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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

Compute the first partial derivatives of the following functions. f(x, y) = ln (1 + e)
Use the method of your choice to evaluate the following limits. y sin x lim (x,y) (0.1) x(y + 1)
The following functions have exactly one isolated peak or one isolated depression (one local maximum or minimum). Use a graphing utility to approximate the coordinates of the peak or depression.
Compute the first partial derivatives of the following functions. f(x, y) = 1 tan¹ (x² + y²)
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
Compute the first partial derivatives of the following functions. f(x, y) = 1- cos (2(x + y)) + cos² (x + y)
Use the method of your choice to evaluate the following limits. x² + xy - 2y² lim (x,y) (1,1) 2x² - xy - y²
The following functions have exactly one isolated peak or one isolated depression (one local maximum or minimum). Use a graphing utility to approximate the coordinates of the peak or depression. g(x,
The following functions have exactly one isolated peak or one isolated depression (one local maximum or minimum). Use a graphing utility to approximate the coordinates of the peak or depression. h(x,
Use the method of your choice to evaluate the following limits. lim (x,y) →(1,0) y ln y X
Compute the first partial derivatives of the following functions. h(x, y, z) = (1 + x + 2y)²
Use the method of your choice to evaluate the following limits. |xy| lim (x,y) →(0,0) xy
Use the method of your choice to evaluate the following limits. |x - yl lim (x,y) (0,0) |x + yl
The following functions have exactly one isolated peak or one isolated depression (one local maximum or minimum). Use a graphing utility to approximate the coordinates of the peak or depression. p(x,
Use the method of your choice to evaluate the following limits. xyey lim 2 (x,y) (-1,0) x² + y²
Compute the first partial derivatives of the following functions. g(x, y, z) = 4x - 2y – 2z зу - бх 6x - 3z
Find an equation for the family of level surfaces corresponding to f. Describe the level surfaces. f(x, y, z) = 1 2 2 2 x² + y² + z²
Limits at (0, 0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as r → 0 along all paths to (0, 0). Evaluate the following limits or
Use the method of your choice to evaluate the following limits. lim (x,y) → (2.0) 1 - cos y 2 xy²
Find an equation for the family of level surfaces corresponding to f. Describe the level surfaces. f(x, y, z) = x² - y² - z
Find an equation for the family of level surfaces corresponding to f. Describe the level surfaces. 2 f(x, y, z) = x² + y² - z
Limits at (0, 0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as r → 0 along all paths to (0, 0). Evaluate the following limits or
Find an equation for the family of level surfaces corresponding to f. Describe the level surfaces. f(x, y, z) = √x² + 2z²
Limits at (0, 0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as r → 0 along all paths to (0, 0). Evaluate the following limits or
Limits at (0, 0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as r → 0 along all paths to (0, 0). Evaluate the following limits or
Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. f(x, y, z) = ln (z - x² - y² + 2x + 3)
Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. g(x, y, z) 10 2 x² (y + 2)x+ yz
A classical equation of mathematics is Laplace’s equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady state distribution of
Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. f(x, y) = sin¹ (x - y)²
A classical equation of mathematics is Laplace’s equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady state distribution of
A classical equation of mathematics is Laplace’s equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady state distribution of
Evaluate the following limits. lim x²y ln xy (x,y) →(4,0)
A classical equation of mathematics is Laplace’s equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady state distribution of
Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. h(x, y, z) = Vz² √2² xz + yz - xy
Evaluate the following limits. lim (x,y) →(0,2) (2xy)¹
The flow of heat along a thin conducting bar is governed by the one-dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions)where u is a measure
Evaluate the following limits. 1 cos xy lim (x,y) (0,7/2) 4x²y³
The flow of heat along a thin conducting bar is governed by the one-dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions)where u is a measure
The flow of heat along a thin conducting bar is governed by the one-dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions)where u is a measure
The flow of heat along a thin conducting bar is governed by the one-dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions)where u is a measure
Evaluate the following limits. y²-4 2x lim (x,y) → (2,2) xy
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 + 2y +
Find the following derivatives. Z., and Z., where z = xy - 2x + 3y, x = cos s, and y = sin t
Sketch a graph of the following functions. In each case identify the surface, and state the domain and range of the function. = V₁-x² - y² F(x, y) = 2
Find the first partial derivatives of the following functions. f(x, y) = √x²y³
Evaluate the following limits. lim (x,y) (4,5) √x + y − 3 x + y - 9
Find the following derivatives. w, and wt, where w = z=s-t X-Z y + z x = s + t, y = st, and
Evaluate the following limits. Vy - √x + 1 lim (x,y) (1.2) y-x-1
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. h(x, y) = ex-y; P(In 2,
Sketch a graph of the following functions. In each case identify the surface, and state the domain and range of the function. G(x, y) = V1 + x² + y²
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. P(x, y) = ln (4 + x² +
Find the following derivatives. W, W, and w, where w = z = rt √x² + y² + 2²₁ x = st, y = rs, and
Sketch a graph of the following functions. In each case identify the surface, and state the domain and range of the function. √x² + y² H(x, y) = √x² x2 2
At what points of R2 are the following functions continuous? f(x, y) = ху x²y² + 1
Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. z = 3 cos (2x + y); [-2,2] × [-2,2]
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) = x/(x - y);
Consider the following functions and points P.a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P.b. Find a vector that points in a direction of no change in
Use the Two-Path Test to prove that the following limits do not exist. x + 2y lim (x,y) (0,0) x 2y X Z= x + 2y x - 2y y
Sketch a graph of the following functions. In each case identify the surface, and state the domain and range of the function. P(x, y) 2 √x² + y² - 1
Sketch a graph of the following functions. In each case identify the surface, and state the domain and range of the function. g(x, y) = y³ + 1
Match functions a–d with surfaces A–D in the figure. a. f(x, y) = cos xy b. g(x, y) = c. h(x, y) = d. p(x, y) = ln (x² + y²) 1/(x - y) 1/(1 + x² + y²)
Consider the following functions and points P.a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P.b. Find a vector that points in a direction of no change in
Use the Two-Path Test to prove that the following limits do not exist. y4 - 2x² lim (x,y) (0,0) y + x²
Use the Two-Path Test to prove that the following limits do not exist. 4xy lim (x,y) (0,0) 3x² + y² 4xy Z = 3x² + y² x
Consider the following functions and points P.a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P.b. Find a vector that points in a direction of no change in
Use the Two-Path Test to prove that the following limits do not exist. x² - y² lim (x,y) →(0,0) x² + y² 2
Consider the following functions and points P.a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P.b. Find a vector that points in a direction of no change in
Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. z = x² + y²; [-4,4] × [-4,4]
Consider the following functions and points P.a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P.b. Find a vector that points in a direction of no change in
Consider the following functions and points P.a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P.b. Find a vector that points in a direction of no change in
Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x. x²2y² - 1 = 0
Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. z = x - y²; [0,4] × [-2,2]
Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. z = 2xy; [-2,2] × [-2,2] Z
Use the Two-Path Test to prove that the following limits do not exist. lim (x,y) →(0,0) y³ + x³ xy2
Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x. 2 sin xy = 1
Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x. 2 x3 + 3xy² - ys = 0 - =
Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. 2 Vx² + 4y²; [-8,8] × [-8, 8]
Use the Two-Path Test to prove that the following limits do not exist. lim y 2 X (x,y) (0,0) √x² - y²
At what points of R2 are the following functions continuous? 2 f(x, y) = x² + 2xy - y³
Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x. yey2 = 0
Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. z = ex²-2²; [-2,2] × [-2,2] Z
At what points of R2 are the following functions continuous? p(x, y) = 2 4x²y² x² + y²
Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x. √x² + 2xy + y² = 3
Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. z = √/25 - x² - y²; [−6, 6] × [−6, 6] 2
Verify that fxy = fyx for the following functions. f(x, y) = 2x³ + 3y² + 1
At what points of R2 are the following functions continuous? S(x, y) = 2 4x²y² x² + y²
Given the following equations, evaluate dy/dx. Assume that each equation implicitly defines y as a differentiable function of x. y ln (x² + y² + 4) = 3
Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. 2 z = Vy - x² = 1; [-5,5] x [-5,5]
Verify that fxy = fyx for the following functions. f(x,y) = xe
At what points of R2 are the following functions continuous? f(x, y) 2 2 x(y² + 1)
Verify that fxy = fyx for the following functions. f(x, y) = = cos xy
Verify that fxy = fyx for the following functions. f(x, y) = ex+y
At what points of R2 are the following functions continuous? f(x, y) xy x² + y² 0 if (x, y) = (0,0) if (x, y) = (0,0)
Verify that fxy = fyx for the following functions. 3x²y-¹-2x-¹y² f(x, y) = 3x2y
At what points of R2 are the following functions continuous? f(x, y) = x² + y² x(x² - 1)
Find the indicated derivative in two ways:a. Replace x and y to write z as a function of t and differentiate.b. Use the Chain Rule. z' (t), where z H X + 1 , x = 1² + 2t, and y = t³ - 2 y
Verify that fxy = fyx for the following functions. f(x, y) = √xy
At what points of R2 are the following functions continuous? f(x, y) = 4 2x² y4 + x² 0 if (x, y) = (0,0) if (x, y) = (0,0)
At what points of R2 are the following functions continuous? f(x, y) = ex²+y²
At what points of R2 are the following functions continuous? 2 f(x, y) = √x² + y²

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