(a) æ = sin t, y = csc t, 0 1. Thus, the curve is the portion of the hyperbola y = 1/x with y > 1. (1, 1)

Q: 4) Find the critical points of the parametric equations x = t* - 1 y = t3 - t

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Q: Identify the length of the arc having parametric equations x = t3 and y = 2t2 from t = -2 to t = 0

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Q: Show that the curve y = x? sin 3x touches the parabolas y = x andy =-x² alternately at infinite…

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Q: 1. Find the equation of the tangent line (s) to the following set of parametric equations at the…

A:

Q: What is the differential equation of the family of ellipses with their centers origin and their…

A: Let us solve the problem using the basic equation of ellipse  later we will use different methods to…

Q: Find a parametrization of the right branch (x > 0) of the hyperbola(x/a)2−*(y/b)2= 1 using cosh…

A: Given that. xa2-yb2=1     ...(1) Need to evaluate parametrize form of (1).

Q: Chapter 12, Section 12.3, Question 006 Find the arc length of the parametric curve. x = 6 cos t, y =…

A:

Q: 5. Eliminate the parameter to find a Cartesian equation of the following curves. et (a) x = vt+1, y…

A: Eliminate parameter t.

Q: 5. Eliminate the parameter to find the equation of the ellips Sx(t) = 4 cos(t) + 1 ly(t) = 3 sin(t)…

A:

Q: #3. Find parametric equations for the tangent line to thể curve with parametric equations x = (sin…

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Q: 4. (i) Use the parametric equations of an ellipse, x = a cos (t) , y = b sin (t), 0 < t 2m, to find…

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Q: III. Find the parametric equations of normal line to the surface given by f(x, y) = tan 2 V + y? at…

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Q: 15) (10.) Sketch the parametric curve x = 2 + 3sin(t), y = -3+ 3cos(t) for <t<. Find the slope of…

A:

Q: 15. Find the radius of curvature at 0 = ÷n of the parametric equations x = 3(0 + cos 0), y = 3(1 —…

A: Given two parametric equations, x=3(θ+cosθ),y=3(1-cosθ) To find the radius of curvature at θ=π3

Q: 9/Find Parametric equations for the Line that is tangent to the curve r(t) = eti + (Sint); + In…

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Q: 27. Show that the curve with parametric equations x = t cos t, y =t sin t, z = t lies on the cone z²…

A: To Show: i. Show that the curve with the parametric equationx=tcost, y=tsint, z=t  lies on the…

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 equation of the tangent line to right branch of the hyperbola π π ㅠ x = sect, y = tant, st≤…

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Q: Eliminate the parameter to find the cartesian equation of the curve: x = 2 sin 0, y = cos² 0, sos…

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Q: Given parametric equations x = 2 cos 0 and y = sin 0. d²y Then dx2 in the point (V2,) is

A: Given parametric equation is  x=2cosθ and y=sinθ

Q: 9/Find Parametric equations for the Line that is tangent to the curve r(t) = eti + (Sint)j + In…

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Q: 7. The curve at right is given by the parametric equations t2 z = v2t. x = In t, y = Find its length…

A: Given that, x=lnt,y=t22 and z=2t. That is, r→t=lnt,t22 ,2t. It is known that, the length of the…

Q: 6. A particle at (1,0,0) starts moving in space in such a way that its position vector at any time t…

A: The vector equation of the tangent line of the parametric curve r→t at point P is expressed by…

Q: 4. Find the slope of the curve r = cos 20 at the origin 5. A particle moves along a path whose…

A:

Q: 3. A curve has parametric equations r = sin(3t) and y = cos(t) where t takes all real %3D values. dy…

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Q: 1. Find the moving trihedral and the curvature at any point of the curve R(t) = cos t î + sint ĵ+ k.

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Q: Find the exact value of x²y³z ds, where C is the curve with parametric equations x = e-t cos(5t), y…

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Q: 4) Find the critical points of the parametric equations x = t* – 1 y = t3 -t

A: Topic:- application of derivatives

Q: 1. Consider a fence whose base is the path with parametric equations |x(t) = cos³ t, |y(t) = sin³t,…

A: Given, A fence whose base is the path with parametric equations xt=cos3tyt=sin3t for 0≤t≤π2. Let the…

Q: EXAMPLE 3 Find parametric equations for the tangent line to the helix with parametric equations X =…

A: The vector equation of the curve isrt=⟨5cost,2sint,t⟩Therefore r't=⟨−5sint,2cost,1⟩The parameter…

Q: 1 Consider the curve defined by the parametric equations x = ÷ť & y=÷ť for 2<t<3 2 3 Determine the…

A: take derivative

Q: The tangents to the rectangular hyperbola xy = c² and the parabola y = 4ax at their point of…

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Q: (c) (d) Determine the exact length of the curve of the conic section from 0 = 0 to 0 = π/6. Explain…

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Q: A curve C has parametric equations X = tチ+1. dy (i) Find in terms oft. dx (ii) Show that there are…

A: We first find dy/dx

Q: 1. Use the parametric equations of an ellipse, x = a cos 0, y = b sin 0, 0 < 0< 27, to find the area…

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Q: 8. Solve for the critical points of the parametric equations x = t3 – 1,y = t² +t %3D

A: The answer is found in step-2.

Q: 3. Find the equation of the tangent line to the curve C given by the parametric equations x= e', y=…

A:

Q: A curve has parametric equations x = 2 sin t ,y = cos (2t) + 2 sin t for −π/2≤ t ≤−π/2 . Find out…

A: Given : parametric equations x = 2 sin t , y = cos (2t) + 2 sin t for −π/2 ≤ t ≤ π/2     Note :…

Q: B) Find parametric equations for the line that is tangent to the curve r(t) = e'i + (sin t)j + In(1…

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Q: Show that the arclength of the graph of y = sin x on the interval [0, 27] is equal to the circum-…

A: The length of the smooth curve ζ : [ 0 , 2π] →ℝ2  is given by I(ξ) = ∫02π ξ'(t)  dt

Q: 11. C is the curve F(t) = ( In (2 Vt +3 Find parametric equations for the tangent line to C at the…

A: Note: Here, the given point (0,1,2) is incorrect there should be(0,-1,2) because the point is on the…

Q: 2. Find a set of parametric equations for a particle that oscillates on the line y = 6x – 5 between…

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Q: 31. Find parametric equations for the tangent line to the curve of intersection of the cylinders x2…

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Q: 2. A particle is moving in space with position function R(t) = (sin(3t), cos(3t), 3 – 4t). 2.1. Find…

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Q: 9.) a. Find the parametric equations for the tangent line to the Curve at' the Specific point. X =t°…

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Q: 4. (i) Use the parametric equations of an ellipse, z = a cos (t) , y = bsin (t), 0 < t< 2r, to find…

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Q: 7- A sinusoidal curve has a minim um point at (-t/3, 5) and the closest maximum point to the right…

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Q: 1. Find the tangent to the curve given by the parametric equations: X = 1/t; y= VE e-t

A: Appliaction of derivative problem

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: