Find the equation of the tangent line of the vector function F(t) = (e= cos(t), e=' sin(t), e') at the point (1,0, 1). %3D

Q: Find the equation of the tangent plane to z = f(x, y) at the point (1, , 1).

A: Given Data: Surface: z=x2esiny Point: x0, y0, z0=1, π, 1   (a) The first derivative of the given…

Q: Find the tangential and normal components of acceleration for a projectile fired at an angle 0 with…

A: we know that for projectile fired at angle θ with the horizontal tangentialTt and normalNt…

Q: 2. Find the unit tangent vector for the curve r(t) = (t cost, tsin t).

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Q: Compute the derivative of the vector-valued function r(t)=⟨sin2t,et2⟩.

A: In order to compute the derivative of the vector-valued function: Differentiate the vector function…

Q: Consider the curve y = 10 + 4x − x^(2) at (x, y) = (3, 13). Find a vector, v, that has length 4 and…

A: Given the curve y=10+4x-x2 at (x,y)=(3,13).

Q: Find the unit tangent vector to the curve at the specified value of the parameter. r(t): = 7 cos(t)i…

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Q: Find the direction in which the maximum rate of change occurs for the function f(x, y) = 3x sin(xy)…

A: NOTE: Refresh your page if you can't see any equations. . take the partial derivative with respect…

Q: A unit tangent vector to the curve r(t) = [sin(t), cos(t), 2f] at the point A:(0, — 1, 2п) is given…

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Q: Consider the function f(x, y) = y ln x - x ln y - . a Find the directional derivative of f at the…

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Q: Calculate the velocity and acceleration vectors and the speed at the time indicated.r(θ) =〈sin θ,…

A: Consider that: rθ=sin θ, cos θ, cos 3θ at θ=π3. Velocity vector denoted by vθ is vθ=r'θ.  Calculate…

Q: Find the direction in which the maximum rate of change occurs for the function f(x, y) = 3x sin(zy)…

A: The maximum rate of change of a function f(x, y) at (a, b) occurs in the direction ∇f(a, b)  where…

Q: Find the normal vector to the tangent plane of z = 2u at the point (4,8,3) X component y component…

A: We need to find normal vector component.

Q: 0. for the funchion f (x,4) = 3-x?- u?, find a unit tangent vector to the level curve at the point…

A: Given: fx,y=3-x2-y2-3 at P3,5​​​​​​​

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: Find a vector equation of the tangent line of f(t) = (cost, sint, t) at the point Po (0, 1,

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Q: Calculate the velocity and acceleration vectors, and speed for r(t) = (sin(3t), cos(t), cos(: OS…

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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: 2. Find the rate of change of the function f(x, y) = x² sin y at the point (1,) in the direction of…

A: To find out the rate of change in the direction of the given vector.

Q: 3. A unit vector tangent to the curve y x2 at the point (2, 4).

A: We are entitled to solve only one question at a time so we will answer only the first question.…

Q: Find a vector equation for the tangent line to the curve 7(t) = (5 cos(6t))i + (5 sin(6t)) j +…

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Q: Find one normal vector of the tangent plane to f(x, y) = cos(x² +y²) at point P(1,1, cos 2).

A: As we know the expression of the normal vector to the tangent plane at a point on the curve can be…

Q: Find the direction in which the maximum rate of change occurs for the function f(x, y) : 5x sin(xy)…

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Q: Find the unit tangent vector to the curve defined by F(t) = (4 cos(t), 4 sin(t), 3 sin (t)) at t = 3…

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Q: Find the unit tangent vector to the curve defined by 7(t) = (5 cos(t), 5 sin(t), 4 sin (t)) at t = 3…

A: We find unit tangent vector of given curve at t=π/3

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

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Q: Find the normal vector to the tangent plane of z = 6e- Öy at the point (12, 24, 6). x component = y…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: 4. Determine the directional derivatives of $(x, y) = tan"2+3 sin ). at the point (1,1) in the…

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Q: an object is moving in 3-dimensional space with velocity vector v(t)=(sin(t),t,t^2) starting at…

A: Consider the given velocity vector: vt=sint,t,t2 Find the position vector: Integrate velocity vector…

Q: a)What is the slope of the tangent line to the curve y=2sin4x  at x=⊼(pie)/4? b) Find a unit vector…

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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: Find the unit tangent vector to the curve defined by 7 r(t) = (5 cos(t), 5 sin(t), 2 sin² (t)) at t…

A: To find the unit tangent vector to the curve defined by r→t=5 cost, 5 sint, 2 sin2t

Q: Find the unit vector towards the tangent of the curve y = x at x =1

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Q: Let the the position vecor be R(t) = (e' cos(3t) , e' sin(3t),0). Compute the unit tangent vector

A: The given position vector is R(t)=etcos(3t),etsin(3t),0 . compute the unit tangent vector T(t)=?

Q: find the directional derivative of the function in the direction of the vector u = cos i + sin j

A: Consider the given function, fx,y=sin2x+y Here we need to find the directional derivative of the…

Q: Compute the directional derivative of f(r.y) = at the point (1, 1) in a direction normal to the…

A: Given that: f(x,y)=x2y at the point (1,1) Ellipse: x2+2y2=1 at the point (1,0)

Q: find the acceleration of a particle whose position function is x(t)=sin(2t)+cos(t)

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Q: At which point P of the curve defined vectorially with r (t) = (21, 4t + 1, t) is the tangent vector…

A: The general form of the parametric equation of the curve in three dimensional space is given by…

Q: Compute the directional derivative of f (x,y)= at the point (1,1) in a direction normal to the…

A: The directinal derivative is given below

Q: What is the rate of change of the function f(x,y) = ye^(2cos x) + tan(x+3y) at ( 3π/2 , π/2) in the…

A: Given: fx,y=ye2cosx+tanx+3y To Find: Rate of change of fx,y in the direction of the vector 3,4 at…

Q: The directional derivative of f(x, y) = xe + cosxy at P,(2,0) in the direction of the vector 3i - 4j…

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Q: A curve is given by the parametrization r(t)= [²,-t, ť ]. If the tangent vector to this curve at a…

A: The given curve is,  ,

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: Find the derivative of the vector function r(t) = (tan t, sec t,). %3D t2

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Q: Find a vector equation for the tangent line to the curve 7(t) = (5 cos(6t)) i + (5 sin(6t)) j +…

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Q: Choose the direction vector for which the function f defined by f(x, y)= sin(5x) cos(3y) has the…

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Q: Consider the function f(x, y) = y ln x - x ln y - . a Find the directional derivative of f at the…

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Q: Find the tangent plane and normal line equations to x+y +z =1 at the point (1, %3D -2, -4).

A: This question is related to 3D geometry and equations of plane and lines,  we will solve it using…

Q: Find the directional derivative of the function f(x, y) = In(x² + y*) at the point (– 3, – 1) in the…

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Q: Find the directional derivative of the function at the given point in the direction of vector v.…

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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: