Question A1 Consider a vector field F= 2x° yị +x'j., and a curve C defined by the equation y = 2x². Calculate the line integral (F•dr along the curve C starting at x = 0 and finishing at x = 1. 2 1.8 1.4 12 1 0.8 F 0.6 0.4 0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 X-axis

Q: Draw the slope field (direction field) for the following DE: y'=y-t

A: Given DE is Make table as follows:…

Q: Find the vector equation of the tangent line to the curve given by 7(t) = sinti + cos (2t)j + tan tk…

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Q: 1. Let ř(1) = . Find a vector and parametric equation of the line tangent to F(1) at the point (-…

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Q: 3. Consider the following helix with vector equation 7 (t) = 2 cos (t) 7 (t) 7 + tỉ + 2 sin (a)…

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Q: Let r(t) = (sin 3t +2t)i + (teʻ)j+(t° + 2t)k. Find the unit tangent vector at the point where t = 0.…

A: The unit tangent vector can be calculated using the differentiation. First obtain the derivative of…

Q: Find r'(t), r(t), and r'(to) for the given value of to- r(t) = (e², e²t), to = 0 r'(t) = r(to) =…

A: rt=et,e2t,t0=0

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Q: A sodium ion (Na+) moves in the xy-plane with a speed of 2.90 ✕ 103 m/s. If a constant magnetic…

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Q: FInd the tangential and normal components of the acceleration for the curve 8 r(t) = (1 + t) 3/2 i +…

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Q: Find r'(t)'r" (t) given the following vector function. r(t) = 3cos(t)i+ 4sin(t)j cost sint -25cost…

A: Given vector function is, r(t)=3cost i+4sint j

Q: Let 7(t) = (t + 4, – e*, 3 sin(3t)) - Find the unit tangent vector T (t) at the point t = :0 T(0) =

A: We have to find

Q: F.dr. QUESTION 12 Let C be the oriented curve parameterized by r(t)=(t,t> for 0sts2 (so C is…

A: Vector calculus

Q: Find the vector equation of the tangent line to F(t) = (2 sin 2t, cos 2t, – 6t) at the point where…

A: Consider the given vector. rt=2sint,cos2t,-6t   Find the derivative of the vector with respect to t.…

Q: Sketch a direction field for the equation y'= 2y(1-y) by hand. Then use that and sketch a few…

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Q: 2. Let T(x; y) = sin(2x – y) , P (-; ) and Q(0;0) (a) Determine the directional derivative of T at P…

A: Tx,y=sin2x-y, P=-π3,π6,𝐐=0,0                                      P1  P2                𝐐1,…

Q: 6. Let f(r, y, z) = e=² cosh y + z ar ff at the point P in the direction of :

A:

Q: Let r(t) = (-3t³-4, -3e¹, 5 sin( - 3t)) Find the unit tangent vector T' (t) at the point t = 0 T(0)…

A: r(t)=<-3t3-4, -3e4t, -5Sin(-3t) >

Q: 2. Consider the vector-valued function R(t) with Ả(1) = (1,2,−1) and R'(t): = (17/3/7-2) a) Find…

A: According to the guideline, only three subparts of the question can be answered. Please repost other…

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

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Q: Oy" + 3.2 y' + 64 y = 0 Oy" + 3.2 y' +16 y = 0 Oy" + 1.6 y' +16 y = 0

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Q: Find the vector equation of the tangent line to the curve given by 7(t) = sin ti + cos (2t)j + tan…

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: II. Let F and G be vector-valued F(1) = (1,0,2), F'(1) = (1,1,2), G(1) = (2,1,3), G'(1) = (-1,…

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Q: 2. Find the length of the curve r(t) = e'i + e' sin 2tj + e' cos 2tk, 0st< 2n.

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Q: Find a tangent vector of the curve7(t) = (t², 2 sin(t), 2 cos(t)) at (0,0, 2) (0,0,1) (1,0,0) None…

A: Here, Find it’s derivative This is the tangent at any point on the curve.

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…