Find T(t), N(t), at, and an at the given time t for the plane curve r(t). r(t) = ti + 4j, t= 3 %3D T(3) = %3D

Q: r(t) = ti + (2tt2) intersect the paraboloid (2) At what points does the curve r(t) z = x² + y²?

A: We are asked to find the intersection point.

Q: Consider the following. x = t³, y = In(t) X = (a) Eliminate the parameter to find a Cartesian…

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Q: Find a parameterization for the curve y = 7 – 2x that passes through the point (0,7, 3) when t = 2…

A: To find the parameterization of the curve

Q: 1 - Findthe lenght of the Curve a) r(t)* = (3cost)i³ + (3sint)j* + 4tk¬ for 0 < t < 2n b) r(t)* =…

A: Use formula of length of curve

Q: At what points does the curve r(t) = t i + (6t - t2)k intersect the paraboloid z = x2 + y2? (Enter…

A: Given: Equation of curve:                                    r(t)=t i +(6t-t2) k Equation of…

Q: Find T, N, and k for the space curve r(t) = - 2e cos t)i- (2e sin t) j+ 2k.

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Q: find the work done by F in moving a particle once counterclockwise around the given curve. F = 2xy3i…

A: Consider F = 2xy3i + 4x2y2j We have M = 2xy3 and N = 4x2y2

Q: (B) The point on the curve r(t)=Dti + (1 + r)j at which r(t) and r'(t) are perpendicular is (1,0).

A: This question is about application of vector calculas

Q: Find the point in the plane x – 2y + z = 4 that is closest to the origin.

A: Solution)                   Given                           plane  x-2y+z =4

Q: At what points does the curve  r(t) = ti + (5t − t2)k  intersect the paraboloid  z = x2 + y2

A: We have to find the intersection point

Q: Find the maximum height above the xy-plane of a point on r(t) = (e1 , sint, t(4 - t))

A: Maximum height above the x-y plane will be the maximum value of z .So we will take z parameter of…

Q: dy In the xy-plane, the point (0, 2) is on the curve C. If 4t for the curve, which of the following…

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Q: Consider the curve C in R$ given by: r(t) = (e' + t, e' – t, 4e"/²) con 0<t<1 An expression that…

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Q: Find T(t), N(t), aT , and aN at the given time t for the plane curve r(t) r(t)=(t-t3)i+2t2j, t=1

A: The solution is given as follows

Q: Compute f cos(Tæ) dæ + 10xy dy, where y is a part of parabola y = 3x? from the point (0,0) to the…

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Q: (6) Find the points on the curve 5x2 + 6xy + 5y² = 9 which are nearest to and farthest from the…

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Q: The curve r+y-6x2y 0 is symmetric about Oy = 0 Ox = 0 Oy =x None of these

A: We need to check the symmetry of the curve x4+y3-6x2y=0

Q: = Let {B(t), t≥ 0} be a standard Brownian motion and X(t) E [(X(2) + X(4))²]. -3t+2B(t). Find the

A: A Standard brownian motion is B(t) X(t)= -3t +2B(t) X(2)= -6+2B(2) X(4)= -12+2B(4) now we add X(2)…

Q: Given F(x, y) = [arcot(/K)-(secx)* - 2y]i+ 3x-es 2021 evy and the curve C=C,UC,UC, as shown in the…

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Q: Find the work done by F = xyi+yj - yzk over the curve r(t) = ti + t²j + tk, 0≤t≤1

A:

Q: A particle moves along the curve r(t)=⟨t2,t2−4t,t⟩. Find the minimum speed of the particle.

A: A particle moves along the curve The velocity of the particle at any time t is Now, speed of the…

Q: xy*dx + x?dy, where C is the le with vertices (0,0), (4,0), (4,5), and

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Q: Evaluate x*dx C + xy dy, where C is the triangular curve consisting of the line segments from (0, 0)…

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Q: Find the length of the curve y = V- from z =1 to r = e. 3

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Q: Find the equation of the plane tangent to the following surface at the given points. 2 x +y -z - 52…

A: See the details solution in below

Q: Consider the closed curve in the xy-plane given by x² + 2x + y* + 4y = 5. dy -(x+1) (a) Show that dx…

A: Given data:   The expression for the given function is x2+2x+y4+4y=5.   (a) Differentiate the above…

Q: Draw direction field of the DEQ below, by hand, and sketch an approximate solution curve that passes…

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Q: Evaluate [ f (x,y,z) ds given that f(x,y,z) =, and the curve C is ř(t) = 2ti +t²j+÷tk (1<t<∞) %3D…

A: Given: f(x,y,z)=x2xy+6yz r→(t)=2t i→+t2 j→+13t3 k→   1≤t&lt;∞

Q: The length of the curve y = Ey3/2 between x = 0 and X 1 is %3D

A:

Q: Find the coordinates of the point(s) on the curve 2y = 10 – a² that are closest to the origin. а) О…

A: Given that

Q: :. Let x(t) = 2 sinh t , y(t) = cosh(t²) be a curve in the Oxy-plane, given in parametric form. (a)…

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Q: 14. Find the trace of the plane 6x 3y + z = 4 in the xz-plane. 15. Determine whether the curve given…

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Q: Find the position function of a particle moving along a coordinate line that satisfies the given…

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Q: Find the Length L of the curve R(t)= -2cos(4t)i + 2sin(4t) j + t k over the interval [2,6]. L =

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Q: 1. Compute (1/zy, 1/(z + y))- dr along the curve (t, t2). 1<ts4

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Q: (A) Evaluate" [(x² + y)dx + (x + 1) 1+ along the path ABCA shown in figure C(0,3) A(0,1) x=3sint…

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Q: (2² + y² – 1)³ – x²y3 = 0, 2- Given the curve dy find dx -2 2

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Q: determine the parametric coordinate of the critical point of the curve x=t^2+3t=2 and y=t^2-1.

A: Curve will have critical point if it have vertical tangent or horizontal tangent . Slope of…

Q: Find the equation w=T(x,y,z) of the tangent hyperplane at p. F(x,y,z)=3x²-2y²+xz², p=(1,2,-1)

A: General equation for the tangent to the plane is expressed as:…

Q: 2. If F(x, y, z) = y²i+(2xy+e3=)j+3ye³=k. Evaluate F·Î ds, where C is the curve represented by r(t)…

A: Given F→x,y,z=y2i^+2xy+e3zj^+3ye3zk^ and the curve is C given by r^t=cost,t,sint,  0≤t≤π2. Find…

Q: Evaluate | 9+x* dx+xy*dy along the positively oriented curve C where the C is the boundary of the…

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Q: At what points does the curve r(t) = ti + (6t – t2)k intersect the paraboloid z = x2 + y2? (If an…

A: knlk

Q: 24. Evaluate . (x – y) dx + (x + y) dy counterclockwise around the triangle with vertices (0, 0),…

A: Given integral is, It can be represented using Green's Theorem as,

Q: Let y(t) = (1/2)te*/2, 0<t< 2n What is the terminal point of the curve? — 2п

A: Terminal point of curve is obtained at terminal point of path.

Q: 3. Find the equation of the plane tangent to the graph of f (x, y) = 3x + 4y – xy at the point (1,…

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Q: At which points does the curve r(t)=ti+(2t -t² )k intersect the paraboloid z = x² +y? ?

A: Given: r→t=ti^+2t-t2k^ and z=x2+y2

Q: Sketch the plane curve r(t) = t2i + 2tk, and find its length over the given interval [0, 3]

A: We have, r(t)=t2i+2tk  we have to determine the length of the curve over the interval [0,3]

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.