Let R be a differentiable vector-valued function such that R'(t) = (3t2, 2tvt2 + 1, 2t + 3) and R(1) = 4/2 (4, 2, 3). Find R().

Q: Find the point on the graph of z = 3x2 + 2y at which vector (24, 8, 2) is normal to the tangent…

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Q: Find the derivative of the vector function r(t) = ta x (b+tc), where а 3 (—3, —4, — 2), b — (—2, —5,…

A: Topic:- vector and it's derivatives

Q: Find the gradient vector field of f(x, y, z) = 9x5 + 7y3 + 6z?

A: To find the gradient of  f(x,y,z)=9x⁵+7y³+6z²

Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y, z) =…

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Q: Find a vector-valued function whose graph is the indicated surface.

A: Given : The plane x + y + z = 6.   To find a vector-valued function.

Q: Find the derivative of the vector function r(t)=ta×(b+tc), where a=⟨3,−2,−3⟩, b=⟨1,3,−2⟩, and…

A: To find the derivative of the vector function, rt=ta×b+tcWhere, a=3, -2, -3, b=1, 3, -2, and, c=-2,…

Q: Find the gradient vector field of f(x, y) = ln(5x + y)

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Q: 6. Find the unit vector for which the directional derivative Duf(x,y) of f(x,y) = %3D 3x2 + 4y2 at…

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Q: 2. Suppose that f (1,2)= 3 and that the directional derivative of fat point (1,2) in the direction…

A: Given that f1,2=3 and the directional derivative of f at point (1,2) in the direction of the vector…

Q: Find the gradient vector field of f(x, y, z) = 9x² + 8y* + 6z3

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Q: Find the gradient vector field of f(r, y) = In(x + 3y)

A: Solve for the gradient vector

Q: Find the divergence of the vector field F(x, y,z) = x²zi – 2xzj + yzk and the point (2, -1, 3).

A: Divergence of a vector field: The divergence of a vector field F = (P, Q, R) is given by, The given…

Q: V=(x^2)yi-xyj+xyzk

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Q: 2. Suppose that f (1,2) = 3 and that the directional derivative of ƒat point (1,2) in the direction…

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Q: Find a vector tangent to the curve of intersection of the two cyclinders x2+y2=32x2+y2=32 and…

A: Solution: The objective is to find a vector tangent to the curve of intersection of the two…

Q: Find the increase rate of the scalar V at point P from origin to the point P(2,6; 9,8 ; 9,5) where…

A: According to question given that The origin to the point P(2,6 ;9,8 ; 9,5) where V=3, 3xy+ 2, 6xyz

Q: Choose a correct potential function for the conservative vector fieldF = (3x-y+1, –x-4y-2).

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Q: determine where the given vector-valued functions are continuous. V(1) = (Vi-I ,,1 ). V(t): t-1

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Q: Q1. Calculate the directional derivative of f (x, y, z) of vector v =i+2j+3k. = xyz at point P =…

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

Q: Let F(x, y, z) =2x2+3y2+4z2. For which unit vector 0 will the directional derivative DoF at…

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Q: Find the directional derivative of f(x, y, z) (2,1,3) in the direction of a vector making an angle…

A: Given the function f(x,y,z)=zx+y4

Q: nd the directional derivative of the function f(z, Y) In(z +y) at the point (3, – 1) in the rection…

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Q: - (a x r) = a x r' for any constant vector a. dt Show that

A: To prove: ddt(a×r)=a×r'. Consider L.H.S. Differentiate with respect to t using product rule.…

Q: f f(x, y, z) = xy: on of vectorî - %3D

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Q: The directional derivative of the field u(x, y, z) = x² – 3yz in the direction of the - vector (i +…

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Q: 2. Find the equation of the normal line and the tangent to the parametric surface defined by the…

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Q: 2. Prove that the directional derivative of f (x, y, z) = xy sin z at direction v = i+2j+ 2k is .…

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Q: Find the linear approximation to f(x, y, z) = " at the point (2, -1,1): f(x, y, z) z

A: Given the function

Q: (a) Find the directional derivative of the function f(x, y) = 4.xy – x2 – y? in the direction of…

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Q: Find the maximum rate of change of f at the given point and the direction in which it occurs for…

A: Given: fx,y,z=3x+7yz 1,7,-1 To find: Maximum rate of change of f at the given point. The…

Q: Find the derivative of the vector function r(t) = ta x (b+tc), where (5,-4, 1), b = {(-4, –3,1), and…

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Q: Let f(x, y, 2) = cos(ryz) Find the directional derivative of f at the point (,, 7) in the (a)…

A: We have to find Directional Derivative and Largest possible value.

Q: Let f(x, y, z) = x2y− xy2 + z3, P(1,1,2), u is the vector from P to Q(1, −2, −2). Determine the…

A: The objective is to determine the maximum and minimum directional derivative at P and give a vector…

Q: Find the gradient vector field of f(x, y) = ln(4x + 2y)

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Q: 2. Prove that the directional derivative of f(x, y, 2) = ry sin z at (1,2, ) in the direction v = i+…

A: Given:        fx,y,z=xy sinz         x,y,z=1,2,π2    direction v=i+2j+2k   Differentiate the…

Q: Suppose that the directional derivative of a function f at a point P in the direction of the vector…

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Q: Vector Field F = 6xi + (2y – y²)j+(6z – x³)k is Conservative Select one: True False

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Q: Show that whether the functions sint and cost in the vector space C[-n, 1]of continuous functions on…

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Q: The position vector of a particle in the xy-plane is given by for all t > 0. The acceleration…

A: position vector of a particle in the xy-plane is given by < t^2+1, ln(2t+3) > for all t>=0

Q: Given the function of two variables ƒ(x, y) = 4x³ − 60x² + 12xy² − 24y² +3. - - Find and classify…

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Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y, z) =…

A: Given : f(x,y,z)=9x2y+5z2   To Find : Conservative Vector field by finding it's gradient

Q: The largest value of the constant b such that the vector-valued function F(t) =< cos 21It, sin (t-…

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Q: Find the unit vector u along which the rate of change of the function f(x, y) = x 2 + xy3 at the…

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Q: ƏF dF = (dr. V)F+ -dt. at

A: Given: A vector function F depends on both space coordinates x, y, z and time t.  To Find:  To show…

Q: Find the derivative of alpha of t if alpha of t is the measure of the angle between the two given…

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Q: Suppose a vector function F(x, y) = 3xi + 2xyj + y ln zk moves along a path C where x = t2, y = t +…

A: The formula to find the work done by the vector function helps to find the required work done. The…

Q: Consider the vector space of continuous functions over the interval [-1,1] with the inner product…

A: Given that,

Q: Find the derivative of the vector function r(t) = ta x (b + tc), where a = (2,-2, 5), b (2,-5, 3),…

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Q: Find the derivative of the vector function ta x (b tc), where r(t) = (-1,1,-2) а %3 (-2,3, -1), ь 3…

A: Given vector function is,

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: