Find a vector function, r(t), that represents the curve of intersection of the two surfaces. The paraboloid z = 3x2 + y? and the parabolic cylinder y = 2x2 r(t) = (1,32,212 + 9^) ×

Q: Find a vector tangent to the curve of intersection of the two cyclinders x2 + y2 = 50 and y? + 22 =…

A: Find a vector tangent to the curve of intersection of the two cylinderx2+y2=50 and y2+z2=50 at the…

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

A: To find :               Find the vector valued function whose graph is the indicated surface.

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 a parametrization of the curve. The line passing through (1, 0, 4) and (4, 1, 2)

A: Given: A line is passing through the points 1, 0, 4 and 4, 1, 2.

Q: Find a vector function that represents the curve of intersection of the paraboloid z = 7(x^2) +…

A: We have to find vector function that represents the curve of intersection of the paraboloid z =…

Q: Find a parametrization for the cylinder x^2 + y^2 = 4.

A: Given: x2+y2=4

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 a vector tangent to the curve of intersection of the surfaces given by x² + y² + z² = 1 and 2²…

A:

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

A:

Q: Find a vector which is tangent to the curve of intersection of the surfaces z = x² – y? and xyz + 30…

A: We will find out the required tangent vector.

Q: Find the vector equation of the plane tangent to the surface z = x^3 - y^3 at the point (0,1,-1)

A:

Q: Find a vector equation of the normal line (the line orthogonal to the surface) for the surface…

A:

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: What is the length of the acceleration vector of a particle travelingaround a circle of radius 2 cm…

A:

Q: Find the normal vector to the surface z = xe™Y at the point (2,0, 2).

A:

Q: Find the orthogonal trajectories of the family of circles with center at the point (0, 1) and radius…

A:

Q: Find a parametrization of the circle of radius 5 parallel to the yz-plane with center (-1, 4, 6).

A: The general form of a parametrization of the circle of radius r parallel to yz-plane with center…

Q: Suppose ƒ is differentiable at (9, 9), ∇ƒ(9, 9) = ⟨3, 1⟩, and w = (1, -1). Compute the directional…

A: Given that ∇ƒ(9, 9) = ⟨3, 1⟩. Note that w = (1, -1) is not a unit vector. Its unit vector along same…

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: Compute the vector line integral: x – 4 dx + (x – 4)² + y² -y dy :- 4)2 + y² where C is the circle…

A: To find the below vector integral. ∮C−yx−42+y2dx+x−4x−42+y2dy

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: Find the direction in which the maximum rate of change occurs for the function f(z,9)2zsin(ay) at…

A:

Q: What is a parametrization of a curve in the xy-plane?

A: we have to tell what is parametrization of a curve in the xy-plane?

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

A:

Q: 9. Determine the position of the centroid of a sheet of metal formed by the curve y = 4x - x² which…

A:

Q: find the points on the surface + z² = 1 that have their tangent plane perpendicular to the vector…

A: For tangent plane of the surface f(x,y,z)=k at a given point normal vector to tangent plane  will be…

Q: Which of the following vector functions represents the curve of intersection of the cylinder x2 +y =…

A: Given Data: The equation of the curve is: x2+y2=4 ......1 The equation of the plane is: x+z=3…

Q: Find the direction in which the maximum rate of change occurs for the functio f(r, y) = 2x sin(ry)…

A:

Q: Find equations of the normal line and the tangent plane to the parametric surface defined by the…

A: NOTE: Refresh your page if you can't see any equations. . the given point is  compare coordinate on…

Q: A point moves along the intersection of the elliptic paraboloid z = 9x + 7y and the planex = 1. At…

A:

Q: Find First and second derivatives of Funchion . tm - (e-a -a)(e-a+a) Where a is parameber

A: To find a first and second derivative by first simplify the equation and finding it's value at x=0

Q: Find the maximum rate of change of f(x, y) Maximum rate of change: = In(x² + y?) at the point (3, 1)…

A:

Q: Find the two-unit vectors that are parallel to the tangent line to the curve y =x2sin(x) at the…

A:

Q: Find a vector equation for the tangent line to the curve of intersection of the cylinders x2+y2=25…

A:

Q: Find a vector parametrization for the line with the given description. Perpendicular to the…

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Q: Find the moments of inertia about the coordinate axes of a thin rectangular plate of constant…

A:

Q: A point moves along the intersection of the elliptic paraboloid z = 4x + 11y and the plane y = 4. At…

A: given information is z=4x2+11y2 The plane y=4To find :Rate of change of z with respect to x when the…

Q: Find a vector equation for the curve of intersection of the surfaces x^2 + y^2 = 4 and z = xy.

A:

Q: ) Find the points on the ellipsoid 6x² + y² + 3z² = 4 where the tangent plane is parallel to the…

A:

Q: 2dy -2y 2. zt at

A: Solved below

Q: Find a vector function that represents the curve of intersection of the paraboloid z = x^2 + y^2 and…

A: Intersection of paraboloid and cylinder can be obtained by representing y in the form x and z in the…

Q: What is special about the derivatives of vector functions of constant length? Give an example.

A:

Q: Does the the paraboloid z = 1 – 2x² - - y² have a tangent plane at some – x + y + z = 1?

A: Introduction: Let P0=(x0,y0) is a point in the domain of f, and S is a surface defined by a…

Q: Compute the directional derivative and gradient of f(x, y, z) = 4r – yez at (2, -1,0) in the…

A: Given: fx,y,z=4x-y2e3xz       at     2,-1,0 Directional Derivative = Gradient of function x unit…

Q: Find a vector tangent to the curve of intersection of the two cyclinders x2 +y? = 200 and y? + z2 =…

A:

Q: Find a vector function that represents the curve of intersection of the surface 4x+2y-8z^2=16 and…

A: To find a vector function that represents the curve of intersection of the surface 4x+2y-8z2=16 and…

Q: Find a point on the ellipsoid 3x2 + y2 + 3z2 = 1 where the tangent line is parallel to the plane…

A: Given: The equation of ellipsoid is 3x2+y2+3z2=1 and the equation of the plane is -12x+2y+6z=0.…

Q: nit vector that is parallel to the

A: Here have to find the unit vector that is parallel to the line tangent to the parabola y = x2 at the…

Q: Find a vector function that parameterizes the curve of intersection between the surfaces y^2-z^2=x-2…

A: To determine the vector function for the curve of intersection between the surfaces.

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: