A particle moves in the xy-plane in such a way that its path is defined by x = et cos t and y = et sin 2t. Find the speed of the particle when t = /2.

Q: Find the lengthof the curve r = a sin² (0/2), 0 0

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Q: Show that the curves r = a sin 0 and r = a cos 0 intersect at right angles.

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Q: Find the length of the curve r = cos(f) + sin(0), 0sOs

A: The objective is to find the length of the curve.

Q: show that r = cos 2θ is symmetrical with respect to the horizontal axis.

A: show that r = cos 2θ is symmetrical with respect to the horizontal axis.

Q: Find the distance traveled by a particle with position (x, y) as t varies in the given time…

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Q: Suppose that a particle following the path (e' cos(t). sin(), ) flies off on a tangent at lo = 2.…

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Q: Suppose the position function of a particle is given r(t) = (√3 cos (t), t-4, √3 sin(t)). If the…

A: Given : r(t)=3cos(t), t-4, 3sin(t) is the position function of a particle. To find : If the particle…

Q: Find the arclength of the curve a = 8 cos(3t), y = 8 sin(3t) with 0 < t<. 9

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Q: Find T', N and B for the curve r(t) = (2 cos(5t), 2 sin(5t), 4t) at the point t = 0

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Q: find y'' for the curve cosx-siny=4

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Q: Find the value of zrr at the point (0, 3, 2) where x²y? + z² – cos(xyz) = 10. |

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Q: Eliminate the arbitrary constant/s of this equation to obtain the DE. Express the DE in general…

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Q: 8. Given the space curve r(t) = (sin²t, cos't, tant), find an equation of the tangent line in…

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Q: Find the tangent plane to the equation 3y cos(4x – 2y) at the point (2,4,12) Z =

A: Z=3ycos(4x-2y)  Point= (2,4,12)

Q: Sketch the plane curve r(t) = a cos3 ti + a sin3 tj and find its length over the given interval [0,…

A: The curve is r(t)=a cos3ti + a sin3tj   Interval is 0, 2π

Q: Find the residues of sin(2²) at z = 0 and z = π. (sin z)2

A: To find the residue of sinz2sinz2 at z=0 and z=π Let  fz=sinz2sinz2=sinz2csc2z…

Q: Find the equation of the tangent line(s) to the following set of parametric equations at the given…

A: We have to find the equation of tangent from the parametric curve

Q: Given the acceleration and initial velocity of a body moving along a coordinate line at time t, find…

A: The acceleration of a body moving along a coordinate line at time t is given by: a(t) = 8 sin(4t)…

Q: A particle moves in the xy-plane in such a way that its path is defined by x = e^t cos t and y = e^t…

A: Given that, x = e^t cos t y = e^t sin 2t Differentiating x with respect to t we get, dx/dt = e^t cos…

Q: Find T,N and k for the curve: 1- r(t) = (6 sin 2t)i + (6 cos2t) j+5tk.

A: As per our guidelines we are supposed to answer only one question at a time. Kindly repost other…

Q: Find the velocity and acceleration of a particle whose position function is <(1)=sin(21)+ cos(t)

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Q: Graph the curve r2 = 4 cos θ in the Cartesian xy-plane

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Q: 4. Find the parametric equation of the tangent line to r(t) = 4costi – 3tj %3D at Po(2, –n)

A: Given: To find the parametric equation of the given tangent line as follows, r(t) = 4 costi- 3tj…

Q: Find ſo F · dr for F = 8yi – (sin y)j on the curve counterclockwise around the unit circle C…

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Q: Sketch the motion of a particle with position as: x = 5 sint y = ²/-nstsn - St≤n as tvaries in the…

A: We need to plot the motion of the particle, whose position is defined as: x=5sint, y=t2 in the…

Q: Find the equation of the tangent line(s) to the following set of parametric equations at the given…

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Q: 1. Find the length of the curve r(t) = (sin 2t, cos 2t) from t 0 to t = 2n. %3D 2. Find the length…

A: As we know that, The expression for the length of vector-valued curve is: L=∫abx't2+y't2dt   1).…

Q: Use an appropriate local linear approximation to estimate the value of cos(29°). cos(29°) -

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Q: Find the points on the cardioid r = 4(1 – sin0) at which there is a horizontal tangent line, a…

A: According to the given information, it is required to determine the horizontal tangent line,…

Q: Calculate the length of the curve r=a*sin(^3)(x/3) for (x(1)=0, x(2)=3pi)

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Q: Find an equation of the tangent(s) to the curve with paramnetric equations ¤ = 1+ VE, y = e" at the…

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Q: Suppose a curve is traced by the parametric equations 3(sin(t) + cos(t)) = x у 3 18 — 6 соs"(€) — 12…

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Q: A particle moves in the xy-plane so that its position at any time t is given by x= cos(5t) and y =…

A: According to the questionx=cos5ty=t3

Q: Find the slope of the tangent line to polar curve V3 T r = 4 – 6 sin 0 at the point 4 – 6 Preview

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Q: Graph the Plane Curve defined by: I = 3 cos(t) - t + 5. y = sin(t). for t in [0, 37]. x Clear + Undo…

A: Plot the points for different values of t from [0,3π]. Draw the curve passing through the points.

Q: x = 2 cos(3t) – 4 sin( 3t) IT t = 2 y = 3 tan( 6t)

A: given,        x=2cos3t-4sin3t   and y =3tan6t

Q: Calculate the length of the path over the given interval. (sin 3t, cos 3t), 0 <t<+

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Q: Compute the length of the curve where 0 ≤0 ≤ T. x = 2 cos² 0, y = 2 cos 0 sin 0,

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Q: Find the distance traveled by a particle with position (x, y) as t varies in the given time…

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Q: Use an appropriate local linear approximation to estimate the value of cos(59°). cos(59°) ~

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Q: Consider the curve C parametrized by(x, y) = (5 cos (−12t), 5 sin(−12t)), for −π/2 ≤ t ≤ π/2 What is…

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Q: Calculate the length of the curve γ (t) = (cos(2t), sin(2t)) t ∈ [0, π].

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Q: The length of the curve y = In(cos x), 0< x <:

A: To determine the length of the curve, y=lncos x, 0≤x≤π4.

Q: Find the distance traveled by a particle with position (x, y) as t varies in the given time…

A: Differentiate x with respect t. dxdt=ddtcos2t=-2sin t cost  Differentiate y with respect to t.…

Q: Find the length of the parametric curve x = sin-1t,y = In V1– t² for 0 <t<-

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Q: The velocity of a particle moving in the xy plane is given by the parametric equations dx/dt=…

A: The given parametric equations of the particle is dxdt=-2t sin2t and dydt=2t cos2t. Evaluate the…

Q: Find the length of the curve. r(t) = cos(2t) i + sin(2t) j + 2 In(cos(t)) k, osts n/4

A: Given:

Q: Given that r= cos 0 ; dx = - sin 20 and de dy = cos 20. do The slope of the tangent line to the…

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Q: The position of a particle in the xy-plane at time t is given by r(t) = (cos 2t) i+ (2 sin 2t) j…

A: Given: rt=cos2ti+2sin2tj For finding equation in x and y whose graph is the path of the particle, we…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.