Siven the following vector field and orientated curve C, evaluate F•Tds. (x,y) F = on the curve r(t) = (2t",9t), for 1sts3 3/2 (x² +y?)

Q: In words, describe the curve r(t) = (2 sint,e',2 cost) for 0<= t <=6 . Be sure to a -- give enough…

A: To Explain: The parametric curve is r(t)=(2sint,et,2cost) for 0≤t≤6π , so that its graph can be…

Q: 5. Consider the vector-valued function R(t) = (t³ — t, 2t², 2 — 2t²) a. Find ||R'(t)|| (Simplify…

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Q: Find the directional derative of f at the point P0(1,1) in the direction of the vector A = 3i - 2j…

A: Direction derivatives at (1,1) is given in direction of A = 3i - 2j.

Q: Find the domain of r(t) and the interval(s) on which the vector- valued function is continuous. r…

A: We need to find the domain and the interval on which the vector-valued function is continuous.

Q: Define two vector functions 7(t) = 2 sin(t)i + 8 cos(t)j + t°k ü(t) = 8 sin(t)? + 2 cos(t)j + (t? –…

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Q: b) Evaluate S(2y + 4z)dx + (2æ + 3z)dy + (4x + 3y)dz, where C is the curve given by r(t) = for 0 <t…

A: given ∫C2y+4zdx+2x+3zdy+4x+3ydz where C is the curve given by rt=t3,2sinπt2,3cosπt2 for 0≤t≤1…

Q: F(t) = (Cos(Tt), - 1), G(1) = (1, 1, – 1), G'(1) = (2, 3, 2) , Ĝ"(1) = (0, 1, 0) . (F.G) 1) (Gxở)…

A: Given :- F →t =  cos πt , e2t-1 , t2 - 1  , G →1 =  1 , 1 , -1G' →1 =  2, 3, 2G'' →1 =  0 , 1 , 0…

Q: 1. Consider the vector-valued function F(t) = sin(2t) t (2, 1, -2), a) Find the domain of F. b) Show…

A: Disclaimer: There are two questions here and as per Bartleby policy, I am solving only the first…

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

Q: Find the directional derivative of the scalar field = xy + 4xz at the point (2, -2, 4) along the…

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Q: Let r(t) = (-3t5 - 1, 2e³, 4 sin(4t)) Find the unit tangent vector T(t) at the point t = 0 T(0) -

A: If  is vector valued function which is differentiable and then the unit tangent of is given by

Q: write answer as fraction -(1/2)^3

A: We know , (a/b)n = an/bn

Q: 3. [32] −1 4-3 4. 5. [. |_3 -4 2 1 2 -1 4,

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Q: Let r(t)= Determine the interval(s) on which this vector valued function is continuous.

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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)=&lt;-3t3-4, -3e4t, -5Sin(-3t) &gt;

Q: Consider the vector function r(t) = (t, t¹, t³) Compute r' (t) = (1, 4t³, 8t7 3 T(1) = ( r' (t) = (…

A: Recall:  T(t)=r'(t)r'(t)

Q: 3. Let F = (x - y, xy) and C be arc of the circle x? + y? = 4 traversed counter-clockwise from (2,0)…

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Q: Q4 (a) Given a cylinder x' +y² = 1 and a plane x +z = 1. (i) Find a vector-valued function r(t) that…

A: we have,  A cylinder x2+y2=1 and a plane x+z=1 i x2+y2=1x2+y2=cos2t+sin2tx2+y2=cost2+sint2⇒x=cost,…

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: 8. Given the space curve C as the image of the vector valued function r(t) = 2ti+t³j+-t-¹ k OR r(t)…

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Q: In(t + 4) sin2t Given the vector valued function 7(t) = (e-3t, t a) Find the domain of 7(t). b) Find…

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Q: Let r(t)= (212-t,t³ +1²-5t), -2≤t≤ 2. (a) Sketch the curve which is generated by r(t). You may use…

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Q: 31. Ke [<cos (1₂) c² - 3+₁ ²+/=/=) +‡ -1 (-1,19,-27, t = 11. Ŕ(t) (1) show that I is continuous at t…

A: This is a problem of vector calculus.

Q: Show that the vector-valued function shown below describes the function of a particle moving in a…

A: Part - 1 : Given vector-valued function is                                      rt = 5i +5j +3k +…

Q: Given the vector-valued function r (t) = e-t i +t j +(tan t) k Find: a) r'(t) b) r"(t) c) r'(t) .…

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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: True or False? Vector functions r 1 = ti + t 2 j, 0 ≤ t ≤ 1 , and r 2 =(1 — t )i+(1 — t ) 2 j, 0 ≤ t…

A: Given: Vector functions: r1=ti+t2j, 0≤t≤1r2=(1-t)i+(1-t)2j, 0≤t≤1 We are required to find: If the…

Q: consider the vector-valued function r(t) = a) State the domain of r(t) b) Identify where r(t) is…

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