A particle moves in the xy-plane along the curve represented by the vector-valued function r(t) = (4 cos t)i + (7 sin £)j. (a) Describe the curve. O Disk O Torus O Shell O Ellipse O Circle -6 -4 4 -2 2 6 4. -6 2 4 3 (c) Find the minimum and maximum values of ||r'|| and ||r"||. ||r"|| minimum maximum

Q: 1. Use the parameter r =t to find a vector-valued function for the space curve represented by the…

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Q: Find the domain of the vector-value function 7 (t) = (- 2 sin(t), – 5 cos(t), In(t – 1)) Domain: {t…

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Q: 0I. Also, r(t) = (2cos t)i + (2sin t)j + (/5t)k, 0stsn Find the curve's unit tangent vector. T(t) =…

A: Differentiate and take magnitude and use it for unit tangent vector

Q: Let C be the curve of intersection of the paraboloid x² + y² = z and the cylinder + (y − 1)² = 1 (a)…

A: We will use the basic knowledge of vector calculus and differential calculus to answer all three…

Q: Find the Tangent vector, the Normal vector, and the Binormal vector (T, N and B) for the curve 7(t)…

A: We will find out the required value.

Q: A vector normal to the surface y-:=3 at the point (-1-2, 2) is (a) i+2j+k (b) -i+j+2k) (c) (21 -j+k)…

A: Definition used -              For finding a normal vector to the given surface, We have to find…

Q: Find the open interval(s) on which the curve given by the vector-valued function is smooth. (Enter…

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Q: Find a vector parametrization of the curve x = -3z² in the xz-plane. t as the parameter in your…

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Q: 18. Find an equation of the tangent plane to the surface given by the parametric vector equation…

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Q: Describe the curve represented by the vector-valued function r (t) = (2 cos t) i +tj + (2sin t) k…

A: Given: r→(t) =2costi→+tj→+2sintk→. x=2costy=tz=2sint Then, x2=costz2=sint

Q: Reparametrize the curve with respect to arc length measured from the point where t = 0 in the…

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Q: Draw the graph of the vector-valued functions (a) for the plane curve, r(t) = 2cost³i + 3sint³j, 0<t…

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Q: The position vector of a moving particle in space at time t is given by the vector function r (t) =…

A: position vector of particle in space r→(t)=12(5-t)2i→+(2t-1)j→+43(5-t)32k→

Q: Sketch the curve represented by the vector-valued function                               r(t) = ⟨…

A: Given  r(t) = ⟨ cos t, sin t ⟩

Q: 2. Find thé moving trihedral of the curve C traced by the vector-valued function R(t) 4 cost i+4…

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Q: 3. Let the curve C be traced by the vector-valued function R(t) = cos(zt) î + at j+ sin(rt) k. %3D…

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Q: Sketch the vector-valued function r(t) = ⟨cos(t), 1, sin(t)⟩,

A: Sketch the vector-valued function r(t) indicating the orientation of the curve with an arrow.…

Q: Sketch the plane curve represented by the vector-valued function and give the orientation of the…

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Q: Suppose that a motion described by the vector valued function (i.e. position function) r(t) has…

A: Given that :- r't=vt=5cost , 5sint ,-2 and r0=2,1,,1 find position function rt

Q: A particle moves so that its position vector r at timet is r = a cos wt + bsin wt, where w is a…

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Q: A force F acts on a particle that is moving in the plane along the semi-circle parameterized by 7(t)…

A: Given that: γt=−costi+sintj; t∈0,π F=x2+y2τ, τ is the unit vector tangential to the path. To…

Q: Given the vector valued function: T(t) = Then the point 9 units along this curve (in the direction…

A: Tangent vector two a curve always point in the direction of t increasing.

Q: a) Represent the parabola y = x² + 4 as a vector-valued function b) Sketch the vector valued…

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Q: A smooth curve is normal to a surface f(x,y,z) = c at a point of intersection if the curve's…

A: Given: A smooth curve is normal to a surface fx,y,z=c at a point of intersectionif the curve's…

Q: 2. Consider the curve traced out by the vector valued function r(t) = (sin t cos t, cos? t, sin t)…

A: given that r(t)=(sintcost,cos2t,sint) we need to show that the curve lines on the surface of the…

Q: Q5:- a-Solve the initial value problem for r as a vector function of t. d'r (sin t)i- (cost)j + (4…

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Q: f (x, y, z) = arctan(x² + 2y + z) At the point A(0,1,0) along the direction ñ, where n is the…

A: Given that f(x,y,z)=arctan(x2+2y+z)  For a function of three variables f(x,y,z), the directional…

Q: 3. Let C be the space curve given with the vector valued function R(t) = (1+2 sin(2nt))i+ (In t)J +…

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Q: Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t…

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Q: (a) Let f(x,y) = x² – 5xy. (i) Find the directional derivative at (2, 1) in the direction of i = -i…

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Q: 3. Let the curve C be traced by the vector-valued function R(t) = cos(at) î + at j + sin(at) k. Find…

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Q: The path r(t) = (t– sin t) i + (1- cos t) j describes motion on the cycloid x=t- sin t, y = 1- cos…

A: Given:           r(t)=(1−sint)i+(1−cost)j

Q: A force F acts on a particle that is moving in two dimensions along the semicircle x = 1 – cos 0, y…

A: Given the forces on a particle that is moving in 2D along the semicircle x=1-cosθ, y=sinθ, 0≤θ≤π.

Q: 2. (a) Prove that the function f(x, y) = xy is differentiable everywhere in its do- %3D x² + y?…

A: TO solve the question, we use the following facts (a) A functions is differentiable at a point if…

Q: 2. Consider the ellipse x2 y? +. 4 1 - a. Describe this curve with a vector valued function r(t), 0…

A: We have to find vector valued function and tangent vector at point t=π4.

Q: A particle moves along the graph of y = cos r so that the r-component of acceleration is always 2.…

A: Use the given information carefully.

Q: The path r(t) = (t- sin t) i+ (1- cos t) j describes motion on the cycloid x=t- sin t, y=1- cos t.…

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Q: The vector-function gives a particle velocity as function of time: T(t) =, and the particle position…

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Q: a) Find the directional derivative f (x, y, z) = x In(y+z) at point P (2, e, e) in the direction of…

A: We have to find directional derivative of vector f in the direction of vector v:

Q: A force F acts on a particle that is moving in the plane along the semi-circle parameterized by 7(t)…

A: Work done W=∫CF.ds=∫abF(r(t)) . r'(t)dt   Where curve C is given by parametric equation r(t)

Q: The position vector r describes the path of an object moving in the xy-plane. Position Vector Point…

A: The position vector                              rt=4costi+4sintj and a point 22,22. Then we have to…

Q: (b) Let r(t)-ti- +(4-t) k, then the vector equation of the line tangent to the graph of r(e) at the…

A: Given:- r(t) =t2 i -(1+t)-1j +(4-t2)k To find:- Vector Equation of tangent line passing through…

Q: A vector normal to the surface y-:-3 at the point (-1,-2, 2) is (a) i+2j+k (b)-i+j+2k) (e) (21 -j+k)…

A: The given surface is: xy-z=3 And, the given point is: -1, -2, 2 Since, the gradient represents a…

Q: A particle moving along a curve in the xy-plane has position (x(t), y(t)) = (-9 sin(9t), 3t² + 10t),…

A: Solved below

Q: The temperature T, at a point (x, y) in the xy-plane is given by T(x, y) = 4x^2y^3 degrees Celsius.…

A: We have to find a unit vector in the direction in which the temperaturedecreases most rapidly at (2,…

Q: Match the vector-valued functions (a)-(f) with the space curves- (1)–(vi) in Figure 10. (a) r(1) =…

A: (d) Identify the graph of the vector function rt=cos3t, sin3t, sin 2t. From the function, it…

Q: Find the open interval(s) on which the curve given by the vector-valued function is smooth r(0) = 4…

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Q: Sketch the plane curve represented by the vector-valued function and give the orientation of the…

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Q: Find the open interval(s) on which the curve given by the vector-valued function is smooth. (Enter…

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Q: 1. Let C be the curve of intersection of the paraboloid x2 + y? = z and the cylinder 4 - + (y – 1)²…

A: Given: The paraboloid equation is x2+y2=z and the cylinder x24+y-12=1 a) Find the vector-valued…