1. Consider the curve C: r(t) = (t - sin t)i + (1– cos t)j, 0StS 2. %3| | (a) Find an equation of the tangent line to the graph of r(t) at r(7/2). (b) Find the length of C.

Q: 4. Find the moving trihedral of the curve f(t) at the indicated value: †(t) = cos? t î + sin² t ĵ…

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A: Given: For the curve x=1-sin(t) t 0 π/2 π x 1 0 1

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Q: (a) Find the slope of the tangent line to the parametric curve x = t/2, y = t 2 + 1 at t= −1and at t…

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Q: ne if the curve (In t. t In t. 1) is tangent to the surface rz- uz? + cos Tu =1 at (0.0. 1).

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Q: 1. Find the length of the curve given by e' (sin t+cos t) , yY = e'(cos t – sin t), x = 1<t< 4 .

A: The length of a parametric curve is given by L=∫abx't2+y't2dt Now, we have…

Q: Set up an intergal of the Curve for the length 3T Y%3DZE* sin (x), O4 xE 2

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Q: What is the point of intersection of the tangent lines to the curve r(t) = sin(πt)i + 8sin(πt)j +…

A: Let us consider the given curve r(t) the tangent line is obtained as Find the derivative of the…

Q: Consider the graph of f(x, y) = excos(xy²). Find an equation of its tangent plane at (0,0, 1) and…

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Q: 4) Consider the curve given by: cos(xy) = x(y – 7) – 1 %3D Find y' dx a)

A: Given function is cosxy=xy-π-1 Chain rule of differentiation states that: ddxfgx=f'gx·g'x…

Q: 10. For the asteroid given by = 4cos (t), y = 4sin' (t) on 0,2n|, find the slope of the tangent at…

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Q: Consider the curve C shown in the figure, parameterized by: C: r(t) = (1 - cos(t), 1- sin(t), 1 +…

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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: For the implicitly defined curve: cos(7y) 5y sin(x) dy (e.g., find y' ). dx a. Find a formula for dy…

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Q: Find the length L of the curve R(t) = e" cos(2t) i + e* sin(2t) j+2e"k over the interval [1, 4]. L =

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Q: Find the derivative of the parametrically defined curve (t) = 2 cos(4t), y(t) = sin(4t).

A: We have to find dy/dx.

Q: 10. Find the exact length of the curve y = In(sec x),0 < x<T/4 %3D

A: This is a problem related to integration of the function. Based on the integral formula we will find…

Q: 2. Find the arc length of the curve y= from x-2 to xD3 16 2x2

A: The solution is given as

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…

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Q: 7. Find the slope of the tangent to the parametric curve x = t - lnt, y = t + lnt at the point (e +…

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A: The given curve is rt=sinπt,2sinπt,cosπt. Differentiating it with respect to t:…

Q: Find the exact length of the curve. y = ln(cos x),   0≤ x ≤ π/3

A: Formula used: L=∫ab1+dydx2dx

Q: 2. Find the length of the curve r(t) = e'i + e' sin 2tj + e' cos 2tk, 0st< 2n.

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Q: Find an equation of the tangent to the curve at the point corresponding to the given value of the…

A: To find the slope of the tangent we need to find dy/dx Then find dy/dx at t=pi

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Q: 8. Find the length of the curve defined by x = cos(2t), y = sin(2t) from t = 0 to t = n.

A: If x = f(t), y = g(t) in (a, b) then length of the curve is,

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Q: The slope of the tangent line to the curve parametrized by the equations x (t) = -2(cos(t) + t…

A: Slope of tangent is equal to dy/dx.

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

A: 1)- Given: r→t=sin 2t, cos 2t from t=0 to t=2π. The objective is to determine the length of the…

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

A: Since you have posted multiple question in this question which are not interlinked so i have solved…

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Q: Find the length of the curve. r(t) = cos(4t) i + sin(4t)j + 4 In(cos(t)) k, 0 <t< t/4

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Q: Find a parametric equation for the tangent line to the curve defined by r(t) = when t = 1.

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Q: 8. Find parametric equations for the tangent line to the curve r(t) = (t² +3t +2, e' cost, In(t +1)…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

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

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Q: Find the derivative 4cos4t sin4t O2cot4t O Other: -1/2 cot4t of the parametrically defined curve (t)…

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Q: Let C be the graph of the equation In(3y – 2e") = sinh (ma) - tan(y- 1). dy (a) Find da (b) Find an…

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Q: The parametric form of the tangent line to the image of f(t) = (5t², 2,t – 2) at t = -1 is L(t) =

A: Topic = Application of derivative

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A: Topic = Application of integration

Q: 3) Find the exact length of the curve. y = In(cos r), 0 < x<

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Q: Consider a curve parameterized by: c(t)=(sin^3(t),cos(t)) Compute dy/dx at the point ((sqrt(2)/4),…

A: Differentiate the given function as follows.

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Q: Q1. Find the length of the of the curve y = In(cos x) from x = 0 to x = . %3D

A: Explanation of the solution is given below...