1) Given the vector-valued function r(t]=e",-te"",tlnt) Evaluate lim r(t) and Evaluate lim r(t) b- Find r '(t) and evaluater (0) Find r'"(t) c- Evaluate fr(t|dt d-

Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = tan x, a = π

A: The linearization of a function f(x) at x = a is, Now differentiating  given function with respect…

Q: Let y be the curve with vector form r(t) = (1/(t+ 2), sin(sin(sin t)), e-"). a) Determine whether…

A: The parametric curve is r(t)=⟨1t+2, sin( sin( sin t)),e-t⟩. x(t)=1t+2, y(t)=sin( sin( sin t)),…

Q: Find the divergence of the field. F = (2x+7y-8z)i + (5x +3y+6z)j + (-6x-y+2z)H div F =

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Q: b) Show that F - dr is independent of its path, given that y F(r, y, 2) = 1+. + tan-2) i+ tanrj+ k.…

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Q: 6. Let F = (x³ sin y, a* cos y). COS (a) Is F a conservative vector field? (b) If your answer to…

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Q: Match the vector field with its graph. F(x, y) = yi -6+ 世 -5+

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Q: 1. If the parametric curve $x=f(t), y=g(t)$ satisfies $g^{\prime}(1)=0,$ then it has a horizontal…

A: TRUE Slope of the curve, when in parametric form = dy/dx = g'(t) / f'(t)

Q: (b)Find the derivative of g in the direction of a= 2i+ curl w where at point (0,-1,1) g = In- 4 + 2…

A: Given, w=cosxyzi+xeyz+1j+xyzk Then, Curl…

Q: Find the directional derivative of the field f(x, y, z) = In(4 x) at P = (1, –2, –5) in the…

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Q: 1. Sketch the curve of the vector-valued function r(t) the graph. (3sect)i + (2tant)j and give the…

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Q: Consider the vector-field F = (2 — ysinr − 1) + (cos x − y ). – - (a) Show that this vector-field is…

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Q: 2. Determine whether the functions f1(x) = e2", f2(x) = e=2#, and f3(x) = sinh 2x are linearly…

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Q: (a) express dw/ dt as a function of t, both by usingthe Chain Rule and by expressing w in terms of t…

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Q: 10. For a vector field F = <2xy + y³, x² + 3xy²+ 2y7 find a function f such that F = Vr.

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Q: 21 (important) Find the fixed point for F(x) = ax + s. When is it attracting?

A: Given Fx=ax+s

Q: Supposing vector field G is related to the flow of fluid P is one point in the domain of G. Give the…

A: To Explain: The geometrical interpretation of G(P) where G is a vector filed and P is a point in…

Q: 2. Consider the vector-valued function R(t) = { Show that is continuous at t = 1. (el-¹, 1-², In (2-…

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Q: Derive a Dirichlet principle for the elliptic operator L in divergence form with A e C² and b= 0.

A: Given elliptical operator L~ in divergence form with A∈C1 and b=0. Suppose A∈C1Ω of the second order…

Q: Match the vector field with its graph. F(x, y) = yi 1 |||||||||| 三 6 靈

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Q: Given: vector-valued function R(t) whose derivative R'(t) = (4t, 6t, —12t) (b) Reparametrize R using…

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Q: 1. Give an equation of the tangent plane to the surface z = In(4y – x) at the point (3,1,0). 2. Use…

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

Q: (1) (a) Show that the line integral tan(y) dr +x sec (y) dy is independent of path, where C is any…

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Q: Consider the vector-valued function (t - 2)² In(t – 2), sin(t – 2) 12 - 4 12-3t+2 12 +t-6 Find dom R…

A: Here, the domain of R(t) is nothing but an intersection of the domains of each term. Domain of…

Q: Verify if the vector field F = (x° cos(4y), -x sin(4y)) is conservative and evaluate the line…

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Q: 6. Let f(x,y) = sin(x + y) + arcsin (xy) a. Find the linearization L(x,y) at the point (0, ")

A: We have to find linearization of given function f(x,y): Concept: The linearization of a function…

Q: FIND THE DIVERGENCE OF THE VECTOR F GIVEN = sin (8x)i + cos (5y); + z ² k FIELD F(X, Y, 2) BY

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Q: The circuitation of the vector field F(x, y)- (3 + xy´, 4x + xy) along the boundary of the plane…

A: The solution are next step

Q: 4. Let f(z) = 1 23 1 - and find [ f(z) dz, where C a circle of radius 2 about the origin with…

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Q: 5. Suppose that the vector field F = is conservative. Find a potential function of the vector field…

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Q: 3 2) Integrarte the vector field hlxry,2)= xyi + ya J+xa k over the twisted cubic PIt)=ti+t?g + t3 k…

A: Given that, the vector field        h(x, y, z) = xy i + yz j + xz k        r(t) = t i + t2 j + t3 k…

Q: A scalar field is given by f = x2/3 + y2/3, where x and y are the Cartesian coordinates. The…

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Q: Match the vector field with its graph. F(x, y) = yi -5+ -5+

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Q: Verify by differentiation that a) / t dt : V1+t2 /1+t² + C; _ b) tan(x)dr = – In(cos(x)) +C, –1/2 <…

A: Given to find ∫t1+t2dtPut 1+t2=udifferentiate 2tdt=dutdt=du2therefore…

Q: 1. Use isoclines to sketch the direction field for y' = x+ 2y and sketch the solution curve that…

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Q: Find the divergence of the vector field F div F -6z sin (a + 7) %3D

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Q: Given a scalar function V(x, y) =-(x² +y² ), the directional derivative of V in the direction of the…

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Q: The slope field oY = f(t,y), with initial condition f(0) = 1, is shown. Find limf(t.y). dt - 10 「 ンン…

A: On the slope field draw the graphical solution with initial condition f(0)=1:

Q: 4) Verify that Green's theorem is true for f,xy?dx – x²ydy ,where C consists of y = x² from (-1,1)…

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Q: Show that the function f(x + iy) = Vlæ|ly| satisfies the Cauchy-Riemann equations at the origin but…

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Q: Consider a regularizing term defined by where k!2* and the linear differential operator D is defined…

A: We are given the following regularizing term.                    ∫ℝN0DF(x)2 dx=∑k=0∞ak∫ℝN0DkF(x)2 dx…

Q: If F = (x² − y² + 2xz)i + (xz − xy + yz)j + (z² + x²)k, find the divergence of - curl F and curl of…

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Q: Let R be a vector-valued function defined by VA -1, , In(t + 1) sin 3t if t+0 (2, 3, 0) if t= 0.…

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Q: Consider F(x, y, z)=e-2 yi-y³/²zx²j+ln(22)k and calculate the divergence of F.

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

Q: The divergence of a vector field F = 3x² yz? î + 2yz ĵ – xz² k at he point (1,0, 1) is -2 O 2 O 1

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Q: 3) Find the divergence and curl of f3xi+ 5x} j + 2y²k, at the point (1,2,3)

A: Solution is done in step 2

Q: 7. Let F = (2x sin(y)e", x² cos(y)e*, z² sin(y)e*). Find along the path c:T = (sin(t), t, cos(t))…

A: F=2xsinyez, x2cosyez, x2sinyez curlF=ijk∂∂x∂∂y∂∂z2xysinyezx2cosyezx2sinyez⇒curlF=0 So, F is a…

Q: 4. (a) Evaluate limit of the two variable funcllon. x- y + 5Vx + 5/y lim V + Vy (x.y)-(0,0) (h) The…

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Q: Compute the divergence and curl of the vector field cos (22 ) sin² (z² yz) 10 yz4 – xyz2 at the…

A: We have to find divergence and curl of vector field:

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.