T: P2→R, T(a, + a,x + azx?) = ao

Q: Evaluate the line integral / (x+ 2- rom (0,0) to (3, 0), the path C2 fro co (0,0) by applying the…

A: Consider the provided question, Hello. Since your question has multiple sub-parts, we will solve…

Q: Let f(x, y,z) = x² sin y + y e², where x =t + 3,y = net-1 and z = In 2t. dL at t = 1 is: 1-2n O -12n…

A: Given, fx,y,z=x2siny+yez, where x=t+3,y=πet-1 and z=ln 2t.   The objective is to determine dfdt at…

Q: Let f(x, y,z) = x² sin y +y e, where x t +3,y = net-1 and z = In 2t. %3D at t = 1 is: dt O -12n O…

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Q: (y In a In y) da + dy = 0 O In In y + T Ina += C O y= eCa 1 y%3D O Iny Ce

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Q: Let f(x, y) = xcosy- y'e*y. fry at (1,0)is equal to: %3D | -2/3 -2 1

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Q: dw Use an appropriate form of the chain rule to find dt w = In(10x² – 4y + 5z³); c = t', y = tỉ, z =…

A: From the given function: w=ln10x2-4y+5z3 ; x=t12 ; y=t23 ; z=t-2 As we know that: The Power law:…

Q: D Find the Valve of K So that the fallowing Where u= (XxXz) V= (YY) inner praduct in an.

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Q: 2. What is the relation between dV /dt and dr/dt if V = ()7r? %3D

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Q: Study the analytically of w = L(2) at the points z = 0, z= v2,z = -21

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Q: Prove that A <m AUB where AUB = {0x | r E A}U{ly|y € B}.

A: Solution: Let us consider two sets A and B. It may be possible that A∩B≠∅. But we want to construct…

Q: t (s? + t) ds – s (s² – (s² – t²) dt = 0 %3D - 242 In Let|

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Q: Utt = 8uxx for 0 < x < ™ u(x, 0) = x² u(x, 0) = 1/2cos(8x) Uz(0, t) = u„(T, t) = 0 %3D II

A: The given P.D.E is                            utt=8uxx for…

Q: Find the value of dr/dt at t = 0 if r = (u2 + 7)^(1/3) and u^2t + u = 1.

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Q: AXYZ is isosceles with base YZ and M is any point on YZ. Prove that a. ZXMY > ZY Z. b. XM < XY M

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Q: Hiw Prove that c=z²,'s coutinuous at z=z. O prove that: if Z#Z0 is dis continnity at Z=Zo, where 2o

A: (1) Calculate the RHL and LHL for the function f(z).…

Q: Considering f(, Y, z) = z + x²y and S as part of the cylinder y² + z² = 25 that is between the plans…

A: It is given that surface S is the part of the cylinder y2+z2=25 lies between the planes x = 0 and x…

Q: Let x,(t)-v(t)+sin(r), x,-vt) then x,(t)= xN-cosit) Select ones True, False

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Q: Evaluate | CurlF nds F(x,v,z) = - yzi + xzj – xyk, and S is that part of the cylinder 4= x² + y²…

A: Given function is  Fx,y,z=yzi+xzj-xyk To evaluate ∬SCurlF·nds where S is the part of the cylinder…

Q: aーメ ターメ2.y to い-X to cylindHcal coordin ates

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Q: Evaluate O F.dT where +(3y %3D y + 1 and C is the triangle with vertices: (0,0), (6,0), (3, 5).

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Q: Let w(z, y, z) = 1 + y² + z?, where z = 3', y = Int, and z = = et. Find I using the Chain Rule.

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Q: Use the Chain Rule to find dz/əs and dz/at. z = tan(u/v), u = 8s + 3t, V = 3s 8t az as az Ət II

A: Given data: *z=tan(uv) *u=8s+3t and v=3s-8t The partial derivative of z with respect to s is given…

Q: 2. In A ABC, X is the centroid. A A If CW = 15, find cx and XW. W, B. If BX = 8, find BY and XY.

A: we know that  centroid is located 23 of the distance from the vertex along the segment that connects…

Q: dw Use an appropriate form of the chain rule to find dt In(14.x² – 4y + 523); x = ti, y= tổ, z = t-2…

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Q: Q2// A :-Compute for the following at t=1 z= ze", z =t, y=t1

A: Since you have asked multiple questions in a single request so we will be answering only first…

Q: Numerically integrate y=x-X+4 from a-0 to b= 2 using single segment trapezoidal rule. O 12 0 8 0 6 O…

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Q: Let f(x, y,z) = x² sin y + y e², where x = t + 3,y = net-1 and z = ln 2t. %3D df at t = 1 is: dt…

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Q: Evaluate p(10ey – 9x)dy – (yx – 5 In x)dx by using Green's Theorem where C is a trapezium with…

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Q: 2. Use the Chain Rule to find for z = e¤/y where x = st and y = s2 +t

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Q: Since z = In (x2 + y2), x = eucosv, y = eusinv əz/ðu =? az/dv =?

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Q: The signal (t) = tan(t) satisfies the Dirichlet conditions. O True O False

A: False

Q: 4. (a) Use the trapezoidal rule to numerically integrate from a = 0 to b= 2 /(z)-0.2+25 +3r

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Q: 2. Let i = y?x + xŷ + z²ż, and let S be the paraboloid z = x2 + y? with 0 < z < 1. Show that fci. de…

A: Given  v→=y2x^+xy^+z2z^ and S be the paraboloid defined by  z=x2+y2      , 0≤z≤1 The boundary C of…

Q: 1. Detanmne the rusutt OF a. Gradi ent f(x.y,z) ex sin cy) In (7) b. Divergenie =7? C Curl F=y2 t…

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Q: Suppose the goind densiy Runction Xanef y i ginen by frxy) = 44 (ス-8) ○ZXC0,○ ス %3D otheraise…

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Q: -H.. OProve that fcz) =z²,s coutinuous at z=z, O prove that: if Z#20 is dis continnity at Z=Zo,…

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Q: (2,2) b) Evaluate o x*y dx + x dy, shown in adjacent figure using green's theorem. where c is…

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Q: 10 The double integral 2x dædy, where T is the triangle with vertices (0,0), (3,0) and (6,3), equals

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Q: 4) Find Yp, using D-operator, where Yp (e). %3D (D+1) (D-2)

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Q: Prove that 3x +8 Viv = dr dy: %3D where R [-1,1] x [0, 2).

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Q: dz for z = da z In(zy) + y, y = c os (x² + 1)

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Q: f(z) dz where a inside r= z-a

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Q: 3 Find the bounded Solution of : = xt メ>。,t>。 >y(メノ) where y cx, o) = %3D and y co,t) = 0 . Ansi] y=…

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Q: Let z = f(x, y) and suppose x = 2u – 3v and y = 5u + 6v. Use the chain rule to express dz in terms…

A: x=2u-3vy=5u+6v Now we apply the chain rule  ∂z∂u=∂z∂x·∂x∂u+∂z∂y·∂y∂u Take derivative of x  and y…

Q: X+Z+3Siny X +2y-2

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Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (5,0,0), (0,6,0), and (0,0,3)

A: This question is related to triple integral, We will solve it using given information.

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (2,0,0), (0,4,0), and (0,0,1)

A: We have to find the volume integral of the given function.

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.