37. Finding work in two ways Find the work done by xi + yj F : (x² + y?)³/2 over the plane curve r(t) = (e' cos t)i + (e' sin t)j from the point (1, 0) to the point (e²™, 0) in two ways: a. By using the parametrization of the curve to evaluate the work integral. b. By evaluating a potential function for F.

Q: For an equation " + a? = 0 a) Draw a tangent vector to a solution (2(t), x'(t)) in the phase plane…

A: (a) To solve the given differential equation: Let   p=dxdt, or, x' ⇒dpdt=d2xdt2 ... differentiating…

Q: Show that the curve r= Vti + Vtj + (2t – 1)k is tangent to the surface x +y² – z = 1 when t=1

A: Given that r=ti+tj+2t-1k and f=x2+y2-z-1 Take the derivative for f, ∇f=ddx,ddy,ddzf=2x,2y,-1…

Q: 54. Work done by a radial force with constant magnitude A par- ticle moves along the smooth curve y…

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Q: For an equation x" +x = 0 a) Draw a tangent vector to a solution (x(1), x' (1)) in the phase plane…

A: (a) Solve the differential equation as follows: Let p=dxdt, or, x'. Differentiate it with respect to…

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Q: Calculate the work done by the force F = (x + y)i + (1 y)j along the curve ellipse x^2 /9 + y^2 /4=…

A: We find the work done by the force F = (x + y)i + (1 y)j along the curve ellipse x^2 /9 + y^2 /4= 1…

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Q: Find the directional derivative of Du" f (2,1) in the direction of: a = 3i + 4j To f(x, y) = InGes…

A: Gradient of a function :  Let f be a differentiable function at  x , y  and let f x , y  have…

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Q: F(1)=r cot"(1.4-r')i +1.4(r² -1) j + Jar +1k t = 0 at and is normal to the plane x+y-z 3

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Q: 6) Let f(xxx)=3xy²-x² be a potential function of a fluid velocity field. a) Find the magnitude of…

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Q: The equation of the tangent to the curve y=4+ sin' x at X=0 is O y = 4 O y = 2 O y = 6 O y = 3

A: Since you have posted multiple questions in this question which are not interlinked so I have solved…

Q: Sketch the slope field and solution curves   a) yy' + 4x = 0, (1,1), (0,2)

A: The given equation is yy'+4x=0 for 1, 1 and 0, 2. To find: sketch the slope field and solution…

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: Use hyperbolic functions to parametrize the intersection of the surfaces x' - y 49 and z= (Use…

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Q: 2) Find the work of F = (x² - y² -4,2xy) on the semi-circle of radius 2 centered at the origin…

A: ∮F·dr=∮x2-y2-4dx+2xydy             =∬S∂∂x2xy-∂∂yx2-y2-4 dxdy               By applying Greens…

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Q: (Calculus) The tangent vector to (x(t), y(t)) is (x'(t), y'(t)). The tangent vector's length ||(x',…

A: Given that the tangent vector to and is the speed of the point along the curve at time t .Here also…

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Q: (a) Given a cylinder x' + y = I and a plane x +z = 1. (i) Find a vector-valued function r(t) that…

A: 1) If r(t) is the function of the curve, then the tangent of the curve is given by  T(t) =…

Q: 3. (a) Given the vector valued function r(t) = 4 cos zti + 4 sin atj + tk. Find the position vector,…

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

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Q: Find the general solution to the following 3-by-3 linear system: 1 -5 10 2 -4 8 X. dt 3 -5 8.

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…