the distance function.
Without using a calculator, evaluate, if possible, the following expressions. sin¹1
Assume logb x = 0.36, logb y = 0.56,and logb z = 0.83. Evaluate the following expressions. X logb y
Determine the following limits and justify your answers. 1² + 5 21+ √² + 5 lim
Determine the following limits and justify your answers. lim (√x² + 7)
Determine the following limits and justify your answers. .2 lim (x² 4 + x² - 9) - →-1
Determine the following limits and justify your answers. 4x + 10 2 V 2x - 2 lim
Investigate the following limits. 1 lim tan 0 0/2+ 3
Investigate the following limits. lim (−10 cot x) x-0
Determine the end behavior of the following functions. f(x) = 4x³+1 1-x³ 3
Investigate the following limits. lim csc x 0
Investigate the following limits. lim csc 0 0-0¹
Evaluate the following limits and justify your answer. lim 2x (²x + 1)² X
Find the first partial derivatives of the following functions. f(x, y) = x²y
Describe the appearance of a smooth surface with a local maximum at a point.
Find the first partial derivatives of the following functions. f(x, y) = 3x² + 4y³
Find the first partial derivatives of the following functions. f(x, y) = 3x²y + 2
Compute the gradient of the following functions and evaluate it at the given point P. f(x, y) = = 4x² - 2xy + y²; P(-1,-5)
Describe the usual appearance of a smooth surface at a saddle point.
Compute the gradient of the following functions and evaluate it at the given point P. f(x, y) = 2 + 3x² - 5y²; P(2,-1)
Find the domain of the following functions. f(x, y) = 2xy - 3x + 4y
Interpret the magnitude of the gradient vector at a point.
Find the first partial derivatives of the following functions. 00 f(x, y) = y + 2x6 + 2xy 8
How many axes (or how many dimensions) are needed to graph the function z = f(x, y)? Explain.
Explain why evaluating a limit along a finite number of paths does not prove the existence of a limit of a function of several variables.
Find the first partial derivatives of the following functions. f(x, y) = xey
How many axes (or how many dimensions) are needed to graph the level surfaces w = f(x, y, z)? Explain.
Evaluate the following limits. lim 101 (x,y) → (2.9)
Compute the gradient of the following functions and evaluate it at the given point P. g(x, y) = x² - 4x²y - 8xy²; P(-1,2)
Find the domain of the following functions. f(x, y) = cos (x² - y²)
Find the first partial derivatives of the following functions. f(x, y) = ln (x/y) In
Evaluate the following limits. lim (x,y) →(1,-3) (3x + 4y - 2)
Give two methods for graphically representing a function with three independent variables.
Compute the gradient of the following functions and evaluate it at the given point P. p(x, y) = V12 4x² - y²; P(-1,-1)
Compute the gradient of the following functions and evaluate it at the given point P. f(x, y) = xey; P(1,0)
Find the domain of the following functions. f(x, y) = V25 - x² - y² 2 2
Find the first partial derivatives of the following functions. g(x, y) = cos 2xy
Find the first partial derivatives of the following functions. h(x, y) = (y² + 1) ex
Evaluate the following limits. (4x² - y²) lim (x,y) →(-3,3)
Compute the gradient of the following functions and evaluate it at the given point P. f(x, y) = sin (3x + 2y); P(π, 3/2)
Find the domain of the following functions. 1 Vx² + y² - 25 f(x, y) = 2
Find the first partial derivatives of the following functions. f(x,y) = exy
Evaluate the following limits. (xy8 - 3x²y³) lim (x,y) →(2,-1)
Find the domain of the following functions. X = sin - y f(x, y) =
Find the domain of the following functions. f(x, y): || 12 2 2 y²-x²
Evaluate the following limits. cos xy + sin xy 2y lim (x,y) → (0,π)
Compute the gradient of the following functions and evaluate it at the given point P. F(x, y) = e¯x²-²y²³; P(−–1,2)
Compute the gradient of the following functions and evaluate it at the given point P. h(x, y) = ln (1 + x² + 2y²); P(2, -3)
Find the first partial derivatives of the following functions. f(s, t): || S-t s+ t
Evaluate the following limits. lim (x,y) →(e²,4) In Vxy
Find the domain of the following functions. g(x, y) = ln (x2 - y) In
Find the first partial derivatives of the following functions. f(w, z) = W ,2 W 2 w² + z²
Evaluate the following limits. lim (x,y) (2.0) → x² – 3xy² x + y
Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. f(x, y) = x² - y²;
Find the domain of the following functions. f(x, y) = sin(y - x²)
Find the first partial derivatives of the following functions. (₂x + 2) [x = (2¹x)8
Evaluate the following limits. 10xy - 2y² lim (x,y)-(1-1) x² + y²
Evaluate the following limits. х² - 3ху lim (x,y)→(6,2) x - Зу
Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. f(x, y) = 3x² + y³;
Find the domain of the following functions. g(x,y) = ху Vx² + y² 2
Find the first partial derivatives of the following functions. s(y, z) = z² tan yz
Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. f(x, y) = 10 − 3x² -
Find the first partial derivatives of the following functions. F(p, q) = Vp² + pq + q² pq+q²
Find the following derivatives. Z., and z,, where z = y = s² + 1² sin (2x + y), x = s² - 1², and
Find the domain of the following functions. h(x, y) = √x - 2y + 4 V
Evaluate the following limits. y² + 2xy lim (x,y) (1.-2) y + 2x
Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. g(x, y) = sin 7(2x -
Evaluate the following limits. lim (x,y) →(3,1) 2 x² - 7xy + 12y² x - 3y
Find the following derivatives. Z., and Z.₁, where z = xy = x²y, x = s + t, and y = s - t
Find the first partial derivatives of the following functions. G(s, t) = Vst s+t
Find the following derivatives. Z., and Z., where z sin x cos 2y, x = s + t, and y = s-t
Sketch a graph of the following functions. In each case identify the surface, and state the domain and range of the function. f(x,y) = 3x - бу + 18
Find the first partial derivatives of the following functions. h(u, v) UV u - v
Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. 2 f(x, y) = √4x² –
Sketch a graph of the following functions. In each case identify the surface, and state the domain and range of the function. h(x, y) = 2x² 2x² + 3y² 2
Evaluate the following limits. lim (x,y)-(-1,1) 2x² - xy - 3y² x + y
Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. f(x, y) = 13e; P(1, 0);
Sketch a graph of the following functions. In each case identify the surface, and state the domain and range of the function. = x² - y² 2 p(x, y) = x²
Find the following derivatives. Z., and Z₁, where z exty, x = st, and y = s + t
Find the first partial derivatives of the following functions. f(x, y) = x²y
a. Determine the domain and range of the following functions.b. Graph each function using a graphing utility. Be sure to experiment with the window and orientation to give the best perspective of the
Consider the following functions f, points P, and unit vectors u.a. Compute the gradient of f and evaluate it at P.b. Find the unit vector in the direction of maximum increase of f at P.c. Find the
Evaluate the following limits. lim (x,y,z)→(1,1,1) 2x x² + xy - xzyz 2 - X
Consider the following functions f, points P, and unit vectors u.a. Compute the gradient of f and evaluate it at P.b. Find the unit vector in the direction of maximum increase of f at P.c. Find the
Evaluate the following limits. lim (x,y,z)→(1,-1,1) xz + 5x + yz + 5y x + y
a. Determine the domain and range of the following functions.b. Graph each function using a graphing utility. Be sure to experiment with the window and orientation to give the best perspective of the
a. Determine the domain and range of the following functions.b. Graph each function using a graphing utility. Be sure to experiment with the window and orientation to give the best perspective of the
Consider the following functions f, points P, and unit vectors u.a. Compute the gradient of f and evaluate it at P.b. Find the unit vector in the direction of maximum increase of f at P.c. Find the
a. Determine the domain and range of the following functions.b. Graph each function using a graphing utility. Be sure to experiment with the window and orientation to give the best perspective of the
Consider the following functions f, points P, and unit vectors u.a. Compute the gradient of f and evaluate it at P.b. Find the unit vector in the direction of maximum increase of f at P.c. Find the
Consider the following functions f, points P, and unit vectors u.a. Compute the gradient of f and evaluate it at P.b. Find the unit vector in the direction of maximum increase of f at P.c. Find the
Use the method of your choice to evaluate the following limits. y² lim (x,y) (0,0) x² + y² 8 2

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