5) The position vector of a particle is given by s(t) = 2t2 – 2t + 1.Find the time at which the instantaneous velocity equals the average velocity over the time interval [1, 6). [2 marks]

Q: If a particle moves in the xy-plane so that at time t > 0 its position vector is (In (t2 + 2t) ,…

A: As we know that the expression for the velocity is; v=dstdt Given position vector is:…

Q: 2. Solve the initial value problem for r as a vector function of t. (Hint: think anti-derivative) dr…

A: We will integrate the given function w.r.t. t.

Q: 8. Find the orthogonal trajectories of vy = cx

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Q: Find the velocity and acceleration vectors of the particle whose position at time t seconds is given…

A: We have a given position vector of any particle. And we have to find velocity and acceleration…

Q: 2. Find the directional derivative of o = yx2 +x²z + 3xyz in the direction of vector n = î + 2 ŷ -2·

A:

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: 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: 6. A particle moves in the xy-plane in such a way that its velocity vector is (1+1, 1). If the…

A: Find the position vector

Q: 8. Find the orthogonal trajectories of Vy = cx

A:

Q: 1. Find the orthogonal trajectories of the given family of curves: a. y = Ce-mx; with m held fixed

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Q: (b) Find orthogonal trajectory of x2 - y = K. (Orthogonal trajectories]

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Q: 9. An object moving along a curve in the xy-plane has position (x(t),y(t)) at dy dx :=1+ tan(r²) and…

A: According to Bartleby q&a guidelines I will attempt only one question

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

A:

Q: Suppose z is given implicitly by the equation ln (yz2) + x3z = 1 in a neighborhood of the point P…

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Q: 1. Find a parametrization for the line segment from (2, 1, 3) to (4,3, 2). 2. Find the velocity and…

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Q: 2) A particle is moving along the path given by ~c(t) = (t^4 − 4t, t^3 + t). (a) Compute the…

A:

Q: 5. Suppose a particle is moving along a smooth path r(t) with constant speed. If a(t) + 0, show that…

A: Two vectors is perpendicular if their dot product is zero.

Q: 2. Find a function f(x, y) which is NOT linear in either a or y and for which the tangent plane at…

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Q: YouTrylt 4 Consider the path of a particle given by tangential and normal components of acceleration…

A: The position vector of the particle is given by  r(t) = 2t2, t3, -1

Q: 25. A particle is located at (1, 1,0) at time t = 0 and it follows a path whose velocity vector is…

A: Let's find.

Q: Find the maxcimum rate of change of f at the given point and the direction in which it occurs. F(x,…

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Q: 2. At time t sec, the positions of two particles A andB on a horizontal coordinate line are sA = 3t3…

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Q: Find the orthogonal trajectories of a - =1; where 2 is an arbitrary parameter. a + 2

A: This is a question of the Differential equation.

Q: Q/1: Find the derivative of the non-uniform B-Spline curve with (k=3, n=5) Basis Functions N2.

A:

Q: Given the one-parameter family of curves y² = Cx³ + 5. Find the family of orthogonal trajectories.…

A:

Q: 3. Find the length of the trajectory of the vector valued function f(t) = (7V2.1,7",e-7) from 0 to…

A: The detailed solution of the problem is given in the next steps.

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

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

A:

Q: 2. If a particle moves in the xy-plane so that at any time t > 0, its position vector is (In(r² +…

A: Solution-

Q: Q1: Find the derivative of f(x,y,z)=x^3-xy^2-2z at point P_o (1,1,0) in the direction of vector…

A: Solution:-

Q: For time t > 0, the position of a particle moving in the xy-plane is given by the vector (, et).…

A: Position of particle = (1/t, e3t) Velocity = derivative of position of particle = (-1/t2 , 3e3t)

Q: In(3x² – 2y + 4z³); x = t², y = t³ , z =t-l. Find the maximum derivative the function f(x, y) = x² –…

A: Given that w=ln3x2-2y+4z3 The objective is to find dwdt By chain rule,…

Q: A bee with velocity vector r' (t) starts out at (2, 5, -2) at t = 0 and flies around for 1 seconds.…

A: We can answer first question as per the honor policy. Please resubmit other parts as well to know…

Q: Suppose z is given implicitly by the equation x3y-yz2+z/x=4 in a neighborhood of the point P (1,…

A:

Q: A ball is bounced directly south, with an initial velocity of 10 m/s off the ground, with an angle…

A: Given initial velocity of ball is 10 m/sec. Angle of elevation is 30°. The ball moves at a constant…

Q: In a horse race, horse 2 passes horse 1 at point A, where the two velocities are v₂ = m 7™ and v₁ =…

A: To determine the distance from point A to B and the position of horse1 relative to horse 2 when…

Q: 5) The position vector of a particle is given by s(t) = 3t2 – 4t + 5.Find the time at which the…

A:

Q: Find a vector that gives the direction of maximum rate of increase for the e2y cosx at (n/4,0).…

A: Consider a function . Then the maximum rate of change of this function occurs at the direction of…

Q: 3. Evaluate the gradient of S(1, y, 2) = In(cosh z+ y) - arctan| at (1,y) = (7/2, 1).

A: Given that fx,y,z=lncosh x+y-z2 arctanx2y2+z2 To find the gradient of f(x,y,z) :…

Q: 5) A ball is bounced directly south, with an initial velocity of 10 m/s off the ground, with an…

A: Consider the provided question, sketch a diagram according to the given question,

Q: 8. Find the directional derivative of f (x, y, z) = xy²z³ at P(2,1,1) in the direction of Q(0, –3,5)

A:

Q: 3. Find the length of the trajectory of the vector valued function f(t) = (7V2 t, e",e-) from 0 to…

A:

Q: (6) Find the orthogonal trajectories of the family of curves r= 5C cos 8, where C being parameter.

A: we have to find the orthogonal trajectory for the given function

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

A:

Q: 1. Find the orthogonal trajectories of the family of curve z + 3y = cy,where cis an arbitrary…

A: We will first find the differential equation of the of the given family of curves

Q: 9. Find the directional derivative of f(x , y , z) = 3x° +y° +z° - 9x + 9y – 9z + 4 at the point 9.…

A:

Q: 4. If a particle moves in the xy-plane so that at time t its position vector is sin 31 ,3r2 find the…

A:

Q: 6. Suppose that the temperature T at point (x, y,z) is given by the function T = 2x2 – y? + 4z², and…

A: Given temperature function is: T=2x2-y2+4z2 The tiny bug is at point 1,-2,1 Now, to find in which…