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mathematics
precalculus 1st

Calculus For Scientists And Engineers Early Transcendentals 1st Edition William L Briggs, Bernard Gillett, Bill L Briggs, Lyle Cochran - Solutions

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. 2 f(x, y) = x²y² - 8x² - y² + 6
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 rate of change of the function in the direction of maximum increase at P.d. Find the directional
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, y) = = (x²-x-2)(y² + 2y)
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, y) = 1 e-(x² + y²²-2x)
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, y) = 2 + |x-1| + y - 1|
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 state that they do not exist. x - y lim 2 (x,y) (0,0) √x² + y²
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 state that they do not exist. x² lim (x,y) (0,0) x² + y² 2
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 state that they do not exist. (x - y)² lim 2 (x,y) (0,0) x² + xy + y² 2
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 state that they do not exist. (x - y)² lim (x,y) →(0,0) (x² + y²)³/2
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 heat in a conducting medium. In two dimensions, Laplace’s equation isShow that the following
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 heat in a conducting medium. In two dimensions, Laplace’s equation isShow that the following
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 heat in a conducting medium. In two dimensions, Laplace’s equation isShow that the following
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 heat in a conducting medium. In two dimensions, Laplace’s equation isShow that the following
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 of the temperature at a location x on the bar at time t and the positive constant k is related to the
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 of the temperature at a location x on the bar at time t and the positive constant k is related to the
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 of the temperature at a location x on the bar at time t and the positive constant k is related to the
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 of the temperature at a location x on the bar at time t and the positive constant k is related to the
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 + 5; P(1, 2); (-3,4)
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, In 3); (1, 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) = 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 + y²); P(−1, 2); (2,1)
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); P(4,1); (-1,2)
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 the function at P. 2 f(x, y) = x² - 4y² – 9; P(1,-2)
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 the function at P. f(x, y) = x² + 4xy - y²; P(2, 1)
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 the function at P. f(x, y) = x4 – x²y + y² + 6; P(−1, 1)
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 the function at P. p(x, y) 2 V20 + x² + 2xy - y²; P(1,2)
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 the function at P. F(x, y) = ex²/²-²¹/2; P(-1, 1)
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 the function at P. f(x, y) = 2 sin (2x - 3y); P(0, π)
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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