A particle at (1,0,0) starts moving in space in such a way that its position vector at any time t> 0 is R(t) = (cost +t sin t)i + (sin t – tcos t)j +t²k,t > 0. (a) Find parametric equations for the line tangent to the trajectory of the particle at the point where t = 7/2. (b) Calculate the acceleration of the particle at time t = 7 7/2. (c) Calculate the total distance traveled by the particle in the time interval 0

Q: A particle moves along a circular path in such a way that is x- and y- coordinates at time t are…

A:

Q: At time t, a particle moving in the xy-plane is at position (x(t), y(t)), where x(t) and y(t) are…

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: Find the parametric equations of the tangent line to the curve represented by the vector function…

A:

Q: Find parametric equations for the tangent line to the curve of intersection of the paraboloid z =x²…

A: Tangent line to ellipse will be perpendicular to normals of both the surfaces Therefore,  The…

Q: 8. Find the flux of F = (2x – 5y², 3y, x) out of the sphcre of radius 2 centered at the origin.

A: The flux of the vector space is given by the formula, ∫S∇.F dS where S is the sphere of radius 2…

Q: Find the equation for the principal curvatures, the differential equation of the lines of curvature,…

A: Given equation is 2z=x2a+y2b2p=2xa, where ∂z∂x=p and 2q=2yb, q=∂z∂ya=xp and b=yq2z=x2xp+y2yq2z=px+qy

Q: An object is moving in the xy-plane along a path whose curvature at time t is given by the function…

A:

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: Express the velocity and the acceleration of a moving point p(r,0)

A: We have given a curve in polar coordinate : r=1+sin(θ) with θ=t2…

Q: The velocity field for a fluid is given by V=(X2t3)i+(2XYT)j. The convective acceleration in the…

A: Given, V=x2t3i^+2xytj^ Here, u=x2t3 and v=2xyt Convective acceleration in y-direction is given by:…

Q: give the position vectors of particles moving alongvarious curves in the xy-plane. In each case,…

A:

Q: Find the derivative of ƒ(x, y, z) = xyz in the direction of the velocity vector of the helix r(t) =…

A: Concept: A vector is a directed line segment whose length is the magnitude of the vector and with an…

Q: The parametric equations of a curve are given by x = In (VE +1) and y = cosh-' (2t). dy when t = 1.…

A: Given x=lnt+1 and y=cosh−12t We have to find the dydx when t=1 We know that dydx=dydtdxdt.....1…

Q: at a unit-speed regular parametrised curve 3: (-7, 7)→ R² with T, N) has constant curvature k> 0.…

A: Sol

Q: 1. Find a vector which is tangent to the curve of intersection of the surfaces z = x2 – y? and xyz +…

A:

Q: Find the tangential and normal components of accelerationfor a particle moving along the circular…

A: We know that, tangential and normal components of acceleration for a particle moving along the…

Q: Determine the parametric equation for a line that is tangent (intersection) of the two fields below:…

A: We have to find out parametric equation for a line that is tangent (intersection)of the two fields…

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: Compute for the directional derivative of f (x,y,z) = tan (ax – 2ny)e4y-z at the point (,0, ln 4) in…

A:

Q: 7. Find the directional derivative of the function f(x, y, z)=(y+28) at (2, 1, 1) in the direction…

A:

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

A:

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 an equation for the plane tangent to thesurface z = ƒ(x, y) at the given point. z = ln (x2 +…

A: Given z=ln(x2+y2)

Q: Find the moment of inertia: y = x³; x = 1; y = 0 about x = -1.

A:

Q: 3. Find the directional derivative of f(x, y, z) =x*y-yz' +z at the point (1, -2,0) in the direction…

A:

Q: The parametric equations of a curve are given by x = In (VE +1) and Vt + y = cosh-(2t). X = dy when…

A:

Q: Find the derivative of f (x, y,z) = xyz in the direction of the velocity vector of the helix…

A: Answer and explanation is given below...

Q: Find the equation of the osculating plane of the helix x = sin 2t, y = t, z = cos 2t at the point…

A: The helix is given as x=sin 2t, y=t, z=cos 2t. We have to find the equation of the osculating plane…

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

A:

Q: A particle moving along a curve in the xy-plane has position (x(1),y(t)) at time dy dx t with dt =…

A:

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

A:

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

A:

Q: 5. Compute the directional derivative and gradient of f(x,y,z) = 4x – y²e3xz at (2. –1.0) in the…

A:

Q: V3 t², a cost + at sin t) 2 r(t) = (a sint – at cos t,

A:

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: A particle moves through 3-space in such a way that its acceleration is a(t) = 8sin 2ti +8 cos…

A:

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…

A:

Q: 2. Calculate the instantaneous rate of change of g at the point (4, 1,0) in the direction of the…

A: Please see the attachment

Q: 1. Let f(r, y) = arctan(ry). (a) Find the unit vector in the direction where the directional…

A:

Q: A particle moves through 3-space in such a way that its acceleration is a(t) = 8 sin 2ti + 8 cos…

A: Gg

Q: Find the line integral with respect to arc length Is (3x + 5y)ds, where C is the line segment in the…

A:

Q: Find the equation for the principal curvatures, the differential equation of the lines of curvature,…

A:

Q: 10. Find the work done by the force F=vx +2y y - 7 for a particle that moves between points (-1,0,1)…

A:

Q: Find the moment of inertia of the solid formec y2 = 2x and th

A: Introduction: If a solid is formed by revolving an area about the y-axis, then the moment of inertia…

Q: 3. Find the directional derivative of the function f (x, y)= sin (xy) at a point -2i +j in the…

A: The given function: f(x, y)=sin2(xy)P=34, πu→=-2i^+j^5

Q: A particle at (1, 0, 0) starts moving in space in such a way that its position vector at any time t…

A: