A force field F moves a particle with mass m > 0 along a trajectory C in space parametrized by r(t) with a

Q: Let y = V7 – x. %3D Find the differential dy when a = 4 and dx 0.3 Find the differential dy when x =…

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Q: (x,y) = xy i + x+15 yj, on R'. Find a potential function for

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Q: The force exerted by our home planet is GMx if x R, measured on a unit mass at a distance x from the…

A: We are given the function h(x) as the following. Notice that GMxR3 is continuous everywhere. And…

Q: Assume that after few w months, butterf lies papulace in looa is Calculated as: F(w)=12W+Cos (lbw)…

A: Minimum and maximum of a function can be calculated by testing the function against all the values…

Q: Use the total differential dz to approximate the change in z= 6-x² – y² as (x, y)moves from the…

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Q: Let y = tan(2z + 3). Find the differential dy when z = 2 and dr 0.1 %3D Find the differential dy…

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Q: What is the maximum value of f(t) = sin t + cos t in the interval 0<t< =?

A: Given; f(t) = Sin t + Cos t.

Q: 3. For all a? + y + 2? > 0, verify that the velocity potential 1 $(r, y, z) = Vr? + y² + z2 is a…

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Q: The acceleration of a particle moving back and forth on a lineis a = d2s/dt2 = π2 cos pt m/sec2 for…

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Q: Let u = exsin(y), where x = -(1/2)ln(t) and y = (pi)(t). Find du/dt when x = ln(2) and y = (pi/4)

A: Question is solved.

Q: 2. Show that f(t) =1 – sint defines a smooth flow on the manifold S = ( (cost,sint) ), but %3D g(t)…

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Q: Let z = f(x,y)= x^3+ xy+ y^4    (a)  Use differentials to estimate Δz as x changes from 1 to 1.05…

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Q: Calculate the work done by a constant force field F on a particle that moves once uniformly along…

A: Given, path of the curve x^2+y^2=1 &amp; F(x,y)=(αx,αy)

Q: Find f. fy for f(x,y) = xcosy + %3D x5 siny3 ey

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Q: w =sec (xyz) - Yx +y+z

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Q: Consider the slope field shown. (a) For the solution that satisfies y(0) = 0, sketch the solution…

A: Given,  the slope field :

Q: Use the differential dz to approximate the change in z = √4 − x2 − y2 as (x, y) moves from the point…

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Q: Show that F. dr is independent of its path, given that F(л, у, 2) — + tan 1+x² 2) i+ tanrj+ k. 1+2²…

A: We will show the required result.

Q: Find the total differential of the function f(r, y) = (3 y²+6 2):

A: Given,           The function is fx,y=3y2+6x13 .

Q: Find the work done by F(x, y) = (x, Y) over the curve r(t) = (e' cost, e' sin t) (x² + y?)3/2 from…

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Q: integral f t^3 In t dt

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Q: Show that the displacement u(x, t) in an elastic string of length t fireed at both ends which is set…

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Q: Let y= tanx .(a) Find the differential dy .(b) Evaluate  dy and ∆y if x= and π /4 and dx =-0.1.

A: Given function is (a) Differentiating with respect to x, we get

Q: Find the path of a heat-seeking particle placed at point P on a metal plate whose temperature at (x,…

A: Let the required path be γt=xti+ytj . Then the velocity vector in the direction of maximum…

Q: A force of F = 14x – x N stretches a nonlinear spring by x meters. What work (in J) is required to…

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Q: Let y = V5- z. Find the differential dy whenz= 2 and dr = 0.1 %3D Find the differential dy whenz= 2…

A: y=5-x

Q: Find the potential function for F(x,y,z) = (x^-1z)i + (y^-1z)j + ln(xy)k. Show all steps and explain

A: Potential function of  F, is f such that ∇f=F.

Q: Let y = 5,a. %3D Find the change in y, Ay when r = 5 and Ar = 0.2 Find the differential dy when r =…

A: Our problem is let y=5√x First, it is easiest to take the derivative of this when in a different…

Q: a) Plot a direction field and some integral curves  for xy'-2y=-1 on the rectangular region {-1<or…

A: a) given differential equation  xy'-2y=-1 claim- plot a direction field and sole integral curves for…

Q: Let F(x, y, z) = (e* cos y, -eª sin y, 2). (a) Show that F is conservative. (b) Find a potential f…

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Q: The two functions x(t) and y(t) satisfy the simultaneous equations dx/dt - 2y = -sin(t), dy/dt + 2x…

A: Given: dx/dt - 2y = -sin(t), dy/dt + 2x = 5cos(t)

Q: Dete‎rmine if f (x, y) = ={(xsi‎n3y)/(x‎2+y6), (x, ‎y) ≠ (0, ‎0) ={1, (x, y) = (0, 0) is‎…

A: Solution: A graph for a function that's smooth without any holes, jumps, or asymptotes is called…

Q: 1. Show that the problem may be formulated in terms of the velocity potential (r, 0) satisfying r dr…

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Q: (a) Show that the limit xżycos y lim (x.y)→(0,0) 2x+ +y² does not exist. (b) If u = x cos y+ ye»,…

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Q: Using an appropriate linearization, the local linear approximation of f() = T+ cos z at COS I is -…

A: Use linear approximation

Q: Determine if F=(e* cos y + yz)i + (xz-e* siny) j+(xy+z)k is conservative. If it is conservative,…

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Q: Let y(t) be the position function of a particle in rectlinear motion. Suppose at time t = 0 hr, that…

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Q: Assume that a body moving with velocity v encounters resistance of the form dv/dt = -kv3/2. Show…

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Q: Find the total differential. w = e3Y cos(5x) + z5 dw

A: The total differential of a function w= f(x,y,z) with continuous first partial derivatives in a…

Q: The force exerted by our home planet is GMx if x < R, R3 h(x) = GM if x 2 R, measured on a unit mass…

A: In the question we have to identify if the function is continuous.

Q: Using an appropriate linearization, the local linear approximation of is f(x) = T+ cos r at 2.

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Q: Use the differential dy to approximate Ay whenx changes from x = -2 to x = -1.96. %3D y = x² + 1

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Q: Using an appropriate linearization, the local linear approximation of f(z)= T+ sin z at 27 is

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Q: Find the displacement of the particle with velocity function v (t) = -3 cos (t+) between t 0: %3D t=

A: Consider the function v(t) and the expand it to simplify v(t). Now, refer the value for displacement…

Q: Verify the linear approximation at ( 0, 0) for e^x cos(xy)

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Q: (b) Calculate the rate of change of the function f(r, y, 2) = ryz along the path r(t) = (sin(t), e“…

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Q: Evaluate [xy dx+(x+y)dy along the curve y=x² from (−1,1) to (2,4). [Verify using Mathematica]

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Q: 2. Show that f(t) = 1 – sint defines a smooth flow on the manifold S = ( (cost, sint) ), but g(t) =t…

A: The given question can be solved as shown in step2

Q: The volume V of a right circular cylinder of radius r and height h is V=πr2h (a) How is dV/dt…

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