find an antiderivative Fon of the function fox)=5sinX such that FCO)=1.

Q: ri + yj F(r,y) (x2 + y2 + 1)?

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Q: Find the formula for the function represented by the integral dt 6t+3 %3D

A: use the substitution method  for finding the integral

Q: Calculate the line integral f -y²dx + xdy where ŕ is the path y² = 2x − x² such that x>1 and y > 0.

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Q: Let /:R'R. S(x. y.2) = (e' tan y,x' In y). Compute D(f).

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Q: Consider the functions z =2e* In y, x In (u cos v), and y= u sin v. dz (a) Express and du dz as…

A: Since you have posted a question with multiple sub-parts, we will solve first two subparts for you.…

Q: Let y = 2x + 2x +3. If Ar = 0.2 at a = 2, use linear approximation to estimate Ay %3D Ay Submit…

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Q: Let H(x) integral ,for -2 ≤ x ≤2.Find the value of s such that H(x)= sH(-x).

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Q: Compute line integral of F(x,y) = (2x² – 3,₁²³)^² + (2x +34²) ↑ с over the following C F-dr SCF ¾ ç…

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Q: Let g(r, y) = ye-, r(t) = In(t² + 1), and y(t) = vt. Find g(r(t), y(t)).

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: r2 dt V4 + t?

A: The given integral is: I=∫02dt4+t2

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: Given etye -Z+ xy - e-0, use e =0, use impicit partial denvatve to finid 2ž at the Point (1 2t

A: Explanation of the solution is given below....

Q: Find a function v(x, y) such that, given u = 2x(1– y), f(z) = u + jv is analytic in z.

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Q: Show that(i × j) × j = i × (j × j)Conclude that the Associative Law does not hold for cross…

A: Using the cross products of the unit vectors i,j, k, we obtain

Q: c) Evaluate | V In(z)dr by using integration by part.

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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: Evaluate fz² In z dz using integration by parts 2 Onz-+ +c Oz-+ Onz-+c Oz² Inz - 2 + c

A: ∫z2lnzdz=∫(lnz)z2dz=(lnz)z33-∫1zz33dz+c=(lnz)z33-∫z23dz+c=(lnz)z33-13z33+c=(lnz)z33-z39+c=z33(lnz)-z…

Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=1,…

A: Given DE is 2x2xdx+ydy=8xdy-ydx..........(1) We have following formulae, dyx=xdy-ydxx2 and…

Q: (B) Find the potential function V = V(r, y) such that %3D + yi+ + r)j = V¥ 1+ x2 + y? 1+ x? + y? and…

A: We have to find the potential function ψ=ψx,y such that x1+x2+y2+yi+y1+x2+y2+xj=∇ψ and also evaluate…

Q: If F (r, s, t) = s? î + (2rs + e3t) ĵ + 3se3tk, find potential function f (r, s,t) such that F = Vf.

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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: + (xytl) Determine the Linearity of this DE. * ya dy y dy O Linear O Non-Linear

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Q: Find over xlapping area two equatione also gemetrical nepresentahon 2. 4'+v?=n, u?+cV- vn)=1 whore n…

A: Given curves are,…

Q: Find the line integral of F =9xy*ï +(x² + 3),yj over the path L.

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Q: Let f(x, y) = x² + In(2x – y). Find the gradient Vf(x, y) at the point (1, 1). O O O O

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Q: Find div F and curl F if F(x, y, z) = 4y³z'i – 11a"z°j – 4xy°k. - -

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Q: Suppose F = ∇ * A, where A = (y + sqrt(z))i + e^(xyz) j + cos(xz) k. Determine the flux of F outward…

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Q: If y and z are functions of u and v, then the Jacobian of the function transforming (x, Y, z) → (x,…

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Q: f .5 p(t)dt = 24 3 and .5 | p(t)dt = 9.6 vhat does the following integral equal? p(t)dt = 3

A: Us definite integration property

Q: Find the defivativesof y o x with resPect Ysxě_eX

A: To find the derivative

Q: | Let u = x*y + y°23 where x = rse', y = rs²e=t, = 2, 8 = 1 and t = 0, by using Chain Rule. p2 Sint.…

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Q: Determine the vactor feld of F. F(z, y) = yi - j

A: This is a problem related to vector field calculation. Based on the general concept we will provide…

Q: Find the work done for a force F = N from x =2 to x =3 m. x2

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Q: Use the local linear approximation of (1 – x)2º at xo O to estimate (0.99)2º. %3D

A: The linear approximation of f(x) at x0 is given by L(x) = f(x0) + f'(x0) (x-x0)

Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=1,…

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Q: compute J secx dr. I n trying to integrate

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Q: . Show that the function u = p(ry) + Vry v(y/x) provides the quation = 0. ду? %3D dx²

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Q: Consider the function f (x,y) = x3y3 + xty. Show that it is homogeneous of %3D degree 5 and verify :…

A: A function f(x, y) is said to be a homogeneous function if f(λx, λy)=λnf(x, y) where λ is a constant…

Q: Express dw/ar and aw/as as functions of r and s using chain rule w = In(x² + y² + z²), x = re sin r,…

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Q: EXERCISE 6. dy +y dr where A is an arbitrary constant () - (+*) + A, Given that prove that tan

A: Solve By substitution

Q: linear or nonlinear.

A: A PDE is said to be linear if the dependent variable u and all its partial differential coefficients…

Q: Evaluate by Cauchy’s integral formula: ∫ c e3z/(z+2)4dz where c: mode z = 4.

A: To find - Evaluate by Cauchy's Integral formula : ∫Ce3zz + 24 dz where C : z = 4     Formula used -…

Q: Express the antiderivative F of f satisfying the given initial condition as an integral. f (x) = sec…

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Q: Evaluate the integral. ' 2 (8t i – t³ j + 2t³ k) dt Jo

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Q: $ ((x,y) (E,6) )e product Prove 2 xt + 3yt on e?. Gompute inner IL(,2), where the norm i's…

A: We have to prove that f( (x, y) , (t, t) ) = 2xt + 3yt is an inner product on R2 . 1.) &lt; ( 0, 0 )…

Q: Express dw/dr and dw/as as functions of r and s using chain rule w = In(x + y + z*), x = re sin r, y…

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Q: Qz: IF U=h cx,g )= Lnce+ ě).show that %3D Sy

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Q: Exercise 1 Show that the integral domain Z[V10] = {a + b/10 | a,b € Z} is a FD.

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.