(a) Find the point of intersection between the curv urve r(t) = (cos(at), sin(at), t²) and the sphere r² + y? + 22 = 17.

Q: What is the surface area of the three dimensional solid formed by rotating the following parametric…

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Q: Find the moving trihedral and the curvature at any point of R(t) = (t²,– cos 8t, sin 8t).

A: Moving trihedral are three unit vectors of curve  Unit tangent vector  Unit normal vector  binomial…

Q: Find an equation of the tangent plane to the given parametric surface at the specified point. x = u…

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Q: 1. Find the equation of the plane tangent to the parametric surface S defined by r(u, v) = uvi + u…

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Q: 1. Consider the ellipsoid x² + y² + 4z? = 12. (a) Find an equation of the tangent plane to the…

A: As per bartleby guideline for more than one question, only 1st one is to be answered. Please upload…

Q: 32. At what points does the helix r(t) = (sin t, cos t, t) intersect the sphere x? + y² + z² = 5?

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Q: Find parametric equations for the line tangent to the helix

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Q: 4. Find the parametric equations for the tangent line to the curve of intersection of the y? z1…

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Q: Find the radius of curvature at 0 = t of the parametric equations x = 3(0 + cos 0), y = 3(1 – cos…

A: To find the radius of curvature, first find out dxdθ and d2xdθ2. x=3θ+cos θdxdθ=31-sin θd2xdθ2=-3cos…

Q: Find a rectangular equation for the curve with the given parametric equations. x = 2 sin(t),y = 2…

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Q: (a) Find the tangent line(s) to the parametric curve given by x(t) = t° – 4t", and y(t) = t. at the…

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: Find the parametric equations for the curve of intersection between the cylindrical surface S: x +…

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Q: 2. Given the parametric equations: x = t + sin t,y = t- cost, findand

A: Solution...

Q: x = 2t2 – C : 3t + 1 y = et²-1

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Q: Q2 - A) Find the parametric equations for the line that is tangent to the given curve : r(t) = (a…

A: To find The parametric equations for the line that is tangent to the given curve

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: 32. (sin y, x) • dr, where C is the boundary of the triangle with vertices (0, 0), 0 ), and 2' 2

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Q: Find the slope of the tangent to the parametric curve c(t) = (cos t, sin t) at the point (–1/2,…

A: We need to find the slope of the tangent to the parametric curve c(t)=(cos t, sin t ) at the point…

Q: Consider the parametric equation  x = t2 - 1 and y = t2 + 2t. (a) find (dy)/(dx) and (d2y)/(dx2).…

A: Given parametric equations - x = t2-1t2 = x+1t = x+1         .................(1)y =…

Q: Find of the parametric curve x = sect, y = tant, at t = dx? Select one: -3V3 -1 3V3 3y3 1 3v3 O none

A: Here we find the double derivatives value at t=π/3

Q: Find the area of the surface generated by revolving the parametric curve x = cos² t, y = sin² t (0 ≤…

A: To find: The area of the surface generated by revolving the curve about y-axis. Given: The…

Q: 8. Find the positive constant a(> 0) so that the parametric x = at, y = -t? is a solution to y =…

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Q: Find for the parametric curve x= 2e', y= sin(1+ e') dx

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Q: 1. For the parametric equations: x = 4t3, y= 2 +60t – 8t?, find: a) All values of t at which the…

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Q: B.2. Express to its standard form, determine the center, radius ar 10. x² + y? + 10x – 14y + 58 = 0…

A: # We are entitled to solve one question at a time , please resubmit the other question if you wish…

Q: Consider the parametric equations defined by x(t) = 3t² – 3 and y(t)= 2t +1, for t> 0. d'y at t= 1.…

A: Option A is correct option.

Q: 20. Evaluate Vx2 + y? ds where C is the curve with parametric equations x = 4 cos(2), y = 4 sin(t),…

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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: The position of a particle moving in the xy-plane is given by the parametric equations x(t) =…

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Q: Consider the curve r(t) = (t, 3 cos t, 3 sin t) for t 2 0. Find an equation for the projection of…

A: Equation of projection of curve on xy plane will not contain the term of z.

Q: 29. Find parametric equations for the tangent line to the cur ve of intersection of the paraboloid z…

A: Given paraboloid is z=x2+y2 ellipsoid is x2+4y2+z2=9 and the point (1,-1,2) find the parametric…

Q: Given: R(t) = (cos t + t sin t) î + (sin t – t cos t) ĵ + t?k ,t>0 Find parametric equations for the…

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Q: B. Find the length of the circle of radius r defined parametrically by x = r(v1 – sin t) and y =…

A: We need to find the length of the circle which is parametrically defined as below.

Q: Find parametric equations of the line tangent to the graph r(t) = (t cos 7t)i - (t+ t)j+ sin(t)k at…

A: Given that rt=t+cos 7ti-t2+tj+sintk   Tangents to the curve is given by Tt=ddtrt Tt=ddtt+cos…

Q: Find the equation of the osculating plane of the helix: r(t) = sin(3t), y(t) = 2t, 2(t) = cos(3t)…

A: Given the equation of the helix:     x(t)=sin(3t), y(t)=2t, z(t)=cos(3t) So, rt=sin3t,2t,cos3t. Find…

Q: Find parametric equations for the line tangent to the helix r(t) =(sqrt(2)cos t)i + (sqrt(2) sin t)j…

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Q: 1. Find the equation of the tangent plane to the surface cosh [exp (x²y)] = Z y? at the point where…

A: We will find out the required equation of tangent plane.

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

A: Given x=sin ty=csc t0<t<π2

Q: Find an equation of the plane tangent to the following surface at the given point. z = tan (xy);…

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Q: Find the length of the parametric curve I = cos(t) y = 1 + sin(t) from t = 0 to t =

A: Given- x=cost and y=1+sint To Find- The length of the parametric curve from t=0 to t=π. Identity…

Q: 1. Consider the graph of r(t) = 2t i+ tj +t° k Determine parametric equations of the tangent line to…

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Q: Find parametric equations for the line tangent to the curve of intersection of the surfaces tan-1(x?…

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Q: A curve is defined by the parametric equations z (t) = at and y (t) = bt, where a and b are…

A: Given,  We have  xt = at   and yt = bt  ;  where a and b are constants.  We have to find the length…

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

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Q: Find the parametric equations for the curve of intersection between the cylindrical surface S : x +…

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Q: The surface area of the surface generated by revolving the parametric curve x-2t -2sin t, y-2-2cos…

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Q: (a) Find the point of intersection of the tangent lines to the curve r(t) = (sin 7t, 2 sin Tt, cos…

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Q: Consider the parametric equations x = t2 - 1 and y = t2 + 2t. (a) Find (dy)/(dx) and (d2y)/(dx2).…

A: Consider the parametric equations  x = t2 − 1 and y = t2 + 2t

Q: The surface generated by revolving the curve c(t) = (t2, t) about the x-axis for 0 ≤ t ≤ 1.

A: Given curve,  c(t) = (t2,t)   0 ≤ t ≤ 1. We would like to find the surface generated by revolving…