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mathematics
calculus early transcendentals 9th

Questions and Answers of Calculus Early Transcendentals 9th

Reduce the equation to one of the standard forms, classify the surface, and sketch it.4x2 + y2 + z2 − 24x − 8y + 4z + 55 = 0
Describe in words the region of R3 represented by the equation(s) or inequalities.1 ≤ x2 + y2 ≤ 5
Graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface.−4x2 − y2 + z2 = 1
Describe in words the region of R3 represented by the equation(s) or inequalities.0 ≤ x ≤ 3, 0 ≤ y ≤ 3, 0 ≤ z ≤ 3
Describe in words the region of R3 represented by the equation(s) or inequalities.x2 + y2 + z2 > 2z
Graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface.−4x2 − y2 + z2 = 0
If a · b = √3 and a × b = (1, 2, 2), find the angle between a and b.
Graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface.x2 − 6x + 4y2 − z = 0
The figure shows a line L1 in space and a second line L2, which is the projection of L1 onto the xy-plane.(a) Find the coordinates of the point P on the line L1.(b) Locate on the diagram the points
Show that |a × b|2 = |a|2|b|2 – (a · b)2.
If r = (x, y, z) and r0 = (x0, y0, z0), describe the set of all points (x, y, z) such that |r − r0| = 1.
If r = (x, y), r1 = (x1, y1), and r2 = (x2, y2), describe the set of all points (x, y) such that |r − r1| + |r − r2| = k, where k > |r1 − r2|.
Sketch the graph of the function.f(x, y) = x2 + (y − 2)2
Let(a) Evaluate h(−2, 5).(b) Find and sketch the domain of h.(c) Find the range of h. h(x, y) = ev- Vy-x2
Use the Chain Rule to find dz/dt or dw/dt. x - y x = e", y = e x + 2y
Find the limit. lim (xy - 4y²) (x.リ→(3, 2) ,3
Find an equation of the tangent plane to the given surface at the specified point.z = ex−y, (2, 2, 1)
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f(x, y) → L as(x, y) → (a, b) along
Let F(x, y, z)(a) Evaluate F(3, 4, 1).(b) Find and describe the domain of F. = Vy - Vx – 2z.
Use the Chain Rule to find dz/dt or dw/dt.z = sin x cos y, x = √t , y = 1/t
(a) Newton’s method for approximating a solution of an equation f(x) = 0 can be adapted to approximating a solution of a system of equations f(x, y) = 0 and g(x, y) = 0. The surfaces z = f(x, y)
Find the limit. lim (x, y)-(5, -2) (x'y + 3xy2 + 4)
Find an equation of the tangent plane to the given surface at the specified point.z = y2ex, (0, 3, 9)
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If fx(a, b) and fy(a, b) both exist, then f
Let(a) Evaluate f(4, −3, 6).(b) Find and describe the domain of f. f(x, y, z) = In (z - Vx? + y?).
Use the Chain Rule to find dz/dt or dw/dt.z = √1 + xy, x = tan t, y = arctan t
If the ellipse x2/a2 + y2/b2 = 1 is to enclose the circle x2 + y2 = 2y, what values of a and b minimize the area of the ellipse?
Find the limit. x'y – xy ³ lim (x, y)(-3, 1) x - y + 2
Find the directional derivative of f at the given point in the direction indicated by the angle θ.f(x, y) = arctan(xy), (2, −3), θ = 3π/4
Find an equation of the tangent plane to the given surface at the specified point.z = 2√y /x, (−1, 1, − 2)
If f(x, y) = 16 − 4x2 − y2, find fx(1, 2) and fy(1, 2) and interpret these numbers as slopes. Illustrate with either hand drawn sketches or computer plots.
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f has a local minimum at (a, b) and f is
Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all
Use the Chain Rule to find dz/dt or dw/dt.w = xey/z, x = t2, y = 1 − t, z = 1 + 2t
Find the limit. x*y + xy? lim (x, y)(2,-1) x - y?
(a) Find the gradient of f .(b) Evaluate the gradient at the point P.(c) Find the rate of change of f at P in the direction of the vector u. f(x, y) = x'e', P(3, 0), u =(3i – 4j)
Find an equation of the tangent plane to the given surface at the specified point.z = x/y2, (−4, 2, −1)
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f is a function, then lim (x, y)- (2, 5)
If f(x, y) = √4 − x2 − 4y2 , find fx(1, 0) and fy(1, 0) and interpret these numbers as slopes. Illustrate with either hand drawn sketches or computer plots.
Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all
Use the Chain Rule to find dz/dt or dw/dt. w = In Vx? + y² + z?, x= sin t, y cos t, z= tan t %3| %3D
Find the limit. lim (x, y)(7, 7/2) y sin(x – y)
Find an equation of the tangent plane to the given surface at the specified point.z = x sin(x + y), (−1, 1, 0)
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f(x, y) = ln y, then ∇f(x, y) = 1/y.
Find ∂z/∂s and ∂z/∂t in two ways: by using the Chain Rule, and by first substituting the expressions for x and y to write z as a function of s and t. Do your answers agree?z = x2 + y2,
Find the limit. lim ev2r-y (x, y)(3, 2)
Find an equation of the tangent plane to the given surface at the specified point.z = ln(x − 2y), (3, 1, 0)
Find and sketch the domain of the function.g(x, y) = ln(x2 + y2 − 9)
Find ∂z/∂s and ∂z/∂t in two ways: by using the Chain Rule, and by first substituting the expressions for x and y to write z as a function of s and t. Do your answers agree?z = x2 sin y,
Find the limit. xy- x'y lim (x, y)(1, 1) x²
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f(x, y) = sin x + sin y, then −√2 ≤
Find the first partial derivatives of the function.g(x, y) = x3 sin y
A metal plate is situated in the xy-plane and occupies the rectangle 0 ≤ x ≤ 10, 0 ≤ y ≤ 8, where x and y are measured in meters. The temperature at the point (x, y) in the plate is T(x, y),
Use the Chain Rule to find ∂z/∂s and ∂z/∂t.z = (x − y)5, x = s2t, y = s t2
Find the limit. cos y – sin 2y lim (x, y) (T, 7/2) cos x cos y
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f(x, y) has two local maxima, then f must
Find the first partial derivatives of the function.g(x, t) = ext
Find and sketch the domain of the function. In(2 – x) g(x, y) = 1- x² - y
Use the Chain Rule to find ∂z/∂s and ∂z/∂t.z = tan−1(x2 + y2), x = s ln t, y = tes
Show that the limit does not exist. y2 lim (x, y) (0, 0) x? + y?
Find the first partial derivatives of the function.z = ln(x + t2)
Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all
Use the Chain Rule to find ∂z/∂s and ∂z/∂t.z = ln(3x + 2y), x = s sin t, y = t cos s
Show that the limit does not exist. 2ху lim (x, y)(0, 0) x? + 3y?
Find the first partial derivatives of the function. w
Find and sketch the domain of the function.f(x, y) = sin−1(x + y)
Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all
Use the Chain Rule to find ∂z/∂s and ∂z/∂t.z = √x exy, x = 1 + st, y = s2 − t2
Show that the limit does not exist. (x + y) lim (x, y(0, 0) x + y? .2
Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point.f(x, y) = x3y2, (−2, 1)
Find the first partial derivatives of the function.f(x, y) = yexy
Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all
Use the Chain Rule to find ∂z/∂s and ∂z/∂t.z = (sin θ)/r, r = st, θ = s2 + t2
Show that the limit does not exist. x? + xy? lim (x. y)(0, 0) x* + y?
Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point.f(x, y) = y tan x, (π/4, 2)
Find the first partial derivatives of the function.g(x, y) = (x2 + xy)3
Find the first partial derivatives of the function. f(x, y) = (x + y)?
Sketch the graph of the function.f(x, y) = sin x
Use the Chain Rule to find the indicated partial derivatives. p + q p +r' N = p= u + vw, q- v + uw, r= w + uv; ƏN ƏN aN when u = = 2, v = 3, w = 4 du' dv' dw
Find the limit, if it exists, or show that the limit does not exist. x* + y' + z lim (x, y, 2) (0, 0, 0) x* + 2y2 + z
Find the first partial derivatives of the function.w = y tan(x + 2z)
Find the critical points of f correct to three decimal places. Then classify the critical points and find the highest or lowest points on the graph, if any.f(x, y) = y6 − 2y4 + x2 − y2 + y
Use the Chain Rule to find the indicated partial derivatives. u = xe", x= a*B, y= B'y, t = y'a; ди ди ди when a 3D - 1, В 3 2, у 3D 1 da' aB' dy
Use the Squeeze Theorem to find the limit. 1 lim xy sin- (x, y) (0, 0) x? + y?
Find the first partial derivatives of the function. p = Vr* + u? cos v
Find the points on the hyperboloidx2 + 4y2 − z2 = 4where the tangent plane is parallel to the plane2x + 2y + z = 5
Find the critical points of f correct to three decimal places. Then classify the critical points and find the highest or lowest points on the graph, if any.f(x, y) = x4 + y3 − 3x2 + y2 + x − 2y +
Find the critical points of f correct to three decimal places. Then classify the critical points and find the highest or lowest points on the graph, if any.f(x, y) = 20e−x2−y2 sin 3x cos 3y, |x|
Use the Squeeze Theorem to find the limit. xy* lim 4 (x, y)- (0, 0) x* + y*
Find the absolute maximum and minimum values of f on the set D.f(x, y) = x2 + y2 − 2x, D is the closed triangular region with vertices (2, 0), (0, 2), and (0, − 2)
Use Equation 5 to find dy/dx.tan−1(x2y) = x + xy2 ƏF dy ax 5 dx ƏF F, ду ||
Consider the problem of maximizing the function f(x, y) = 2x + 3y subject to the constraint √x + √y = 5.(a) Try using Lagrange multipliers to solve the problem.(b) Does f(25, 0) give a larger
Use the Squeeze Theorem to find the limit. lim (x, y, 2)(0, 0, 0) x2 + y? + z?
Find the first partial derivatives of the function. ax + By? Ф(х, у, z, г) — z18 + zk
Use Equation 5 to find dy/dx.ey sin x = x + xy ƏF dy ax 5 dx ƏF F, ду ||
Use a graph of the function to explain why the limit does not exist. lim (x, y)- (0, 0) 2x? + 3xy + 4y? 3x2 + 5y?
Find the first partial derivatives of the function. u=Vx구 + xg +.+ x ... + x
If u = x2y3 + z4, where x = p + 3p2, y = pep, and z = p sin p, use the Chain Rule to find du/dp.
Find the absolute maximum and minimum values of f on the set D.f(x, y) = x2 + y2 + x2y + 4, D = {(x, y) | |x| ≤ 1, |y| ≤ 1}
Use Equations 6 to find ∂z/∂x and ∂z/∂y.x2 + 2y2 + 3z2 = 1 ƏF az ax Fx az ду Fy ax ƏF F. ду F: az az

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