.Determine if F(x, y) = (2xy + y-2)i + (x² – 2xy-3)j for y > 0 is conservative.

Q: If f(x,y,z)=xe^{4y}\sin(6z), then the gradient is

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Q: is The antiderivative of the function f(x) =. 2x а. 2 In x + c b. 1

A: Givenf(x)=12xAntiderivative of f(x)=?Antiderivative is nothing but integral.

Q: 3. Find fz and fy where f(x,y) = In(x + y²).

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Q: x+y If Z = x.n. f (***), then show that: X. Zx + y.Zy = z

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Q: (b) f(x) = where A; E S™ – (det(Ao + x1A1 + ...+ xn An))/m, on {x | Ao + ¤1A1 ++xnAn > ► 0},

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Q: If f(x, y, z) = ln(xª + yl0 + 2³) + V (x10 + y8 + z4), then find fr, fy and f; at (1, 1, 1).

A: The given function is fx, y, z=lnx4+y10+z8+x10+y8+z4 and the given point is (1, 1, 1). Evaluate fx…

Q: 1. Let F(x, y) = 25x³ + 120x²y + 192xy? + 102y³ – 300x – 450y. Find the following: | ƏF (i). OF…

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Q: Find fæ, fy, and fz. 4,5 f(x, y, 2) = 4z In (a*y° cos(5z))

A: We can answer all the three question as below.

Q: Find an upper bound for ||L|| where L: R² → R³ with L(x, y) = (2x + y,x + 3y, 3x – 2y) -

A: Here we use the definition of boundedness of a linear operator.

Q: Let f(x, y) = - 4xy² – 2x² + 8y². Find and classify the critical points of f.

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Q: 2. If f (x) = Sn* et? 0, dt, find f'(x) V2t+sect

A: The solution is given as

Q: Find fzyr. f(x, y, z) = x³y5₂6 + xy³ + yºz

A: Given: fx,y,z=x3y5z6+xy3+y6z We have to find fzyx For fzyx we first have to find the partial…

Q: Let f(r, y, z) = ln(x² + z) + y²e#² – cos(2yz). Find fzyr- %3|

A: Consider,                                                                 f(x, y, z) = ln(x² + z) +…

Q: Let fn(x) = E s;(x) Y; be a linear smoother. Find a condition on the weights s;(x) so that fn(x) is…

A:

Q: Let z = f(x, y) = 5x – x³ + 2xy – y². - Find and classify all the critical point(s) of this…

A: Differentiate the function fx,y=5x-x3+2xy-y2 partially with respect to x and y, to obtain fx and fy.…

Q: Find and classify the critical points of f(x, y) = 6xy – 4x³ – 3y?

A: According to our guidelines, we can answer only one question or three subparts of a question. other…

Q: Let f(x,y)=-x³+4xy-2y²+1; find the critical points of f and test them using D=D(x,y).

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Q: Evaluate fSS 6e(²+y²+z²)/² dV over B where B= {(x, y, 2) : x² + y² + z² < 1}. Take T = 3.

A: Since we have    x2 + y2 + z2  expression in the integral and region B is the sphere of radius 1…

Q: 1. If f(x, y)=3x' cos y– y sin x– 2.x In y- yln x-4x²y, find ôf ôf d²ƒ _ô²ƒ

A: According to question given thatf(x,y)=3x3 cosy-y2 sinx-2x ln(y)-y ln(x)-4x2y

Q: Let f(x,y)=3x2-2xy+y2-8y  Find the absolute extrema over [0,1]x[0,2]

A: Consider the given function fx,y=3x2-2xy+y2-8y First, find the partial derivatives of the function…

Q: 6. Consider the function f(x, y) = cot ¹(1+x) — 2e¹+³y If x = s²-t² and y = 3t+tan(1-s), use the…

A: Let's evaluate given partial derivative.

Q: 25. For y = f(x) = x2 - 3, x = 5, and Ax = 0.1, find Ay and f'(x) Ax. %3D

A: Given: fx=x2-3 ,x=5 and ∆x=0.1

Q: Find the linear approximation to f(x) = cos(x) near a =- [Where L(x) = f(a) + f'(a)(x – a) ] 3

A: Solution: The objective is to find the linear approximation

Q: 교꾸가고 (a, y) + (0,0) (x, y) = (0,0) f(x, y) = x2+y2 is discontinuous at (0,0).

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Q: Determine the linear approximation of f(x) = x' at x = 1. О а. уз- 1 х-0 О Б. у3 1x +0 O c. Not in…

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Q: Find the linearization L(x) of f(x) = cot(x) at x = π/4.

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Q: Let f(x,y)=3x2-2xy+y2-8y  Find the absolute extrema over [0,1]x[0,2].

A: We have f(x, y)=3x2-2xy+y2-8y find critical points ∂f∂x=6x-2y=0 ... (1)∂f∂y=-2x+2y-8=0  ...(2)solve…

Q: Let f (x, y) = x³ + y? – 4xy + 3x + 2y + 5. Find and classify all critical points of f.

A: *concept:- maxima and minima in partial derivatives.

Q: Find f, and f,. f(x, y) = 13x In xy

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Q: d) The function +(xy)-r*r2y) exp G 18xy is homogencous in of degree X and

A: The given function is: f(x,y)=(x2+2y)e2xy+8xy...(1) The function f(x,y) is said to be homogeneous…

Q: ) Find the linearization of f(x) = sinx at x = 0.(b) Compute dy if y = x2 +x and x changes from x =…

A: Since you have submitted two questions, we'll answer the first question. For the second question…

Q: 1. Use the Chain Rule to find u and ou for w = x²y3 where x = dw at s2 +t2 and y = ds

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Q: The linear approximation at x = and where B is: 0 to 1 X is A + Bx where A is:

A:

Q: Verify that the indicated function y = p(x) is an explicit solution of the given first-order…

A: This question can be solved using the concept of derivatives.

Q: (4) Let f(x, y) = e-y. Find the linearization of f at an appropriate point and use it to…

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Q: Let f(x, y, z) = ln(x² + z) + y²e#z – cos(2yz). Find fzya.

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Q: Find the linearization L(x) of f(x) = cot x at x = 4

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Q: - Find the total differential df of f (x, y) = (x² + y²) 3/2.

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Q: Find the linearization of the function f (x)= x³ at x = 1. qb |a| sin (a) L (x) = Using this…

A: Given, fx=x5 We know that, If we have a differentiable function f defined near "a" the linearization…

Q: Find f, and f, for f(x,y)=y In (3x +4y). fx fy

A: When the partial differentiation is done with respect to a specific variable, then all other…

Q: Q2- If Z = f(x, y) = sin(x³y5) Find fyy and fyx

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Q: 1) Let f(x, y) = 6x, 0 < x < y < 1 and 0 elsewhere. Find d) f(ylx) e) E(Y|x)

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Q: -2 In(x) Consider the function f(x, y) 3y

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Q: 10.Show that F(x, y) = (1 + xy)e*yi + (ev + x?e*y)j is conservative and find a function "f" such…

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Q: If f(x, y, z) = In(x³ + yl² + 2®) + V (x12 + y6 + 23), then find fr, fy and f; at (1, 1, 1).

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Q: (ii)If y=(x+Jx* +1. prove that y'= Vx +1

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Q: Suppose f(x, y)= g(x)h(y), where g and h are continuously differentiable functions of one variable…

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Q: The function f(x, y) = 2xy + yx + 11 has two critical points. Select one: True O False

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Q: Suppose that the functions f and g, with linear approximations L, and Lz at x = a as shown below.…

A: Limit problem

Q: If f(x) - g(x) = 1, then f and g have the same antiderivative. %3D O True False

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Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.