4. (16 points) Consider the vector field: (2/27 +2ry) j F(x, y) = (xy + y²) i + ( 22 (a) Is F conservative? If so, give a function f(x, y) such that F = Vf. If not, explain how you know. (b) Let C be the smooth curve defined by the vector function r(t) = ti + (-10t2 + 29t - 15)j, with endpoints 1

Q: 2. Consider the floor function [.] whose output is the largest integer less than or equal to the…

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Q: 4. (a) Determine the directional derivative of f(x, y, z) = zety at the point (1, −1, 1) in the…

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Q: Given r(t) be a vector function that satisfies: r'(1)= T'(1)= 1) Determine the unit tangent…

A: Given, r'(1)=<-3,0,4> T'(1)=<2,2,1>

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

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

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

Q: Let r(t) = 5e7t + e2t and y(t) = 3e2t + 1le (i) Rewrite the given functions in the vector form Y(t)…

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A: The gradient vector field of given function is -2x i + 2y j

Q: . Let n 2 3 be an odd number and f(a, y) = (x" + y")'/n. (a) Determine if f(x, y) is differentiable…

A: We will only use the basic knowledge of the differentiability of functions of two variables to…

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

Q: Consider the space curve C given by the vector function f(t) = (t² + 1)i + (2t4 – 3t2)j +t³k where t…

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Q: Let f(x,y) = sin x+sin y. (NOTE: You may use software for any part of this problem.) %3D (a) Plot a…

A: (A) plot the contour of f.

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Q: 2.31 Let A and B be vector functions of position vector x with continuous first and second…

A: Let A and B be vector functions of position vector x with continuous first and second derivatives,…

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

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Q: Suppose that over a certain region of space the electrical potential V is given by the following…

A: The given potential function is Vx,y,z=2x2-5xy+xyz. (a) To Find: Rate of change of the function at…

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: 7. For the vector-valued function √t-1 ln(2t - 1) t² - 1 r(t) = (VI- t-1 2 COS πt rt), (t (t = 1):…

A: Let us solve above problem

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: Find r'(t), r(t), and r'(t,) for the given value of to: Then sketch the space curve represented by…

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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: ar function = (1, 1, 2) i

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Q: Plot the gradient vector field of f together with its contour map. (a) f(x,y)=4- -4-4-4 (b) f(x, y)…

A: Given:  a fx, y=4-x24-y24b fx, y=x2+y2 To Plot the gradient vector field and contour map.

Q: Consider the R − R 2 function r defined by r (t) = ￾ t, t2
; t ∈ [−3, 3] . (a) Determine the…

A: This is a problem of vector calculus.

Q: 2. (a) Prove that the function f(x, y) = xy is differentiable everywhere in its do- %3D x² + y?…

A: TO solve the question, we use the following facts (a) A functions is differentiable at a point if…

Q: t) = (et, e24), to = 0 لاخت لا rve represented by the vector-valued function, and sketch the vectors…

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Q: Q1. Find the vector function r(t) given that r(0)= i+2j. and r'(t)= (2et, 3t², 4 cost)

A: The given data is: r0=i+2jr't=2et, 3t2, 4cost Find the vector function rt